150 Challenges for Mathletes : Along with INMO and TST problems

Table of contents :
Contents
Preface
Acknowledgments
Topics of Olympiad Math
150 Challenges For Mathletes
Solved Challenges
Unsolved Challenges
INMO Problems
TST Problems
Solutions for Solved Problems
About the Authors
Recommended Resources for Olympiad Mathematics

Citation preview

150 Challenges For Mathletes Along with INMO and TST Problems

Pranav M. Sawant Piyush K. Jha Anshuman Shukla

Contents Preface

3

Acknowledgments

4

Topics of Olympiad Math

5

I

9

150 Challenges For Mathletes ……………………………… A.

Solved Challenges………………………………………………..

10

B.

Unsolved Challenges………………………………………….…

20

INMO Problems .…………………………..………………………

44

III Team Selection Test Problems ……………………………….

101

IV Solutions of Solved Problems .……………………………….

181

About the Authors

202

Recommended Resources for Olympiad Mathematics

204

II

Preface Olympiad Mathematics is significantly different from what a normal math student would study in standard math courses such as calculus, pre-calculus, algebra etc. This makes it challenging but at the same time, it is a lot of fun indeed! Mathematics, as you all know, is the essence of science, without which nearly every field of higher study would be impenetrable. Mathematics is, in its way, the poetry of logical ideas. This book is intended to be a problem-solving book in mathematics. In the current edition, there are around 150 problems, crafted by me and my team, at the level of the IOQM (Indian Olympiad Qualifier in Mathematics) / AIME (American Invitational Mathematics Examination) although certain questions involve concepts regularly used in exams such as the USAMO (United States of America Mathematical Olympiad) and the INMO (Indian National Mathematical Olympiad). However we have included certain questions of calculus and real analysis as well. These problems are typically at the level of PUTNAM We have not classified the problems according to the difficulty level as we know that difficulty is a subjective concept and problems that may be hard for some might be very easy for others. Likewise, we have not added the questions topic-wise as most of these questions contain concepts from multiple topics and hence require a good level of analysis and problem-solving abilities across different areas. In the current edition of the book, we have 50 problems with their solutions (labeled as solved problems) and 100 unsolved problems, left as an exercise for the readers. In future editions, I will be looking to add more problems and give hints and solutions for them. If you need any hints or solutions to the unsolved problems, feel free to shoot me an email anytime at [email protected]. At the end of this book, there is a special section for Indian students as well, although it is a good resource for all mathletes. It contains the previous year's INMO (1986-2022) and Indian Team Selection Test (TST) Problems (2001-2019). I have also given certain handouts and book suggestions at the end. Constructive criticism is always welcome and feel free to let me know if you find any errors in this edition. Hope you enjoy this book!

Pranav Sawant India

Acknowledgments This book is a small excursion in my mathematics journey and there is no measure of length when it comes to mentioning who helped me in this journey. A special thanks to Scribe T for helping me with LaTeX and a huge thank you to the editor of this book Anshuman Shukla for the wonderful editing and cover page design. I would also like to thank all my mathematics teachers: Paresh Kokney, Chandrakant Choubey, Prabha Verma, Yury Ustinovskiy, Abhishek Das, Alok Kumar, Prashant Jain, Nikhil Nagaria, Valsamma Varghese, Mini Santhosh and of course, my grandmother and grandfather, without whom I probably would have never loved mathematics. And how can I forget my mom and dad who have supported me throughout the way. A big Thank you to all the institutions and persons who provided me invaluable knowledge and supported me through this journey. Also, a big thank you to all of the AOPS users and members, without you, we would never have a brilliant collection of problems and their elegant solutions on AOPS. Pardon me if I forgot someone! The rights to Indian National Mathematics Olympiad problems and Indian Team Selection Test problems are exclusively held by Homi Bhabha Centre For Science Education (HBCSE) and Mathematics Teachers' Association (India) (MTA(I)).

Topics of Olympiad Math Number Theory ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

Euclidean Division Algorithm and Properties of GCD Fundamental Theorem of Arithmetic Sum of divisors and number of divisors Chicken McNugget Theorem Bezout’s Theorem Congruence Modulo Chinese Remainder Theorem and solving basic congruences Fermat’s Little Theorem, Euler’s Totient Function and Euler’s Totient Theorem Wilson’s Theorem and Lucas’s Theorem Order of an Element Primitive Roots Quadratic congruences Quadratic residues, quadratic reciprocity, Legendre’s symbol, Euler’s criterion Cyclotomic Polynomials Lagrange’s Theorem for Polynomials Diophantine equations p-adic valuation and Legendre’s formula Lifting the Exponent Lemma (LTE) Hensel’s Lemma Pell’s equation and its properties Fermat’s last theorem, Catalan Conjecture and Pythagorean triplets Bertrand’s Postulate Zsigmondy’s Theorem

Algebra ● ● ● ● ● ● ● ● ● ● ●

Algebraic Identities Fundamental Theorem of Algebra Polynomial Division and Synthetic Division Veita’s Relations Factor Theorem and Remainder Theorem Rational Root Theorem Binomial Theorem and Multinomial Theorem Complex Numbers Brahmagupta Identity Euler’s Four-Square Identity Sequences and Series

● ● ● ● ● ●

Fibonacci Sequence and its properties Lagrange Interpolation Symmetric Polynomials Chebyshev Polynomials Rouche’s Theorem Intermediate Value Theorem, Lagrange’s Mean Value Theorem, Rolle’s Theorem, Taylor and Maclaurin series. ● Irreducibility Criterions: Gauss’s Lemma, Eisenstien’s and Extended Eisenstien’s Irreducibility criterion, Cohn’s irreducibility criterion and Perron’s Irreducibility criterion. ● Infinite Descent and Vieta’s Root Jumping

Functional Equations ● Domain, co-domain, range, injectivity, surjectivity, bijectivity, involution functions, additive, multiplicative, periodic and cyclic functions ● Cauchy’s Functional Equations ● Jensen’s Functional Equation ● Monotonicity and continuity ● Polynomial Functional Equations

Inequalities ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

Triangle Inequalities QM-AM-GM-HM Weighted Means Jensen’s Inequality Cauchy Schwarz and Titu’s Inequality Rearrangement Inequality Muirhead’s inequality Holder’s inequality Minkowski Inequality Isoperimetric inequalities Chebyshev’s Inequality Schur’s Inequality Karamata’s Inequality Ravi Substitution Lagrange Multipliers

Geometry ● ● ● ●

Congruence and Similarity of Triangles Angle Chasing, length chasing and trig bashing Cyclic Quadrilaterals Centroid, circumcentre, incentre, orthocentre, incircles and excircles

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

Tangents, Power of a Point and Radical Axis Incentre-Excentre Lemma Nine-point circle Homotheties Ceva’s Theorem and Menelaus Theorem Simson and Symmedian Lines Midpoint of altitudes Isogonal/Isotonic conjugates Cartesian and Barycentric Coordinates Curvilinear and Mixtilinear Incircles Special Points; HM Point, Fermat Point, Isodynamic Point, Bevan Point Inversion and angle conservation in inversion Overlays and orthogonal circles Inversion distance formula Cross Ratio Harmonic Bundles and Quadrilaterals Apollonian Circles Pascal’s Theorem and Projective Transformations Polars/Poles and Bruhcard’s Theorem Brianchon’s Theorem Spiral Symmetry Miquel’s Theorem and Miquel Points Gauss-Bodenmiller Theorem Moving Points and Circle Tangency through Homothety Cayley-Bacharach Theorem Napoleon’s Theorem Sawayama-Thebault Theorem

Combinatorics ● Set Theory, relations and Cartesian Product, image and preimage, composition, cardinality, De-Morgan’s Laws, Venn Diagrams and Syllogism ● Basic counting and Fundamental Theorem of Counting ● Circular Permutations, Selection and Division of Objects, Arrangements and Derangements ● Bijections ● Double Counting ● Recursion ● Principle of Inclusion and Exclusion (PIE) ● Pigeonhole Principle ● Permutation groups, Burnside Counting Lemma and Polya’s Theorem ● Hall’s Marriage Theorem ● Dilworth’s Lemma ● Pascal’s Triangle and Pascal’s Identity

● ● ● ● ● ●

Hockey-Stick Identity Vandermonde’s Identity Bayes Theorem Invariants and monovariants Generating Functions Game Theory

Graph Theory ● Graphs, edges, vertices, faces, adjacent, incident, degree, path, cycle, length of path and cycle, walk, connect/disconnect, tree, forest, Hamiltonian path/cycle, Eulerian circuit, complete, planar, bipartite, k-partite graphs. ● Handshake Lemma ● Euler’s Formula for Planar Graphs ● Kuratowski’s Theorem ● Dirac’s Theorem ● Ore’s Theorem ● Cayley’s Theorem ● Turan’s Lemma ● Four color theorem ● Ramsey Theory ● Zarankiewicz’s Lemma

150 Challenges For Mathletes

Solved Challenges Problem 1.

Let f ( x ) = x3 + ax2 + bx + c and g( x ) = x3 + bx2 + cx + a, where a, b, c

are integers with c ̸= 0. Suppose that the following conditions hold: f (1) = 0, the roots of g( x ) = 0 are the squares of the roots of f ( x ) = 0. Find the value of a2023 + b2023 + c2023 .

Problem 2. if α, β, γ are roots of the equation x3 + 2x2 + 3x + 1 = 0 Find last four digits of α35005 + β35005 + γ35005

Problem 3. In △ ABC, let D be the foot of the altitude from A to BC. Construct squares ABWX and CAYZ outside △ ABC. Let M be the midpoint of XY and P be the intersection of BZ and CW. Prove that M, A, P, and D are collinear.

Problem 4. p √ Define a function g : N → R Such that g( x ) = 4x + 4x+1 + 4x+2 + .... Find the sum of last 4 digits in the decimal representation of g(2023). q

Problem 5. Compute the number of ordered quadruples (w, x, y, z) of complex numbers (not necessarily non-real) such that the following system is satisfied: wxyz = 1 wxy2 + wx2 z + w2 yz + xyz2 = 2 wx2 y + w2 y2 + w2 xz + xy2 z + x2 z2 + ywz2 = −3 w2 xy + x2 yz + wy2 z + wxz2 = −1 10

Solved Challenges Problem 6.

The positive reals x, y, z satisfy the relations x2 + xy + y2 = 1 y2 + yz + z2 = 2 z2 + zx + x2 = 3. √ m−n p

The value of y2 can be expressed uniquely as , where m, n, p, q are positive q integers such that p is not divisible by the square of any prime and no prime dividing q divides both m and n. Compute m + n + p + q

Problem 7.

Solve the equation p +

p

q2 + r =

√

s2 + t in prime numbers.

Problem 8. F ind all functions f : R → R, satisfying the condition f ( x f (y) + f ( x )) = 2 f ( x ) + xy for any real x and y.

Problem 9.

How many f : A → A are there satisfying f ( f ( a)) = a for every

a ∈ A = {1, 2, 3, 4, 5, 6, 7}?

21 times

z }| { Problem 10. W hat is the least positive integer n such that f ( f (. . . f (n))) = 2013 √ where f ( x ) = x + 1 + ⌊ x ⌋? (⌊ a⌋ denotes the greatest integer not exceeding the real number a.) 11

Solved Challenges Problem 11.

How many triples of positive integers ( a, b, c) are there such that

a! + b3 = 18 + c3 ?

Problem 12.

Let x, y, z be real numbers such that x + y + z = 2,

xy + yz + zx = 1

Find the maximum possible value of x − y.

Problem 13.

Let ϕ(n) be the number of positive integers less than n that are

relatively prime to n, where n is a positive integer. Find all pairs of positive integers (m, n) such that 2n + (n − ϕ(n) − 1)! = nm + 1.

Problem 14. L et z1 , z2 , z3 be nonzero complex numbers and pairwise distinct, having the property that (z1 + z2 )3 = (z2 + z3 )3 = (z3 + z1 )3 . Show that |z1 − z2 | = | z2 − z3 | = | z3 − z1 | .

Problem 15.

Show: 8 < 9

Z π/2 0

sin(sinx )dx < 1

12

Solved Challenges Let a ∈ (1, ∞) and a countinuous function f : [0, ∞) −→ R having the

Problem 16. property:

lim x f ( x ) ∈ R.

x →∞

R ∞ f (x) Ra
a) Show that the integral 1 x dx and the limit limt→∞ t 1 f x t dx both exist, are finite and equal. R a dx b) Calculate limt→∞ t 1 1+ . xt

For any integer n ≥ 2 denote by An the set of solutions of the equation

Problem 17.

x=

jxk 2

+

jxk 3

+···+

jxk n

.

Determine the set A2 ∪ A3 .

Find all injective functions f : Z → Z that satisfy: | f ( x ) − f (y)| ≤

Problem 18.

| x − y| ,for any x, y ∈ Z.

Determine continuous functions f : R → R such that

Problem 19. a2 + ab + b2


Rb a

Problem 20.

f ( x ) dx = 3

Rb

x2 f ( x ) dx, for every a, b ∈ R .

a

Calculate: lim

n→∞

R1 0

n

e x dx

13

Solved Challenges Problem 21.

Let the matrices of order 2 with the real elements A and B so that

AB = A2 B2 − ( AB)2 and det ( B) = 2. Calculate det ( A + 2B) − det ( B + 2A).

Problem 22.

Find the minimum number of perfect cubes such that their sum is

equal to 346346 .

Problem 23.

Prove that for any integers a, b, the equation 2abx4 − a2 x2 − b2 − 1 = 0

has no integer roots.

Problem 24.

ABCD is a cyclic convex quadrilateral whose diagonals meet at X. The

circle ( AXD ) cuts CD again at V and the circle ( BXC ) cuts AB again at U, such that D lies strictly between C and V and B lies strictly between A and U. Let P ∈ AB ∩ CD. If M is the intersection point of the tangents to U and V at (UPV ) and T is the second intersection of circles (UPV ) and ( PAC ), prove that ∠ PTM = 90o .

Problem 25.

Let ABC be a triangle. Consider the circle ω B internally tangent to

the sides BC and BA, and to the circumcircle of the triangle ABC, let P be the point of contact of the two circles. Similarly, consider the circle ωC internally tangent to the sides CB and CA, and to the circumcircle of the triangle ABC, let Q be the point of contact of the two circles. Show that the incentre of the triangle ABC lies on the segment PQ if and only if AB + AC = 3BC.

14

Solved Challenges 4 x −cos2 x

Problem 26.

Solve the following equation 2sin

Problem 27.

Let a, b, c, d ∈ N∗ such that the equation

4 x −sin2 x

− 2cos

= cos 2x

x2 − ( a2 + b2 + c2 + d2 + 1) x + ab + bc + cd + da = 0 has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.

Problem 28.

Solve the equation 2023x

Problem 29.

2 +x

+ log2023 x = 2023x+1

In a triangle ABC, where a = BC, b = CA and c = AB, it is known

that: a + b − c = 2 and 2ab − c2 = 4. Prove that ABC is an equilateral triangle.

Problem 30.

Prove that the number 1010 can’t be written as the product of two

natural numbers which do not contain the digit "0" in their decimal representation.

Problem 31.

Show that for every natural n > 1 we have: (n − 1)2 | nn−1 − 1

15

Solved Challenges Problem 32.

Let a, b, c > 0 the sides of a right triangle. Find all real x for which

a x > b x + c x , with a is the longest side.

Problem 33.

(a) Show that for every n ∈ N there is exactly one x ∈ R+ so that

x n + x n+1 = 1. Call this xn . (b) Find lim xn . n→+∞

Problem 34.

Show that for p > 1 we have

1 p + 2 p + ... + (n − 1) p + n p + (n − 1) p + ... + 2 p + 1 p = +∞ n→+∞ n2 lim

Find the limit if p = 1.

Problem 35.

An acute triangle ABC (AB > AC) has circumcenter O, but D is

the midpoint of BC. Circle with diameter AD intersects sides AB and AC in E and F respectively. On segment EF pick a point M so that DM ∥ AO. Prove that triangles ABD and FDM are similar.

∞

Problem 36. I f 1 ≤ r ≤ n are integers, prove the identity:

∑

d =1



n−r+1 d



r−1 d−1



=

  n . r

16

Solved Challenges 1

1

1

1

Problem 37.

Show that for every positive integer n, 2 2 · 4 4 · 8 8 · ... · (2n ) 2n < 4.

Problem 38.

Find all non-negative integer x for which

p 3

13 +

√

x+

p 3

13 −

√

x is

an integer.

Problem 39.

Determine all functions f : R → R such that: f (max { x, y} + min { f ( x ), f (y)}) = x + y

for all real x, y ∈ R

Problem 40.

Find all real quadruples ( a, b, c, d) satisfying the system of equations  ab + cd = 6    ac + bd = 3 ad + bc = 2    a + b + c + d = 6.

Problem 41.

If tan( x + y + z) =

sin 3x +sin 3y+sin 3z cos 3x +cos 3y+cos 3z ,

compute difference between

maximum and minimum value of R = cos( x + y + z)(cos 3x + cos 3y + cos 3z) + sin( x + y + z)(sin 3x + sin 3y + sin 3z)

17

Solved Challenges
Given that x2 + y2 = 1 and 4xy 2x2 − 1 = 1. If the largest possible p √ value of x that satisfies these equations can be expressed as 12 a + b. Find a + b. Problem 42.

Problem 43.

For how many rational numbers p is the area of the triangle formed

by the intercepts and vertex of f ( x ) = − x2 + 4px − p + 1 an integer?

Problem 44.

Compute 1 lim A→+∞ A

Problem 45.

Z A 1

1

A x , dx.

a, b, c, d ∈ R, solve the system of equations:  3 a +b = c     b3 + c = d  c3 + d = a    3 d +a=b

Problem 46.

f , g : R → R find all f , g satisfying ∀ x, y ∈ R: g( f ( x ) − y) = f ( g(y)) + x.

18

Solved Challenges Problem 47.

Find all pairs of positive integers x, y satisfying the equation y x = x50

Problem 48.

In the isosceles triangle ABC the angle BAC is a right angle. Point D

lies on the side BC and satisfies BD = 2 · CD. Point E is the foot of the perpendicular of the point B on the line AD. Find the angle CED.

Problem 49.

Let k,m and n be three different positive integers. Prove that 

Problem 50.

1 k− k



1 m− m



1 n− n



≤ kmn − (k + m + n).

If R and S are two rectangles with integer sides such that the perimeter

of R equals the area of S and the perimeter of S equals the area of R, then we call R and S a friendly pair of rectangles. Find all friendly pairs of rectangles.

19

B. Unsolved Challenges

Problem 51.  The sum 44

∑ 2𝑠𝑖𝑛𝑥𝑠𝑖𝑛1[1 + 𝑠𝑒𝑐(𝑥 − 1)𝑠𝑒𝑐(𝑥 + 1)] 𝑥=2 4 𝑛=1

( ) ( ) , where Φ, Ψ are trigonometric functions and θ1,

𝑛 Φ θ𝑛 Ψ θ𝑛

can be written in the form ∑ (− 1)

θ2, θ3, θ4 are degrees ∈ [0,  45]. Find θ1 + θ2 + θ3 + θ4.

Problem 52  A function 𝑓 from the positive integers to the positive integers is called INMO if it satisfies 𝑔𝑐𝑑(𝑓(𝑓(𝑥)), 𝑓(𝑥 + 𝑦)) = 𝑔𝑐𝑑(𝑥, 𝑦) for all pairs of positive integers 𝑥 and 𝑦. Find all positive integers 𝑚 such that 𝑓(𝑚) = 𝑚 for all INMO functions 𝑓.

Problem 53  In △𝐴𝐵𝐶, let 𝐷 be the foot of the altitude from 𝐴 to 𝐵𝐶. Construct squares 𝐴𝐵𝑊𝑋 and 𝐶𝐴𝑌𝑍 outside △𝐴𝐵𝐶. Let 𝑀 be the midpoint of 𝑋𝑌 and 𝑃 be the intersection of 𝐵𝑍 and 𝐶𝑊. Prove that 𝑀, 𝐴, 𝑃, and 𝐷 are collinear.

Problem 54  If the maximum value of 𝑓(θ) = 𝑠𝑖𝑛θ + 𝑠𝑖𝑛3θ + 𝑠𝑖𝑛5θ Across all 𝑅 is of the form

𝑎 𝑏 𝑐

where 𝑔𝑐𝑑(𝑏, 𝑐) = 1 and 𝑎, 𝑏, 𝑐 are positive integers, Where point of maxima is of

the form θ = 𝑎𝑟𝑐𝑠𝑖𝑛

( ) and ℓ, 𝑚 are positive integers. Compute the distance of the point ℓ 𝑚

(ℓ, 𝑚) From the line 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 (if ans is a irrational report its greatest integer function)

20

B. Unsolved Challenges Problem 55  Suppose 𝑋, 𝑌, 𝑍 are collinear points in that order such that 𝑋𝑌 = 1 and 𝑌𝑍 = 3. Let 𝑊 be a point such that 𝑌𝑊 = 5, and define 𝑂1 and 𝑂2 as the circumcenters of triangles △𝑊𝑋𝑌 and △𝑊𝑌𝑍, respectively. What is the minimum possible length of segment 𝑂1𝑂2?

Problem 56  If

Then find

(

− 𝑥 + 𝑦)

−2024

Problem 57  The line 𝑦 = 𝑘𝑥 (where 𝑘 is a positive real number) makes an acute angle of ◦

70 with the 𝑥-axis. Point 𝑂 is at the origin and point 𝐴 is at (0, 4). Point 𝑃 is a point on segment 𝑂𝐴 and 𝑀 and 𝑁 are points on the line 𝑦 = 𝑘𝑥. Let

𝑎 𝑏 𝑐

be the minimum possible

value of 𝐴𝑀 + 𝑀𝑃 + 𝑃𝑁 where 𝑎 and 𝑐 are relatively prime and 𝑏 is squarefree. Find 𝑎 + 𝑏 + 𝑐.

Problem 58  Let sequence

and

for all 𝑥, 𝑦 ϵ [𝑎, 𝑏]. Define a

. Show that 𝑥𝑛 converges to a fixed point of 𝑓.

21

B. Unsolved Challenges

Problem 59  Define 𝑓: 𝑅 → 𝑅 be a function such that 𝑁

𝑓(𝑥) = ∑ (𝑖 − 𝑥)(2𝑖 − 𝑥) 𝑖=1

if 𝑆(𝑁) be the set of minimum values of 𝑓(𝑥) for different 𝑁 find number of 𝑁 such that +

subset of 𝑆(𝑁) are also the subset of 𝑍

Problem 60  Consider points 𝐷, 𝐸 and 𝐹 on sides 𝐵𝐶, 𝐴𝐶 and 𝐴𝐵, respectively, of a triangle 𝐴𝐵𝐶, such that 𝐴𝐷, 𝐵𝐸 and 𝐶𝐹 concurr at a point 𝐺. The parallel through 𝐺 to 𝐵𝐶 cuts 𝐷𝐹 and 𝐷𝐸 at 𝐻 and 𝐼, respectively. Show that triangles 𝐴𝐻𝐺 and 𝐴𝐼𝐺 have the same areas.

Problem 61  Let 𝐴𝐵𝐶 be a triangle with 𝐼 as incenter. The incircle touches 𝐵𝐶 at 𝐷. Let 𝐷′ be the antipode of 𝐷 on the incircle. Make a tangent at 𝐷′ to incircle. Let it meet (𝐴𝐵𝐶) at 𝑋, 𝑌 respectively. Let the other tangent from 𝑋 meet the other tangent from 𝑌 at 𝑍. Prove that (𝑍𝐵𝐷) meets 𝐼𝐵 at the midpoint of 𝐼𝐵

Problem 62  For a positive integer 𝑛, denote by 𝑔(𝑛) the number of strictly ascending triples chosen from the set {1, 2, ..., 𝑛}. Find the least positive integer 𝑛 such that the following holds: The number 𝑔(𝑛) can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference 336.

{ }𝑛≥1 and {𝑏𝑛}𝑛≥1 be two infinite arithmetic progressions, each of which

Problem 63  Let 𝑎𝑛

the first term and the difference are mutually prime natural numbers. It is known that for

22

B. Unsolved Challenges

(

2

)(

2

2

2

)

any natural 𝑛, at least one of the numbers 𝑎𝑛 + 𝑎𝑛+1 𝑏𝑛 + 𝑏𝑛+1 or

(𝑎

2 𝑛

2

)(

2

2

)

+ 𝑏𝑛 𝑎𝑛+1 + 𝑏𝑛+1 is a perfect square. Prove that 𝑎𝑛 = 𝑏𝑛, for any natural 𝑛.

Problem 64  Let .

Find

Problem 65  A tournament is played between 𝑛 people. Everyone plays with everyone else, and no game ends in a draw. A number 𝑘 is said to be 𝑛-good if there exists such a tournament in which there is, for every 𝑘 people, a player who has lost all of them. a) Prove that 𝑛 ≥ 2

𝑘+1

− 1 b) Give all 𝑛 for which 2 is 𝑛-good.

Problem 66  For a positive integer 𝑛, two players 𝐴 and 𝐵 play the following game: Given a pile of 𝑠 stones, the players take turn alternatively with 𝐴 going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of 𝑛 stones. The winner is the one who takes the last stone. Assuming both 𝐴 and 𝐵 play perfectly, for how many values of 𝑠 the player 𝐴 cannot win?

Problem 67  Let 𝑀(𝑛) = {𝑛, 𝑛 + 1, 𝑛 + 2, 𝑛 + 3, 𝑛 + 4, 𝑛 + 5} be a set of 6 consecutive integers. Let’s take all values of the form 𝑎 𝑏

+

𝑐 𝑑

+

𝑧 𝑤

=

+

𝑒 𝑓

with the set {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓 = 𝑀(𝑛)}. Let 𝑥 𝑢

+

𝑦 𝑣

𝑥𝑣𝑤+𝑦𝑢𝑤+𝑧𝑢𝑣 𝑢𝑣𝑤

be the greatest of all these values.

23

B. Unsolved Challenges a) show: for all odd 𝑛 hold: 𝑔𝑐𝑑(𝑥𝑣𝑤 + 𝑦𝑢𝑤 + 𝑧𝑢𝑣, 𝑢𝑣𝑤) = 1 iff 𝑔𝑐𝑑(𝑥, 𝑢) = 𝑔𝑐𝑑(𝑦, 𝑣) = 𝑔𝑐𝑑(𝑧, 𝑤) = 1. b) for which positive integers 𝑛 hold 𝑔𝑐𝑑(𝑥𝑣𝑤 + 𝑦𝑢𝑤 + 𝑧𝑢𝑣, 𝑢𝑣𝑤) = 1?

𝑛

Problem 68  A polynomial 𝑝(𝑥) of degree 1000 is such that 𝑝(𝑛) = (𝑛 + 1)2 for all nonnegative integers 𝑛 such that 𝑛 ≤ 1000. Given that 𝑏

𝑝(1001) = 𝑎 · 2 − 𝑐, where 𝑎 is an odd integer, and 0 < 𝑐 < 1001, find 𝑐 − (𝑎 + 𝑏))

Problem 69  Call a convex quadrilateral angle-Pythagorean if the degree measures of its angles are integers 𝑤 ≤ 𝑥 ≤ 𝑦 ≤ 𝑧 satisfying 2

2

2

2

𝑤 +𝑥 +𝑦 =𝑧 . Determine the maximum possible value of 𝑥 + 𝑦 for an angle-Pythagorean quadrilateral.

Problem 70  One can define the greatest common divisor of two positive rational numbers as follows: for 𝑎, 𝑏, 𝑐, and 𝑑 positive integers with 𝑔𝑐𝑑(𝑎, 𝑏) = 𝑔𝑐𝑑(𝑐, 𝑑) = 1, write 𝑔𝑐𝑑

(

𝑎 𝑏

,

𝑐 𝑑

)=

𝑔𝑐𝑑(𝑎𝑑,𝑏𝑐) 𝑏𝑑

.

For all positive integers 𝐾, let 𝑓(𝐾) denote the number of ordered pairs of positive rational numbers (𝑚, 𝑛) with 𝑚 < 1 and 𝑛 < 1 such that 𝑔𝑐𝑑(𝑚, 𝑛) =

What is

1 𝐾

.

?

24

B. Unsolved Challenges

Problem 71  Euclid places a morsel of food at the point (0, 0) and an ant at the point (1, 2). Every second, the ant walks one unit in one of the four coordinate directions. However, whenever the ant moves to (𝑥, ± 3), Euclid’s notorious friend uncle chipotle picks it up and puts it at (− 𝑥, ∓ 2), and whenever it moves to (± 2, 𝑦), his cousin uncle john puts it at 𝑝 (∓ 1, 𝑦), If 𝑝 and 𝑞 are relatively prime positive integers such that 𝑞 is the expected number of steps the ant takes before reaching the food, find 𝑝 + 𝑞.

Problem 72  Determine all the triples {𝑎, 𝑏, 𝑐} of positive integers coprime (not necessarily pairwise prime) such that 𝑎 + 𝑏 + 𝑐 simultaneously divides the three numbers 12

𝑎

12

+𝑏

23

12

+ 𝑐 ,𝑎

23

+𝑏

23

+𝑐

11004

and 𝑎

11004

+𝑏

11004

+𝑐

Problem 73  In a group of 2021 people, 1400 of them are squid game runners. James Bond wants to find one squid game runner. There are some missions that each need exactly 3 people to be done. A mission fails if at least one of the three participants in that mission is a squid game runner. In each round James chooses 3 people, sends them to a mission and sees whether it fails or not. What is the minimum number of rounds he needs to accomplish his goal?

Problem 74  𝑛 > 1 is an odd number and 𝑎1, 𝑎2, ···, 𝑎𝑛 are positive integers such that

(

)

𝑔𝑐𝑑 𝑎1, 𝑎2, ···, 𝑎𝑛 = 1. If

(

𝑛

𝑛

𝑛

)

𝑑 = 𝑔𝑐𝑑 𝑎1 + 𝑎1 · 𝑎2 ··· 𝑎𝑛, 𝑎2 + 𝑎1 · 𝑎2 ··· 𝑎𝑛, ···, 𝑎𝑛 + 𝑎1 · 𝑎2 ··· 𝑎𝑛 find all possible values of 𝑑.

25

B. Unsolved Challenges

Problem 75  A complete number is a 9 digit number that contains each of the digits 1 to 9 exactly once. The difference number of a number 𝑁 is the number you get by taking the differences of consecutive digits in 𝑁 and then stringing these digits together. For instance, the difference number of 25143 is equal to 3431. The complete number 124356879 has the additional property that its difference number, 12121212, consists of digits alternating between 1 and 2. Determine all 𝑎 with 3 ≤ 𝑎 ≤ 9 for which there exists 𝑎 complete number 𝑁 with the additional property that the digits of its difference number alternate between 1 and 𝑎.

◦

Problem 76 𝐴𝐵𝐶 is a right angled triangle with ∠𝐴 = 90 and 𝐷 be the midpoint of 𝐵𝐶. A point 𝐹 is chosen on 𝐴𝐵. 𝐶𝐴 and 𝐷𝐹 meet at 𝐺 and 𝐺𝐵 ‖ 𝐴𝐷. 𝐶𝐹 and 𝐴𝐷 meet at 𝑂 and 2 𝐴𝐹 = 𝐹𝑂. 𝐺𝑂 meets BC at 𝑅. Find The sides of 𝐴𝐵𝐶 if the area of 𝐺𝐷𝑅 is 15

Problem 77  Shishimaru and hattori play a game. 2021 stones lie on a table. Starting with shishimaru, they alternatively remove stones from the table, while obeying the following rule. At the 𝑛-th turn, the active player (shishimaru if 𝑛 is odd, hattori if 𝑛 is even) can remove from 1 to 𝑛 stones. Thus, shishimaru first removes 1 stone; then, hattori can remove 1 or 2 stones, as she wishes; then, shishimaru can remove from 1 to 3 stones, and so on. The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player’s moves?

Problem 78  You have a 3 × 2021 chessboard from which one corner square has been removed. You also have a set of 3031 identical dominoes, each of which can cover two adjacent chessboard squares. Let 𝑚 be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the remainder when 𝑚 is divided by 19?

Problem 79  Let 𝑀 be the midpoint of segment 𝐵𝐶 of △𝐴𝐵𝐶. Let 𝐷 be a point such that 𝐴𝐷 = 𝐴𝐵, 𝐴𝐷 ⊥ 𝐴𝐵 and points 𝐶 and 𝐷 are on different sides of 𝐴𝐵. Prove that: 𝐴𝐵 · 𝐴𝐶 + 𝐵𝐶 · 𝐴𝑀 ≥

2 2

𝐶𝐷. 26

B. Unsolved Challenges

Problem 80  Let 𝑇1 = 8, 𝑇2 = 8, 𝑇𝑛 = (𝑇𝑛−1 + 𝑇𝑛−2)𝑚𝑜𝑑 10. Then find the value of 𝑇42.

Problem 81  Doraemon is a robot who can move freely on the unit circle and its interior, but is attached to the origin by a retractable cord such that at any moment the cord lies in a straight line on the ground connecting doraemon to the origin. Whenever his movement is counterclockwise (relative to the origin), the cord leaves a coating of black paint on the ground, and whenever his movement is clockwise, the cord leaves a coating of black paint on the ground, and whenever his movement is clockwise, the cord leaves a coating of orange paint on the ground. The paint is dispensed regardless of whether there is already 2

paint on the ground. The paints covers 1 liter/unit  , and doraemon starts at (1, 0).Each second, he moves in a straight line from the point (𝑐𝑜𝑠(θ), 𝑠𝑖𝑛(θ)) to the point ◦

(𝑐𝑜𝑠(θ + 𝑎), 𝑠𝑖𝑛(θ + 𝑎)), where 𝑎 changes after each movement. 𝑎 starts out as 253 and ◦

decreases by 2 each step. If he takes 89 steps, then the difference, in liters, between the amount of black paint used and orange paint used can be written as

𝑎− 𝑏 𝑐

◦

𝑐𝑜𝑡1 , where 𝑎, 𝑏

and 𝑐 are positive integers and no prime divisor of 𝑐 divides both 𝑎 and 𝑏 twice. Find 𝑎 + 𝑏 + 𝑐. Problem 82  Cynthia loves Pokemon and she wants to catch them all. In victory Road, there are a total of 80 Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations: 1. Every Pokemon in Victory Road is enemies with exactly two other Pokemon. 2. Due to her inability to catch Pokemon that are enemies with one other, the maximum number of the Pokemon she can catch is equal to 𝑛. What is the sum of all possible values of 𝑛?

Problem 83  In △𝐴𝐵𝐶, 𝐴𝐵 = 4, 𝐵𝐶 = 5, and 𝐶𝐴 = 6. Circulars arcs 𝑝, 𝑞 and 𝑟 of measure ◦

60 are drawn from 𝐴 to 𝐵, from 𝐴 to 𝐶, and from 𝐵 to 𝐶, respectively, so that 𝑝, 𝑞 lie completely outside △𝐴𝐵𝐶 but 𝑟 does not. Let 𝑋, 𝑌, 𝑍 be the midpoint of 𝑝, 𝑞, 𝑟, respectively. If 𝑠𝑖𝑛∠𝑋𝑍𝑌 =

𝑎 𝑏+𝑐 𝑑

, where 𝑎, 𝑏, 𝑐, 𝑑 are positive integers, 𝑔𝑐𝑑(𝑎, 𝑐, 𝑑) = 1, and 𝑏 is not

divisible by the square of a prime, compute 𝑎 + 𝑏 + 𝑐 + 𝑑.

27

B. Unsolved Challenges

Problem 84  On a table near the sea, there are 𝑁 glass boxes where 𝑁 < 2021, each containing exactly 2021 balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let 𝑆 be the sum of all values of 𝑁 for which Sowdha has a winning strategy and let 𝑅 be the sum of all values 𝑅−𝑆 of 𝑁 for which Rafi has a winning strategy. What is the value of 10

Problem 85  A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. Thus, Student 3 will close the third locker, open the sixth, close the ninth Student 5 then goes through and "flips" every 5th locker. This process continues with all students with odd numbers 𝑛 < 100 going through and "flipping" every 𝑛-th locker. How many lockers are open after this process?

Problem 86  Say there is a polynomial with integral coefficients such that there exists four distinct integers such that 𝑓 𝐼1 = 𝑓 𝐼2 = 𝑓 𝐼3 = 𝑓 𝐼4 = 2021, find sum of all such

( )

( )

( )

( )

integers such 𝐼′ such that 𝑓(𝐼′) = 2023. [note 𝐼𝑖 is integer for 𝑖 = 1, 2, 3, ··· 𝑛]

Problem 87  There is a table with 𝑛 rows and 18 columns. Each of its cells contains a 0 or a 1. The table satisfies the following properties: 1) Every two rows are different. 2) Each row contains exactly 6 cells that contain 1. 3) For every three rows, there exists a column so that the intersection of the column with the three rows (the three cells) all contains 0. What is the greatest possible value of 𝑛?

28

B. Unsolved Challenges

Problem 88  On a party with 99 guests, hosts shin chan and masao play a game (the hosts are not regarded as guests). There are 99 chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair 𝑐. If some chair adjacent to 𝑐 is already occupied , the same host orders one guest on such chair to stand up (if both chairs adjacent to 𝑐 are occupied, the host chooses exactly one of them). All orders are carried out immediately. shinchan makes the first move; her goal is to fulfill, after some move of hers, that at least 𝑘 chairs are occupied. Determine the largest 𝑘 for which shinchan can reach the goal, regardless of masao play.

Problem 89  Find all positive integer 𝑚 such that there exist an infinite AP (𝑎𝑛) and an infinite GP of positive integer such that 𝑚|𝑎𝑛 − 𝑔𝑛 for all 𝑛 ≥ 1 and 𝑚|𝑎2 − 𝑎1.

Problem 90  Vertices of a triangle are taken from the set A,B,C in the same order and its sides are extended to vertices P,Q,R in same order. if BP is thrice of AB, RC is twice of AC and 𝑎 BQ is just half of BC. if ratio of △𝐴𝐵𝐶 to △𝑃𝑄𝑅 is in form 𝑏 where a and b are positive 2

2

integers and 𝑔𝑐𝑑(𝑎, 𝑏) = 1, find 5𝑎 + 𝑏

Problem 91  201 positive integers are written on a line, such that both the first one and the last one are equal to 19999. Each one of the remaining numbers is less than the average of its neighboring numbers, and the differences between each one of the remaining numbers and the average of its neighboring numbers are all equal to a unique integer. Find the second-to-last term on the line

Problem 92  Points 𝑋 and 𝑌 are the midpoints of arcs 𝐴𝐵 and 𝐵𝐶 of the circumscribed circle of triangle 𝐴𝐵𝐶. Point 𝑇 lies on side 𝐴𝐶. It turned out that the bisectors of the angles 𝐴𝑇𝐵 and 𝐵𝑇𝐶 pass through points 𝑋 and 𝑌 respectively. What angle 𝐵 can be in triangle 𝐴𝐵𝐶 ?

29

B. Unsolved Challenges

Problem 93  In △𝐴𝐵𝐶 the median 𝐴𝑀 is drawn. the foot of perpendicular from 𝐵 to the angle bisector of ∠𝐵𝑀𝐴 is 𝐵1 and the foot of perpendicular from 𝐶 to the angle bisector of ∠𝐴𝑀𝐶 is 𝐶1. Let 𝑀𝐴 and 𝐵1𝐶1 intersect at 𝐴1. Find

𝐵1𝐴1 𝐴1𝐶1

.

Problem 94  Kalia has 3 red color ice cream and 3 black color ice cream. Find the number of distinct ways that kalia can place these checkers in stacks. Two ways of stacking ice creams are the same if each stack of the rest way matches a corresponding stack in the second way in both size and color arrangement. So, for example, the 3 stack arrangement 𝑅𝐵𝑅, 𝐵𝑅, 𝐵 is distinct from 𝑅𝐵𝑅, 𝑅𝐵, 𝐵, but the 4 stack arrangement 𝑅𝐵, 𝐵𝑅, 𝐵, 𝑅 is the same as 𝐵, 𝐵𝑅, 𝑅, 𝑅𝐵.

Problem 95  A water bottle (cylindrical in shape) stands upon a horizontal table. from a point on this plane, a man stares the cap of the water bottle, from which four of its corner ◦

◦

◦

points are visible, their angular elevations from the eye of the observer are 30 , α , 30 and ◦

60 . assuming the cap of the bottle to be perfectly circular and ratio of circumference of bottle cap to circular base of cylinder to be 1:1. find distance of point from the observer ◦

which subtends the α , from the observer’s eye, also find alpha and sum of distance of the ◦

◦

distance between the eye and points that subtended 30 , 60 given that the point which subtend angle α is in the extended line of sight of first point from the observer, horizontal distance between contact point and point at which the angle α is subtended is 6 units, and distance between first point and the point at which the first point subtends angle is 4 units

Problem 96  Find the number of pairs (𝑛, 𝑞), where 𝑛 is a positive integer and 𝑞 a 𝑛! { 2} { 2000 }

non-integer rational number with 0 < 𝑞 < 2000, that satisfy 𝑞 =

30

B. Unsolved Challenges

Problem 97  Suppose 𝐴𝐵𝐶𝐷 is a trapezoid with 𝐴𝐵 ‖ 𝐶𝐷 and 𝐴𝐵 ⊥ 𝐵𝐶. Let 𝑋 be a point on segment 𝐴𝐷 such that 𝐴𝐷 bisects ∠𝐵𝑋𝐶 externally, and denote 𝑌 as the intersection of 𝐴𝐶 and 𝐵𝐷. If 𝐴𝐵 = 10 and 𝐶𝐷 = 15, compute the maximum possible value of 𝑋𝑌.

Problem 98  21 bandits live in the city of Wasseypur, each of them having some enemies among the others. Initially each bandit has 240 bullets, and duels with all of his enemies. Every bandit distributes his bullets evenly between his enemies, this means that he takes the same number of bullets to each of his duels, and uses each of his bullets in only one duel. In case the number of his bullets is not divisible by the number of his enemies, he takes as many bullets to each duel as possible, but takes the same number of bullets to every duel, so it is possible that in the end the bandit will have some remaining bullets. Shooting is banned in the city, therefore a duel consists only of comparing the number of bullets in the guns of the opponents, and the winner is whoever has more bullets. After the duel the police take the bullets of the winner and as an act of protest the loser shoots all of his bullets into the air. What is the largest possible number of bullets the police can have after all of the duels have ended? Being someone's enemy is mutual. If two opponents have the same number of bullets in their guns during a duel, then the police take the bullets of the bandit who has the wider hat among them. Example: If a bandit has 13 enemies then he takes 18 bullets with himself to each duel, and they will have 6 leftover bullets after finishing all their duels.

Problem 99  Define 12

𝑃(𝑥) = 𝑥

11

+ 12𝑥

10

+ 66𝑥

9

8

7

6

5

4

3

2

+ 220𝑥 + 495𝑥 + 792𝑥 + 924𝑥 + 792𝑥 − 159505𝑥 + 220𝑥 + 66𝑥 + 12𝑥 + 1

Find sum of digits of integers

𝑃(19) 4

20

.

Problem 100 Triangle 𝐴𝐵𝐶 is inscribed in circle ω with 𝐴𝐵 = 5, 𝐵𝐶 = 7, and 𝐴𝐶 = 3. The bisector of angle 𝐴 meets sid 𝐵𝐶 at 𝐷 and circle ω at a second point 𝐸. Let γ be the circle

31

B. Unsolved Challenges 2

with diameter 𝐷𝐸. Circles ω and γ meet at 𝐸 and a second point 𝐹. Then 𝐴𝐹 =

𝑚 𝑛

, where 𝑚

and 𝑛 are relatively prime positive integers. Find 𝑚 + 𝑛.

Problem 101  Let 𝑚 ≥ 𝑛 be positive integers. MOTU is given 𝑚𝑛 posters of patlu with different integer dimensions of 𝑘 × 𝑙 with 1 ≥ 𝑘 ≥ 𝑚 and 1 ≥ 𝑙 ≥ 𝑛. He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on the wall or on a spot where it entirely covers a single visible poster and does not overlap any other visible poster. Determine the minimal area of the wall that will be covered by posters.

Problem 102  Bheem and raju are playing a game. raju has 𝑘 +

( ) cards with their front 𝑘 2

sides face down on the table. The cards are constructed as follows: For each 1 ≤ 𝑛 ≤ 𝑘, there is a blue card with 𝑛 written on the back, and a fraction

(

)

𝑎𝑛 𝑏𝑛

written on the front,

where 𝑔𝑐𝑑 𝑎𝑛, 𝑏𝑛 = 1 and 𝑎𝑛, 𝑏𝑛 > 0. For each 1 ≤ 𝑖 < 𝑗 ≤ 𝑘, there is a red card with (𝑖, 𝑗) written on the back, and a fraction

𝑎𝑖+𝑎𝑗 𝑏𝑖+𝑏𝑗

written on the front. It is given that no two

cards have equal fractions. In a turn bheem can pick any two cards and raju tells bheem which card has the larger fraction on the front. Show that, in fewer than 10000 turns, bheem can determine which red card has the largest fraction out of all of the red cards.

Problem 103  There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and 𝑘 meters per second, respectively, where 𝑘 is some positive integer with 7 ≤ 𝑘 ≤ 1000. Suppose that at some point in time, all three flies meet at a location different from their starting point. How many possible values of 𝑘 are there?

32

B. Unsolved Challenges

Problem 104  The number of sequences 𝑎𝑛 of 2𝑛 terms can be formed using exactly 𝑛(1’s) 𝑘

(

)

ans exactly 𝑛(-1’s) whose partial sums are always non negative: ∑ 𝑎𝑖 ≥ 0 , 1 ≤ 𝑘 ≤ 2𝑛 𝑖=1

Problem 105  Is there a number 𝑛 such that one can write 𝑛 as the sum of 2022 perfect squares and (with at least) 2022 distinct ways?

Problem 106  Let 𝑀 be a set of six distinct positive integers whose sum is 60. These numbers are written on the faces of a cube, one number to each face. A move consists of choosing three faces of the cube that share a common vertex and adding 1 to the numbers on those faces. Determine the number of sets 𝑀 for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.

Problem 107  While running from an unrealistically rendered zombie, uncle chipotle runs into a vacant lot in the shape of a square, 100 meters on a side. Call the four corners of the lot corners 1, 2, 3, and 4, in clockwise order. For 𝑘 = 1, 2, 3, 4, let 𝑑𝑘 be the distance between chipotle and corner 𝑘. Let (a) 𝑑1 < 𝑑2 < 𝑑4 < 𝑑3, (b) 𝑑2 is the arithmetic mean of 𝑑1 and 𝑑3, and (c) 𝑑4 is the geometric mean of 𝑑2 and 𝑑3. 2

If 𝑑1 can be written in the form

𝑎−𝑏 𝑐 𝑑

, where 𝑎, 𝑏, 𝑐, and 𝑑 are positive integers, 𝑐 is

square-free, and the greatest common divisor of 𝑎, 𝑏, and 𝑑 is 1, find the sum of all possible remainder when 𝑎 + 𝑏 + 𝑐 + 𝑑 is divided by 1000.

33

B. Unsolved Challenges

Problem 108  To any triangle with side lengths 𝑎, 𝑏, 𝑐 and the corresponding angles α, β, γ (measured in radians), the 6-tuple (𝑎, 𝑏, 𝑐, α, β, γ) is assigned. Find the minimum possible number 𝑛 of distinct terms in the 6-tuple assigned to a scalene triangle.

Problem 109  Rama and bholi play a game writing numbers on a board as follows: The two players play in turns and if in the board is written the positive integer 𝑛, the player whose turn is chooses a prime divisor 𝑝 of 𝑛 and writes the numbers 𝑛 + 𝑝. In the board, it is written at the start number 2 and Rama plays first. The game is won by whoever shall be first able to write a number bigger or equal to 31. Find who player has a winning strategy, that is who may write the appropriate numbers may win the game no matter how the other player plays.

Problem 110  Find all positive integers 𝑛 such that the number 6

3

𝑛 + 5𝑛 + 4𝑛 + 116 is the product of two or more consecutive numbers.

Problem 111  Laxman has a standard four-sided die. Each roll, he gains points equal to the value of the roll multiplied by the number of times he has now rolled that number; for example, if his first rolls were 3, 3, 2, 3, he would have 3 + 6 + 2 + 9 = 20 points. Find the expected number of points laxman will have after the die 25 times.

Problem 112  Raiyan stands on the bottom-left square of a 2022 by 2022 grid of squares, where each square is colored either black, gray, or white according to the pattern as depicted to the right. Each second he moves either one square up, one square to the right, or both one up and to the right, selecting between these three options uniformly and independently. Noting that he begins on a black square, find the probability that Raiyan is still on a black square after 2022 seconds.

34

B. Unsolved Challenges

Problem 113  Find number of integer from 0 to 2022 such that

(𝑛(𝑛+1)(2𝑛+1))! is an ((𝑛−1)𝑛(2𝑛+5))!

integer.

Problem 114  Simplify 𝑛

∑

(2𝑛)! 2

2

𝑘=0 (𝑘!) ((𝑛−𝑘)!)

.

2

2

Problem 115  Find all pairs of positive integers (𝑚, 𝑛) such that 𝑚 − 𝑚𝑛 + 𝑛 + 1 divides both numbers 3

𝑚+𝑛

3

+ (𝑚 + 𝑛)! and 3

3

𝑚 +𝑛

+ 𝑚 + 𝑛.

Problem 116  𝑂 is the circumcenter of △𝐴𝐵𝐶 and 𝐶𝐷 is the median to 𝐴𝐵. 𝐺 is the centroid of △𝐴𝐶𝐷. Prove that 𝑂𝐺 is perpendicular to 𝐶𝐷 only and only if △𝐴𝐵𝐶 is isosceles with 𝐴𝐵 = 𝐴𝐶.

Problem 117  let be a natural number 𝑛, and 𝑛 real numbers 𝑎1, 𝑎2, ···, 𝑎𝑛. Prove that there exists a real number 𝑎 such that 𝑎 + 𝑎1, 𝑎 + 𝑎2, ···, 𝑎 + 𝑎𝑛 are all irrational.

Problem 118 In a ∆𝑃𝑄𝑅 𝑋ϵ𝑃𝑄 𝑎𝑛𝑑 𝑌ϵ𝑃𝑅, if ∠QPR = 30° such that PQ = 7 and PR = 8, computer minimum value of the distance QY + XR + XY.

35

B. Unsolved Challenges

Problem 119  given a 4-digit number (𝑎𝑏𝑐𝑑)10 such that both (𝑎𝑏𝑐𝑑)10 and (𝑑𝑐𝑏𝑎)10 are multiples of 7, having the same remainder modulo 37. Find 𝑎, 𝑏, 𝑐, 𝑑

3

Problem 120  Let 𝑁 be the number of ordered triples (𝑎, 𝑏, 𝑐) ∈ {1, ···, 2022} such that 2

2

2

𝑎 + 𝑏 + 𝑐 = 0 (​ 𝑚𝑜𝑑 2023). What are the last three digits of 𝑁?

𝑘

Problem 121  Find 𝑘 where 2 is the largest power of 2 that divides tha product 2022 · 2023 · 2024 ··· 4048.

4

Problem 122  Vijay picks two random distinct primes 1 ≤ 𝑝, 𝑞 ≤ 10 . Let 𝑟 be the 2205403200

probability that 3 , 𝑓 are decimal digits.

≡ 1​​ 𝑚𝑜𝑑 𝑝𝑞. Estimate 𝑟 in the form 0. 𝑎𝑏𝑐𝑑𝑒𝑓, where 𝑎, 𝑏, 𝑐, 𝑑, 𝑒

𝑝−1 2

{

2

}

Problem 123  Determine the number of primes 𝑝 < 100 such that ∑ 𝑘 𝑝 is an integer, 𝑘=1

where {𝑥} = 𝑥 − [𝑥].

36

B. Unsolved Challenges

Problem 124  Let 𝑓(𝑥) = Φ(𝑥)Ψ(𝑥), where Φ(𝑥) and Ψ(𝑥) are monic polynomials of positive degree with integer coefficients. Then prove at least one of the polynomials Φ(𝑥) and Ψ(𝑥) is recursive.

Problem 125  Find all positive integer solutions (𝑎, 𝑏, 𝑐) to the function 2

2

2

𝑎 + 𝑏 + 𝑐 = 2005, where 𝑎 ≤ 𝑏 ≤ 𝑐.

Problem 126  Let 𝑆 be a set of 𝑛 distinct real numbers. Let 𝐴𝑆 be the set of numbers that occur as averages of two distinct elements of 𝑆. For a given 𝑛 ≥ 2, what is the smallest possible number of elements in 𝐴𝑆?

Problem 127  The incircle of a triangle ABC touches the sides BC and AC at point D and E, respectively. Suppose P is the point on the shorter arc DE of the incircle such that Angle APE = Angle DPB. The segments AP and BP meet the segment DE at points K and L, respectively. If KL = 4 find DE.

Problem 128  A sequence of positive integers 𝑎1, 𝑎2, 𝑎3, 𝑎4 ··· 𝑎𝑛 (necessarily not in same order or order of ascending or descending), such that 𝑛 ≤ 99, if

(

)2 − 4(𝑎𝑛)(𝑎𝑛+1) + (𝑎𝑛)2 = 0, find number of possible values of 𝑎1.

4 𝑎𝑛+1

Problem 129  There are three distinct positive integers, 𝑎, 𝑏, 𝑐 where 1 ≤ 𝑎, 𝑏, 𝑐 ≤ 100. 𝑐

𝑐

How many ways are there for 𝑎 + 𝑏 to be divisible by 130.

37

B. Unsolved Challenges

Problem 130  Let 𝑋 = {1, 2, 3,..., 𝑛} where 𝑛ϵ𝑁 define Compute number of injective functions possible from 𝑋 → 𝑆.

.

Problem 131  In triangle △𝐴𝐵𝐶, the points 𝐴′, 𝐵′, 𝐶′ are on sides 𝐵𝐶, 𝐴𝐶, 𝐴𝐵 respectively. Also, 𝐴𝐴′, 𝐵𝐵′, 𝐶𝐶′ intersect at the point 𝑂 (they are concurrent at 𝑂). Also, 𝐴𝑂 𝐵𝑂 𝐶𝑂 𝐴𝑂 𝐵𝑂 𝐶𝑂 + 𝑂𝐵′ + 𝑂𝐶′ = 92. Find the value of 𝑂𝐴′ × 𝑂𝐵′ × 𝑂𝐶′ . 𝑂𝐴′

Problem 132  Let 𝐼 be the incenter of a triangle 𝐴𝐵𝐶. 𝐷, 𝐸, 𝐹 are the symmetric points of 𝐼 with respect to 𝐵𝐶, 𝐴𝐶, 𝐴𝐵 respectively. Knowing that 𝐷, 𝐸, 𝐹, 𝐵 are concyclic, find all possible values of ∠𝐵.

Problem 133  Given is a triangle 𝐴𝐵𝐶 and points 𝐷 and 𝐸, respectively on 𝐵𝐶 and 𝐴𝐵. 𝐹 it is intersection of lines 𝐴𝐷 and 𝐶𝐸. We denote as |𝐶𝐷| = 𝑎, |𝐵𝐷| = 𝑏, |𝐷𝐹| = 𝑐 and |𝐴𝐹| = 𝑑. |𝐵𝐸| Determine the ratio |𝐴𝐸| in terms of 𝑎, 𝑏, 𝑐 and 𝑑

Problem 134  Jamuna lowers five matching wooden discs over bars placed on the vertices of a regular pentagon. Then she leaves five smaller congruent checkers these rods drop. Then she stretches a ribbon around the large discs and a second ribbon around the small discs. The first ribbon has a length of 56 centimeters and the second one of 50 centimeters. jamuna looks at her construction from above and sees an area demarcated by the two ribbons, what is the area of that area

38

B. Unsolved Challenges

Problem 135  Find the number of pairs of integers (𝑥, 𝑦) such that: 2

6(𝑥! + 3) = 𝑦 + 5

Problem 136  How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?

Problem 137  Consider the following three lines in the Cartesian plane: {ℓ1: 2𝑥 − 𝑦 = 7 ℓ2: 5𝑥 + 𝑦 = 42 ℓ3: 𝑥 + 𝑦 = 14 and let 𝑓𝑖(𝑃) correspond to the reflection of the point 𝑃 across ℓ𝑖. Suppose 𝑋 and 𝑌 are

( (

)) = 𝑌. Let 𝑡 be the length of

points on the 𝑥 and 𝑦 axes, respectively. such that 𝑓1 𝑓2 𝑓3(𝑋) 2

segment 𝑋𝑌; what is the sum of all possible values of 𝑡 ?

Problem 138  Let 𝐴𝐵𝐶 be a triangle. Let Ω denote the incircle of △𝐴𝐵𝐶 having radius 𝑟𝑜. Draw tangents to Ω which are parallel to the sides of 𝐴𝐵𝐶. Let Ω1, Ω2, Ω3 be the inradii of the three corner triangles so formed each having equal radii of

1 9

. Also, tangents to these three

circles are drawn which are also parallel to the sides of the inner corner triangles. Find the minimum value of perimeter of triangle 𝐴𝐵𝐶.

Problem 139  Suppose that in a certain society, each pair of persons can be classified as either amicable or hostile. We shall say that each member of an amicable pair is a friend of the other, and each member of a hostile pair is a foe of the other. Suppose that the society has 𝑛 persons and 𝑞 amicable pairs, and that for every set of three persons, at least one pair 39

B. Unsolved Challenges

is hostile. Prove that there is at least one member of the society whose does include

(

2

𝑞 1 − 4𝑞/𝑛

) or fewer amicable pairs.

Problem 140 Consider a function 𝑓: 𝑁→𝑁 Suppose that for all 𝑚, 𝑛∈𝑁, exactly one of 𝑓(𝑚 + 1), 𝑓(𝑚 + 2), ⋯, 𝑓(𝑚 + 𝑓(𝑛)) is divisible by 𝑛. Prove that 𝑓 has an infinite number of fixed points (inputs that get mapped to themselves).

Problem 141  Two trains start from point A and point B simultaneously towards each other. Their initial speed is 0 after which they move with some uniform speed. Then, they accelerate for a while (acceleration of both trains is different) before attaining some uniform speed again. The ratio of speeds during uniform motion is 4:3. At the time of their meeting, the speeds of the train were equal and they arrived at point A and B simultaneously. If the ratio of the accelerations of the train is given by 2

𝑎 2

𝑏+𝑘

, where a,b are

3

twin prime and k is an integer, calculate 𝑎 + 𝑏 + 𝑘 .

Problem 142 The circumference of a circle is divided into 𝑝 equal parts by the points 𝐴1, 𝐴2, ⋯𝐴𝑝, where 𝑝 is an odd prime number. How many different self-intersecting 𝑝-gons are there with these points as vertices if two 𝑝 -gons are considered different only when neither of them can be obtained from the other by rotating the circle? (A self-intersecting polygon is a polygon such that some of its sides intersect at other points besides the vertices).

Problem 143  If A and B are two rectangles with integer sides such that perimeter of A = area of B and perimeter of B = area of A, then we call A and B friendly pair of rectangles.

40

B. Unsolved Challenges

Set S contains all {𝑎, 𝑏, 𝑐, 𝑑} such that (𝑎 * 𝑏, 𝑐 * 𝑑) are dimensions of friendly pairs of rectangles. Call (𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖)∀ 1 ≤ 𝑖 ≤ 𝑛(𝑆) where 𝑛(𝑆) is the cardinality of set S cyclic if (𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖) given in order are sides of a cyclic φquadrilateral. Denote 𝐴𝑟(𝑖) as the area of cyclic quadrilateral having sides cyclic. Denote Ω(𝑖) as , where ɸ and φ are adjacent angles of cyclic quadrilateral having sides cyclic. Denote ω(𝑖) as . (Note: i is assorted in increasing order of perimeter of cyclic quadrilateral, i.e. indice 1 is assigned to cyclic quadrilateral having minimum perimeter and indice 𝑛(𝑆) is assigned to cyclic quadrilateral having maximum perimeter and equal to x)

is greatest integer less than or

Let XYZ be an equilateral triangle, extend XY beyond Y to a point 𝑌' so that 𝑌𝑌' = . XY, similarly extend YZ beyond Z to a point 𝑍' so that 𝑍𝑍' = Ω(𝑖). YZ and extend XZ beyond X to a point 𝑋' so that 𝑋𝑋' = ω(𝑖) · 𝑋𝑍. If area ∆𝑋𝑌𝑍 =

3 4

remains constant but length

of 𝑋𝑋', 𝑌𝑌', 𝑍𝑍' changes for each value of i, then sum of all possible values of area of ∆𝑋'𝑌'𝑍' is K. Compute

.

Problem 144 Two touching circles with fixed center 𝐴 and 𝐵 respectively having same radii 𝑟. 𝐴 third circle touching both of circles is drawn with center 𝑂 and radius 𝑟1. another circle which is moving in the plane with center 𝐶 and radius 𝑅 is drawn externally tangent to circle with center 𝑂. Again, two circles with center 𝐷 and 𝐸 are drawn such that they are externally tangent to circles with center 𝐴, 𝑂 and 𝐵, 𝑂 respectively. if the minimum perimeter of the pentagon 𝐴𝐵𝐸𝐶𝐷 is obtained for 𝑅 = 𝑘 · 𝑟1. find value of k

2

2

Problem 145  x and y are real numbers such that 6 − 𝑥, 3 + 𝑦 , 11 + 𝑥, 14 − 𝑦 are greater than zero. Find the maximum of the function

41

B. Unsolved Challenges

Problem 146  Fix an integer 𝑛 ≥ 4. Let 𝐶𝑛 be the collection of all 𝑛-point configurations in the plane, every three points of which span a triangle of area strictly greater than 1. For each configuration 𝐶 ∈ 𝐶𝑛 let 𝑓(𝑛, 𝐶) be the maximal size of a sub configuration of 𝐶 subject to the condition that every pair of distinct points has distance strictly greater than 2. Determine the minimum value 𝑓(𝑛) which 𝑓(𝑛, 𝐶) achieves as 𝐶 runs through 𝐶𝑛.

Problem 147  Consider the sequence 𝑥𝑛 > 0 defined with the following recurrence relation 𝑥1 = 0 and for 𝑛 > 1 2 2

(

𝑛

𝑛+1

)

(𝑛 + 1) 𝑥𝑛+1 + 2 + 4 (𝑛 + 1)𝑥𝑛+1 + 2

2𝑛−2

+2

2 2

= 9𝑛 𝑥𝑛 + 36𝑛𝑥𝑛 + 32.

Show that if 𝑛 is a prime number larger or equal to 5, then 𝑥𝑛 is an integer

Problem 148  A jalebi is a loop of 2𝑎 + 2𝑏 + 4 unit squares which can be obtained by cutting a concentric 𝑎 × 𝑏 hole out of an (𝑎 + 2) × (𝑏 + 2) rectangle, for some positive integers 𝑎 and 𝑏. (The side of length 𝑎 of the hole is parallel to the side 𝑎 + 2 of the rectangle). Consider an infinite grid of unit square cells. For each even integer 𝑛 ≥ 8, a bakery of order 𝑛 is a finite set of cells 𝑆 such that, for every 𝑛-cell jalebi 𝐵 in the grid, there exists a congruent copy of 𝐵 all of whose cells are in 𝑆. (The copy can be translated and rotated). We denote by 𝑓(𝑛) the smallest possible number of cells in a bakery of order 𝑛. Find a real number α such that, for cell sufficiently large even integers 𝑛 ≥ 8, we have 1 100


0 then 𝑓 𝑘2 < 𝑐𝑜𝑠 20 < 𝑓 𝑘1

Problem 150  Define 𝑓: 𝑁→𝑁 ∀𝑛 ∊ 𝑁

Computer remainder when A is divided by 2023.

43

Indian National Mathematics Olympiad (INMO) Problems (1986 - 2022)

44

Problem 1 A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged places, when did he go out ? INMO 1986

Problem 2

Solve

INMO 1986

Problem 3 Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that INMO 1986

Problem 4 Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number. INMO 1986

Problem 5

If 

 is a polynomial with integer coefficients and  ,  ,  , three distinct

integers, then show that it is impossible to have 

, 

,  INMO 1986

45

Problem 6 Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal. INMO 1986

Problem 7

If  ,  ,  ,   are integers greater than 1 such that   and   have no common

factor except 1 and  than 1.

 show that 

, 

 for some integer   greater INMO 1986

Problem 8 Suppose   are six sets each with four elements and   are   sets each with two elements, Let  . Given that each elements of   belongs to exactly four of the  's and to exactly three of the  's, find  . INMO 1986

Problem 9 Show that among all quadrilaterals of a given perimeter the square has the largest area. INMO 1986

Problem 10 Given  that

 and   as relatively prime positive integers greater than one, show

is not a rational number. INMO 1987

46

Problem 11 Determine the largest number in the infinite sequence INMO 1987

Problem 12 Let 

 be the set of all triplets 

 of integers such that 

 For each triplet   in  , take number  . Add all these numbers corresponding to all the triplets in  . Prove that the answer is divisible by 7. INMO 1987

Problem 13 If  ,  ,  , and   are natural numbers, and  relation   does not hold.

 then prove that the INMO 1987

Problem 14 Find a finite sequence of 16 numbers such that: (a) it reads same from left to right as from right to left. (b) the sum of any 7 consecutive terms is  , (c) the sum of any 11 consecutive terms is  . INMO 1987

Problem 15 Prove that if coefficients of the quadratic equation  odd integers, then the roots of the equation cannot be rational numbers.

 are INMO 1987

47

Problem 16 Construct the  , the median from the vertex 

, given 

, 

 (the altitudes from 

 and 

) and 

. INMO 1987

Problem 17 Three congruent circles have a common point   and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the common point   are collinear. INMO 1987

Problem 18 Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles. INMO 1987

Problem 19 Let 

 be a rearrangement of the numbers 

. Suppose that   is odd. Prove that the product an even integer.

is INMO 1988

Problem 20 Prove that the product of 4 consecutive natural numbers cannot be a perfect cube. INMO 1988

48

Problem 21 Five men,  ,  ,  ,  ,   are wearing caps of black or white color without each knowing the color of his cap. It is known that a man wearing black cap always speaks the truth while the ones wearing white always tell lies. If they make the following statements, find the color worn by each of them:  : I see three black caps and one white cap.  : I see four white caps  : I see one black cap and three white caps  : I see your four black caps. INMO 1988

Problem 22 If   and   are positive and 

, prove that

INMO 1988

Problem 23 Show that there do not exist any distinct natural numbers  ,  ,  ,   such that   and  . INMO 1988

Problem 24 If 

 are the coefficients of the polynomial show that 

 is even. INMO 1988

49

Problem 25 Given an angle   and a point   outside the angle  . Draw a straight line through   meeting   in   and   in   such that the triangle   has a given perimeter. INMO 1988

Problem 26 A river flows between two houses   and  , the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from   to  , using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks. INMO 1988

Problem 27 Show that for a triangle with radii of circumcircle and incircle equal to  ,   respectively, the inequality   holds. INMO 1988

Problem 28 Prove that the Polynomial  be expressed as a product   , where   and  polynomial with integral coefficients and with degree at least  .

 can't  are both INMO 1989

Problem 29 Let 

 and   be any four real numbers, not all equal to zero. Prove that

the roots of the polynomial 

 can't all be real. INMO 1989

50

Problem 30 Let   denote a subset of the set  property that no two elements of   add up to   elements.

. Prove that 

 having the  can't have more than  INMO 1989

Problem 31 Determine all   divides  .|

 for which

 is not the square of any integer, INMO 1989

Problem 32 For positive integers  , define  positive integers   for which (a) 

 is an even number,

(b) 

 is a multiple of  .

 to be 

. Determine the sets of

INMO 1989

Problem 33 Triangle   has incentre   and the incircle touches   at   respectively. Let   meet   at  . Show that   is perpendicular to  . INMO 1989

Problem 34 Let   be one of the two points of intersection of two circles with centers   respectively.The tangents at   to the two circles meet the circles again at  . Let a point   be located so that   is a parallelogram. Show that   is also the circumcenter of triangle  . INMO 1989 51

Problem 35 Given the equation has four real, positive roots, prove that (a)  (b)  with equality in each case holding if and only if the four roots are equal. INMO 1990

Problem 36 Determine all non-negative integral pairs 

 for which INMO 1990

Problem 37 Let   be a function defined on the set of non-negative integers and taking values in the same set. Given that

(a)  (b) 

 for all non-negative integers  ; ,

find the possible values that   can take. (Notation : here   refers to largest integer that is 

, e.g. 

). INMO 1990

52

Problem 38 Consider the collection of all three-element subsets drawn from the set  . Determine the number of those subsets for which the sum of the elements is a multiple of 3. INMO 1990

Problem 39 Let  ,  ,   denote the sides of a triangle. Show that the quantity

must lie between the limits 

 and 2. Can equality hold at either limit? INMO 1990

Problem 40 Triangle   is scalene with angle   having a measure greater than 90 degrees. Determine the set of points   that lie on the extended line  , for which

where 

 refers to the (positive) distance between 

 and 

. INMO 1990

Problem 41 Let   be an arbitrary acute angled triangle. For any point   lying within the triangle, let ,  ,   denote the feet of the perpendiculars from   onto the sides  ,  ,   respectively. Determine the set of all possible positions of the point   for which the triangle   is isosceles. For which position of   will the triangle   become equilateral? INMO 1990

53

Problem 42 Find the number of positive integers   for which (i) 

;

(ii) 6 is a factor of 

. INMO 1991

Problem 43 Given an acute-angled triangle  , let points   be located as follows:   is the point where altitude from   on   meets the outwards-facing semicircle on   as diameter. Points   are located similarly. Prove that  the area of triangle 

 where 

 is

. INMO 1991

Problem 44 Given a triangle 

Prove that 

 let

. INMO 1991

54

Problem 45 Let 

 be real numbers with 

, 

, 

, and 

. Prove that 

. INMO 1991

Problem 46 Triangle   has an incenter  . Let points  ,   be located on the line segments  ,   respectively, so that   and  . Given that the points   lie on a straight line, find the possible values of the measure of angle  . INMO 1991

Problem 47 (i) Determine the set of all positive integers   for which  ; (ii) Prove that 

 does not divide 

 divides 

 for any positive integer  . INMO 1991

Problem 48 Solve the following system for real 

INMO 1991

55

Problem 49 There are   objects of total weight  , each of the weights being a positive integers. Given that none of the weights exceeds   , prove that the ten objects can be divided into two groups that balance each other when placed on 2 pans of a balance. INMO 1991

Problem 50 Triangle   has an incenter   l its incircle touches the side   at  . The line through   parallel to   meets the incircle again at   and the tangent to the incircle at   meets   at points   respectively. Prove that triangle   is similar to triangle  . INMO 1991

Problem 51 For any positive integer   , let 

 denote the number of ordered pairs 

 of positive integers for which  integers for which 

 . Determine the set of positive INMO 1991

Problem 52 In a triangle 

Problem 53 If  that each of   or  ?

, 

 such that   lies in the closed interval 

. Prove that 

 and 

. INMO 1992

, then show

. Can   attain the extreme value  INMO 1992

56

Problem 54 Find the remainder when 

 is divided by 92. INMO 1992

Problem 55 Find the number of permutations   of   such that for any  ,   does not form a permutation of  . INMO 1992

Problem 56 Two circles   and   intersect at two distinct points   in a plane. Let a line passing through   meet the circles   and   in   and   respectively. Let   be the midpoint of   and let   meet the circles   and   in   and   respectively. Show that   is also the midpoint of  . INMO 1992

Problem 57 Let   be a polynomial in   with integer coefficients and suppose that for five distinct integers   one has  . Show that there does not exist an integer   such that  . INMO 1992

Problem 58 Let   be an integer. Find the number of ways in which one can place the numbers   in the   squares of a   chess board, one on each, such that the numbers in each row and in each column are in arithmetic progression. INMO 1992

57

Problem 59 Determine all pairs  perfect square.

 of positive integers for which 

 is a INMO 1992

Problem 60 Let 

 be an   -sided regular polygon. If  , find  . INMO 1992

Problem 61 Determine all functions 

 such that

INMO 1992

Problem 62 The diagonals   and   of a cyclic quadrilateral   intersect at  . Let   be the circumcenter of triangle   and   be the orthocenter of triangle  . Show that the points   are collinear. INMO 1993

Problem 63 Let   be a quadratic polynomial with  any integer   , show that there is an integer   such that 

. Given INMO 1993

58

Problem 64 If 

 and 

, show that INMO 1993

Problem 65 Let   be a triangle in a plane  . Find the set of all points  from   ) in the plane   such that the circumcircles of triangles   have the same radii.

 (distinct ,  ,  INMO 1993

Problem 66 Show that there is a natural number   such that  notation ends exactly in 1993 zeros.

 when written in decimal INMO 1993

Problem 67 Let   be a triangle right-angled at   and   be its circumcircle. Let   be the circle touching the lines   and  , and the circle   internally. Further, let   be the circle touching the lines   and   and the circle   externally. If   be the radii of   prove that  . INMO 1993

Problem 68 Let   and   be a subset of   having   elements. Show that   has 2 distinct elements   and   whose sum is divisible by  . INMO 1993

59

Problem 69 Let   be a bijective function from  there is a positive integer   denotes the composition 

 to itself. Show that

 such that 

 for each   in   

, where 

 times. INMO 1993

Problem 70 Show that there exists a convex hexagon in the plane such that (i) all its interior angles are equal; (ii) its sides are 

 in some order. INMO 1993

Problem 71 Let   be the centroid of the triangle   in which the angle at   is obtuse and   and   be the medians from   and   respectively onto the sides   and  . If the points  . If further   is a point on the line  show that triangle   and 

Problem 72 If 

 and   are concyclic, show that   extended such that   is a parallelogram,  are similar. INMO 1994

 prove that 

. INMO 1994

Problem 73 In any set of   square integers, prove that one can always find a subset of   numbers, sum of whose elements is divisible by  INMO 1994

60

Problem 74 Find the number of nondegenerate triangles whose vertices lie in the set of points   in the plane such that  ,  ,   and   are integers. INMO 1994

Problem 75 A circle passes through the vertex of a rectangle   and touches its sides   and   at   and   respectively. If the distance from   to the line segment   is equal to   units, find the area of rectangle  . INMO 1994

Problem 76 Find all real-valued functions   on the reals such that   for all  , and 

, 

 for  INMO 1994

Problem 77 In an acute angled triangle   is the midpoint of  . On the line  . Show that  .

,  ,  , take a point 

 is the orthocenter, and   such that  INMO 1995

Problem 78 Show that there are infinitely many pairs  (not necessarily positive) such that both the equations integer roots.

 of relatively prime integers have INMO 1995

61

Problem 79 Show that the number of  element subsets   with   is less than the number of those with 

 of  INMO 1995

Problem 80 Let   be a triangle and a circle   be drawn lying outside the triangle, touching its incircle   externally, and also the two sides   and  . Show that the ratio of the radii of the circles 

 and   is equal to  INMO 1995

Problem 81 Let  that 

. Let   for 

 be   real numbers all less than   and such . Show that

INMO 1995

Problem 82 Find all primes   for which the quotient

is a square. INMO 1995

Problem 83 a) Given any positive integer  , show that there exist distinct positive integers   and   such that   divides   for  ; b) If for some positive integers   and  ,  , prove that 

 divides 

 for all positive integers  INMO 1996 62

Problem 84 Let   and   be two concentric circles in the plane with radii   and   respectively. Show that the orthocenter of any triangle inscribed in circle   lies in the interior of circle  . Conversely, show that every point in the interior of   is the orthocenter of some triangle inscribed in  . INMO 1996

Problem 85 Solve the following system for real 

:

INMO 1996

Problem 86 Let   be a set containing   elements. Find the number of ordered triples   of subsets of   such that   is a subset of   and   is a proper subset of  . INMO 1996

Problem 87 Define a sequence   for  this sequence.

 by   and  . prove that for any 

 and   , 

 is also a term in INMO 1996

63

Problem 88 There is a   array (matrix) consisting of   and   and there are exactly   zeroes. Show that it is possible to remove all the zeros by deleting some   rows and some   columns. INMO 1996

Problem 89 Let   be a parallelogram. Suppose a line passing through   and lying outside the parallelogram meets   and   produced at   and   respectively. Show that INMO 1997

Problem 90 Show that there do not exist positive integers 

 and   such that

INMO 1997

Problem 91 If   are three real numbers and real number  , prove that 

for some INMO 1997

Problem 92 In a unit square one hundred segments are drawn from the center to the sides dividing the square into one hundred parts (triangles and possibly quadrilaterals). If all parts have equal perimeter  , show that 

. INMO 1997

64

Problem 93 Find the number of   array whose entries are from the set   and which are such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by  . INMO 1997

Problem 94 Suppose   and   are two positive real numbers such that the roots of the cubic equation   are all real. If   is a root of this cubic with minimal absolute value, prove that INMO 1997

Problem 95 In a circle   with center  , let   be a chord that is not a diameter. Let   be the midpoint of this chord  . Take a point   on the circle   with   as diameter. Let the tangent to   at   meet   at  . Show that  . INMO 1998

Problem 96 Let   and   be two positive rational numbers such that   is also a rational number. Prove that   and   themselves are rational numbers. INMO 1998

Problem 97 Let  integer   such that  integer   such that 

 be four integers such that   is not divisible by  . If there is an  is divisible by 5, prove that there is an  is also divisible by 5. INMO 1998

65

Problem 98 Suppose  unit. If 

Problem 99 Suppose 

 is a cyclic quadrilateral inscribed in a circle of radius one , prove that   is a square. INMO 1998

 are three real numbers such that the quadratic equation

has roots of the form   and   are real numbers. Show that (i) The numbers   are all positive. (ii) The numbers 

 where 

 form the sides of a triangle. INMO 1998

Problem 100 It is desired to choose   integers from the collection of   integers, namely,   such that the average of these   chosen integers is itself an integer and as minimum as possible. Show that this can be done for each positive integer   and find this minimum value for each  . INMO 1998

Problem 101 Let   be an acute-angled triangle in which   are points on   respectively such that  ; ; and   bisects   internally, Suppose   meets   and   in   and   respectively. If  ,  ,  , find the perimeter of  . INMO 1999

66

Problem 102 In a village   persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equal to   feet. For this purpose, the field was divided into   equal parts. If each part had an integer area, find the length and breadth of the field. INMO 1999

Problem 103 Show that there do not exist polynomials   and   each having integer coefficients and of degree greater than or equal to 1 such that INMO 1999

Problem 104 Let   and   be two concentric circles. Let  equilateral triangles inscribed in   and   respectively. If  points on   and   respectively, show that

 and   be any two  and   are any two

INMO 1999

Problem 105 Given any four distinct positive real numbers, show that one can choose three numbers   from among them, such that all three quadratic equations

have only real roots, or all three equations have only imaginary roots. INMO 1999

67

Problem 106 For which positive integer values of   can the set   be split into   disjoint  -element subsets   such that in each of these sets  . INMO 1999

Problem 107 The incircle of   touches  ,  ,   at  ,  ,   respectively. The line through   parallel to   meets   at  , and the line through   parallel to   meets   at  . Show that the line   bisects   and bisects  . INMO 2000

Problem 108 Solve for integers 

: INMO 2000

Problem 109 If 

 are real numbers such that 

 and then prove that 

Problem 110 In a convex quadrilateral  . Prove that 

, 

, 

. INMO 2000

 and  INMO 2000

68

Problem 111 Let   be three real numbers such that  that if   is a root of the cubic equation  then 

. prove  (real or complex), INMO 2000

Problem 112 For any natural numbers  , (  ), let   denote the number of congruent integer-sided triangles with perimeter  . Show that (i)  (ii) 

; . INMO 2000

Problem 113 Let   be a triangle in which no angle is  . For any point   in the plane of the triangle, let   denote the reflections of   in the sides   respectively. Prove that (i) If 

 is the incenter or an excentre of  ;

, then 

 is the circumcenter of 

(ii) If 

 is the circumcentre of 

, then 

 is the orthocentre of 

(iii) If 

 is the orthocentre of  .

, then 

 is either the incentre or an excentre of 

;

INMO 2001

Problem 114 Show that the equation  infinitely many solutions in integers 

 has . INMO 2001

69

Problem 115 If 

 are positive real numbers such that 

, Prove that INMO 2001

Problem 116 Show that given any nine integers, we can find four,   such that  is divisible by  . Show that this is not always true for eight integers. INMO 2001

Problem 117 , show that 

 is a triangle.   is the midpoint of  .  , and  . Show that   is obtuse. If   is the circumcenter of   is equilateral. INMO 2001

Problem 118 Find all functions  all 

 such that 

 for INMO 2001

70

Problem 119 For a convex hexagon  unequal, consider the following statements. (

) 

 is parallel to 

. (

)

(

) 

 is parallel to 

. (

)

(

) 

 is parallel to 

. (

) 

 in which each pair of opposite sides is

. . .

 Show that if all six of these statements are true then the hexagon is cyclic.  Prove that, in fact, five of the six statements suffice. INMO 2002

Problem 120 Find the smallest positive value taken by  integers  ,  ,   . Find all  ,  ,   which give the smallest value

 for positive

INMO 2002

Problem 121 If  ,   are positive reals such that 

 show that 

. INMO 2002

Problem 122 Is it true that there exist 100 lines in the plane, no three concurrent, such that they intersect in exactly 2002 points? INMO 2002

71

Problem 123 Do there exist distinct positive integers  ,  ,   such that  ,  ,  ,  ,  ,  ,   form an arithmetic progression (in some order). INMO 2002

Problem 124 The numbers  ,  ,   are arranged in an   array, so that the numbers in each row increase from left to right, and the numbers in each column increase from top to bottom. Let   be the number in position  . Let   be the number of possible values for 

. Show that INMO 2002

Problem 125 Let   be an interior point of an acute-angled triangle  . The line   meets the line   at  , and the line   meets the line   at  . The lines   and   intersect each other at  . Let   be the foot of the perpendicular from the point   to the line  . Show that the line   bisects the angle  . INMO 2003

Problem 126 Find all primes 

 and even  .

 such that  INMO 2003

Problem 127 Show that  for all real  . Find the sum of the non-real roots.

 has at least one real root INMO 2003

72

Problem 128 Find all  -digit numbers which use only the digits   and   and are divisible by  . INMO 2003

Problem 129 Let a, b, c be the side lengths and S the area of a triangle ABC. Denote  , 

 and 

. Prove that there exists a triangle with side

lengths x, y, z, and the area of this triangle is 

. INMO 2003

Problem 130 Each lottery ticket has a 9-digit numbers, which uses only the digits  ,  ,  . Each ticket is colored red, blue or green. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket   is red, and ticket   is green. What color is ticket   ? INMO 2003

Problem 131 sides  , 

, 

 is a convex quadrilateral.  ,  ,  ,  .   bisects   at  . 

. Prove that 

, 

 are the midpoints of the  , and

 is a square INMO 2004

Problem 132

 is a prime. Find all integers  ,  , such that  . INMO 2004

73

Problem 133 If   is a real root of 

, show that  INMO 2004

Problem 134 If 

 is a triangle, with sides  ,  ,   , circumradius  , and exradii  ,  ,  , show that  ,  ,  , and  . INMO 2004

Problem 135 S is the set of all ( ,  ,  ,  ,  ,  ) where  ,  ,  ,  ,  ,   are integers such that  . Find the largest   which divides abcdef for all members of  .” INMO 2004

Problem 136 Show that the number of 5-tuples ( ,  ,  ,  ,  ) such that   is odd INMO 2004

Problem 137 Let   be the midpoint of side   of a triangle  . Let the median   intersect the incircle of   at   and   being nearer to  than  . If  , prove that the sides of triangle   are in the ratio   in some order. INMO 2005

74

Problem 138 Let   and   be positive integers such that  minimum possible value of  .

. Find the INMO 2005

Problem 139 Let 

 be positive real numbers, not all equal, such that some two of

the equations

have a common root, say  . Prove that

   is real and negative;  the remaining third quadratic equation has non-real roots. INMO 2005

Problem 140 All possible  -digit numbers, in each of which the digits occur in nonincreasing order (from left to right, e.g.  ) are written as a sequence in increasing order. Find the  -the number in this sequence. INMO 2005

Problem 141 Let   be a given positive integer. A sequence   of positive integers is such that  , for  , is obtained from   by adding some nonzero digit of  . Prove that a) the sequence contains an even term; b) the sequence contains infinitely many even terms. INMO 2005

75

Problem 142 Find all functions 

 such that for all 

. INMO 2005

Problem 143 In a non equilateral triangle   the sides   form an arithmetic progression. Let   be the incentre and   the circumcentre of the triangle   Prove that (1) 

 is perpendicular to 

;

(2) If   meets   in  , and  ,   are the midpoints of  then   is the circumcentre of triangle  .

, 

 respectively INMO 2006

Problem 144 Prove that for every positive integer   there exists a unique ordered pair   of positive integers such that

INMO 2006

Problem 145 Let   by Find all triples 

 denote the set of all triples 

 of integers. Define 

 such that INMO 2006

76

Problem 146 Some 46 squares are randomly chosen from a   chess board and colored in red. Show that there exists a   block of 4 squares of which at least three are colored in red. INMO 2006

Problem 147 In a cyclic quadrilateral  ,   and  . Prove that (1) 

, 

, 

, 

;

(2) 

. INMO 2006

Problem 148(a) Prove that if   is a integer such that 

 then there exists an

integer   such that (b) Find the smallest positive integer 

 for which whenever an integer   is such that 

then there exists an integer   such that INMO 2006

Problem 149 In a triangle   right-angled at   , the median through  angle between   and the bisector of  . Prove that

 bisects the

INMO 2007

77

Problem 150 Let   be a natural number such that  numbers  . Prove that where  one of 

's , 

's ,  's are all nonzero integers. Further, if   does not divide at least  prove that   can be expressed in the form  , where   are natural numbers none of which is divisible by  . INMO 2007

Problem 151 Let   and   be positive integers such that  roots   and  . Prove that   and   are integers if and only if  integer. (Here 

 for some natural

 has real

 is the square of an

 denotes the largest integer not exceeding  ) INMO 2007

Problem 152 Let   be permutation of  . A pair   is said to correspond to an inversion of   if   but  . How many permutations of  ,  , have exactly two inversions? For example, In the permutation  the pairs 

, there are 6 inversions corresponding to . INMO 2007

78

Problem 153 Let   be a triangle in which  . Let   be the midpoint of   and   be a point on  . Suppose   is the foot of perpendicular from   on  . Define

Prove that

Hence show that 

 and 

 if and only if 

 is equilateral. INMO 2007

Problem 154 If  ,  ,   are positive real numbers, prove that INMO 2007

Problem 155 Let   be triangle,   its in-center;   be the reflections of   in   respectively. Suppose the circum-circle of triangle   passes through  . Prove that   are concyclic, where   is the in-center of triangle  . INMO 2008

Problem 156 Find all triples   and   are natural numbers.

 such that 

, where   is a prime and  INMO 2008

79

Problem 157 Let   be a set of real numbers such that   has at least four elements. Suppose   has the property that   is a rational number for all distinct numbers   in  . Prove that there exists a positive integer   such that   is a rational number for every   in  . INMO 2008

Problem 158All the points with integer coordinates in the  -Plane are coloured using three colors, red, blue and green, each color being used at least once. It is known that the point   is red and the point   is blue. Prove that there exist three points with integer coordinates of distinct colors which form the vertices of a right-angled triangle. INMO 2008

Problem 159 Let   be a triangle;   be three equal, disjoint circles inside   such that   touches   and  ;   touches   and  ; and   touches   and  . Let   be a circle touching circles   externally. Prove that the line joining the circum-center   and the in-center   of triangle   passes through the center of  . INMO 2008

Problem 160 Let   be a polynomial with integer coefficients. Prove that there exist two polynomials   and  , again with integer coefficients, such that (i)   is a polynomial in   , and (ii)   is a polynomial in  . INMO 2008

80

Problem 161 Let 

 be a triangle and let   be an interior point such that  .Let   be the mid points of   respectively.Suppose  .Prove that   are collinear. INMO 2009

Problem 162 Define a a sequence 

 as follows

, if number of positive divisors of   is odd , if number of positive divisors of   is even (The positive divisors of   include   as well as  .)Let  number whose decimal expansion contains   in the  -th place, .Determine,with proof,whether   is rational or irrational.

 be the real

INMO 2009

Problem 163 Find all real numbers   such that:

(Here 

 denotes the largest integer not exceeding  .) INMO 2009

Problem 164 All the points in the plane are colored using three colors.Prove that there exists a triangle with vertices having the same color such that either it is isosceles or its angles are in geometric progression. INMO 2009

81

Problem 165 Let   be an acute angled triangle and let   be its ortho center. Let   denote the largest altitude of the triangle  . Prove that:

INMO 2009

Problem 166 Let 

 be positive real numbers such that 

.Prove that:

. INMO 2009

Problem 167 Let   be a triangle with circum-circle  . Let   be a point in the interior of triangle   which is also on the bisector of  . Let   meet   in   respectively. Suppose   is the point of intersection of   with  ; and   is the point of intersection of   with  . Prove that   is parallel to  . INMO 2010

Problem 168 Find all natural numbers 

Problem 169 Find all non-zero real numbers  equations:

 such that 

 does 

 divide  . INMO 2010

 which satisfy the system of

INMO 2010

82

Problem 170 How many 6-tuples   is from the set  for 

 (where 

 are there such that each of   and the six expressions

 is to be taken as 

) are all equal to one another? INMO 2010

Problem 171Let   be an acute-angled triangle with altitude  . Let   be its ortho-center and   be its circum-center. Suppose   is an acute-angled triangle and   its circum-center. Let   be the reflection of   in the line  . Show that   lies on the line joining the mid-points of   and  . INMO 2010

Problem 172 Define a sequence 

 by 

, 

 and

for   For every 

 and 

 prove that 

 divides

.  Suppose 

 divides   for some natural numbers   and  . Prove that 

 divides 

INMO 2010

Problem 173 Let   be points on the sides  triangle   such that   and   Show that   is equilateral.

 respectively of a

INMO 2011

83

Problem 174 Call a natural number   faithful if there exist natural numbers   such that   and   and   Show that all but a finite number of natural numbers are faithful.  Find the sum of all natural numbers which are not faithful. INMO 2011

Problem 175 Let 

 and 

 be two polynomials with integral coefficients such that   is a prime and   and   Suppose that there exists a rational number   such that   Prove that  INMO 2011

Problem 176 Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium. INMO 2011

Problem 177 Let   be a cyclic quadrilateral inscribed in a circle   Let   be the midpoints of arcs   of   respectively. Suppose that   Show that   are all concurrent. INMO 2011

Problem 178 Find all functions 

 satisfying For all 

. INMO 2011

84

Problem 179 Let 

 be a quadrilateral inscribed in a circle. Suppose 

 and   subtends  maximum possible area of  .

 degrees at the center of the circle . Find the INMO 2012

Problem 180 Let  numbers, such that  . Prove that   divides 

 and   and 

 be two sets of prime . Suppose   and 

. INMO 2012

Problem 181 Define a sequence 

 of functions by for 

. Prove that each 

 is a polynomial with integer coefficients. INMO 2012

Problem 182 Let   be a triangle. An interior point   of   is said to be good if we can find exactly   rays emanating from   intersecting the sides of the triangle   such that the triangle is divided by these rays into   smaller triangles of equal area. Determine the number of good points for a given triangle  . INMO 2012

Problem 183 Let   be an acute angled triangle. Let   be points on   such that   is the median,   is the internal bisector and   is the altitude. Suppose that   and   Show that   is equilateral. INMO 2012 85

Problem 184 Let 

for all 

 be a function satisfying 

, 

 and

, simultaneously.

 Find the set of all possible values of the function  .  If 

 and 

, find the set of all integers   such that 

. INMO 2012

Problem 185 Let   and   be two circles touching each other externally at   Let   and   be the centers of   and   respectively. Let   be a line which is tangent to   at   and passing through   and let   be the line tangent to   at   and passing through   Let   If   then prove that the triangle   is equilateral. INMO 2013

Problem 186 Find all 

 and primes 

 satisfying INMO 2013

Problem 187 Let 

 such that  . Show that the equation   has no integer solution. INMO 2013

86

Problem 188 Let   be an integer greater than   and let   be the number of non empty subsets   of   with the property that the average of the elements of   is an integer.Prove that   is always even. INMO 2013

Problem 189 In an acute triangle  orthocenter. Let  midpoint of   If the triangles  possible values of 

 let   and 

 be its circumcentre, centroid and  Let   be the  have the same area, find all the INMO 2013

Problem 190 Let   and 

 be six positive real numbers satisfying   and   Further, suppose that   Prove that   and  INMO 2013

Problem 191 In a triangle  triangles 

 and 

, let   be the point on the segment   such that  . Suppose that the points  ,   and the centroids of  lie on a circle. Prove that  . INMO 2014

Problem 192 Let   be a natural number. Prove that,

is even. INMO 2014

87

Problem 193 Let   be natural numbers with  . Suppose that the sum of their greatest common divisor and least common multiple is divisible by  . Prove that the quotient is at most 

. When is this quotient exactly equal to  INMO 2014

Problem 194 Written on a blackboard is the polynomial  . Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of   by  . And at this turn, Hobbes should either increase or decrease the constant coefficient by  . Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning strategy. INMO 2014

Problem 195 In a acute-angled triangle  , a point   lies on the segment  . Let   denote the circumcenter of triangles   and   respectively. Prove that the line joining the circumcentre of triangle   and the orthocentre of triangle   is parallel to  . INMO 2014

Problem 196 Let   be a natural number. Let  , and define   to be the set of all those elements of   which belong to exactly one of   and  . Show that  , where   is a collection of subsets of   such that for any two distinct elements of   of   we have  . Also find all such collections   for which the maximum is attained. INMO 2014

88

Problem 197 Let   be a right-angled triangle with  . Let   is the altitude from   on  . Let   and  be the incenters of triangles   and   respectively.Show that circumcenter of triangle   lie on the hypotenuse  . INMO 2015

Problem 198 For any natural number  example we write 

 write the finite decimal expansion of 

 as its infinite decimal expansion not 

length of non-periodic part of the (infinite) decimal expansion of 

 (for

. Determine the . INMO 2015

Problem 199 Find all real functions  .

 such that  INMO 2015

Problem 200 There are four basketball players  . Initially the ball is with  . The ball is always passed from one person to a different person. In how many ways can the ball come back to   after   moves? (for example  , or  . INMO 2015

89

Problem 201 Let   intersect at  . Let   and 

 be a convex quadrilateral.Let diagonals   and   and   are altitudes from   on the side   respectively. Show that   has a incircle if and only if 

INMO 2015

Problem 202 Show that from a set of 

 square integers one can select six numbers 

 such that 

Problem 203 Let 

. INMO 2015

 be a triangle in which 

the triangle lies on the incircle. Find the ratio 

. Suppose the orthocentre of . INMO 2016

Problem 204 For positive real numbers  necessarily implies 

 which of the following statements

: (I) 

, (II)   ? Justify your answer. INMO 2016

Problem 205 Let   denote the set of natural numbers. Define a function   and 

. We write   for any 

(i) Show that for each 

 by   and in general 

.

, there exists   such that 

.

90

(ii) For  . Prove that 

, let 

 denote the number of elements in the set  , for  . INMO 2016

Problem 206 Suppose   points of the circumference of a circle are colored red and the remaining points are colored blue . Given any natural number  , prove that there is a regular  -sided polygon all of whose vertices are blue INMO 2016

Problem 207 Let   be a right-angle triangle with  . Let   be a point on   such that the inradii of the triangles   and   are equal. If this common value is   and if   is the inradius of triangle  , prove that

INMO 2016

Problem 208 Consider a non constant arithmetic progression  . Suppose there exist relatively prime positive integers   and 

 such that 

 and   are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers. INMO 2016

91

Problem 209 In the given figure,   is a square sheet of paper. It is folded along   such that   goes to a point   different from   and  , on the side   and   goes to  . The line   cuts   in  . Show that the inradius of the triangle   is the sum of the inradii of the triangles   and  .

INMO 2017

Problem 210 Suppose 

 is an integer and all the roots of   are integers. Find all possible values of  . INMO 2017

Problem 211 Find the number of triples 

 where   is a real number and  , 

 belong to the set   such that  denotes the fractional part of the real number  . (For example  .)

where 

INMO 2017

92

Problem 212 Let 

 be a convex pentagon in which   and the side lengths are five consecutive integers in some order. Find all possible values of  . INMO 2017

Problem 213 Let 

 be a convex pentagon in which   and the side lengths are five consecutive integers in some order. Find all possible values of  . INMO 2017

Problem 214 Let 

 be an integer and consider the sum

Show that   form the sides of a triangle whose area and inradius are also integers. INMO 2017

Problem 215 Let   be a non-equilateral triangle with integer sides. Let   and   be respectively the mid-points of   and   ; let   be the centroid of  . Suppose,  ,  ,  ,   are concyclic. Find the least possible perimeter of  . INMO 2018

93

Problem 216 For any natural number  , consider a   rectangular board made up of   unit squares. This is covered by   types of tiles :   red tile,   green tile and   domino. (For example, we can have   types of tiling when   : red-red ; red-green ; green-red ; green-green ; and blue.) Let   denote the number of ways of covering   rectangular board by these   types of tiles. Prove that,   divides  . INMO 2018

Problem 217 Let   and   be two circles with respective centers   and   intersecting in two distinct points   and   such that   is an obtuse angle. Let the circumcircle of   intersect   and   respectively in points   and  . Let the line   intersect   in   ; let the line   intersect   in  . Prove that, the points   are concyclic. INMO 2018

Problem 218 Find all polynomials with real coefficients   divides 

 such that 

. INMO 2018

Problem 219 There are   girls in a class sitting around a circular table, each having some apples with her. Every time the teacher notices a girl having more apples than both of her neighbors combined, the teacher takes away one apple from that girl and gives one apple each to her neighbors. Prove that this process stops after a finite number of steps. (Assume that the teacher has an abundant supply of apples.) INMO 2018

94

Problem 220 Let   denote set of all natural numbers and let  such that  for all   divides 

 be a function

;  for all 

.

Prove that, there exists an odd natural number   such that 

Problem 221 Let   be a triangle with  segment   and   be a point on line  circumcircle of triangle   at   and   and  . Determine 

 for all   in  INMO 2018

. Let   be a point on the  such that   is tangent to the  is perpendicular to  . Given that   in degrees. INMO 2019

Problem 222 Let 

 be a regular pentagon.For  , let   be the pentagon whose vertices are the midpoint of the sides  . All the   vertices of each of the   pentagons are arbitrarily coloured red or blue. Prove that four points among these   points have the same color and form the vertices of a cyclic quadrilateral. INMO 2019

Problem 223 Let 

 be distinct positive integers. Prove that Further,

determine when equality holds. INMO 2019

95

Problem 224 Let   and  are   distinct primes  .

 be positive integers such that   such that   divides 

. Prove that there  for all  INMO 2019

Problem 225 Let   be the diameter of a circle   and let   be a point on   different from   and  . Let   be the foot of perpendicular from   onto  .Let   be a point on the segment   such that   is equal to the semi perimeter of  .Show that the excircle of   opposite   is tangent to  . INMO 2019

Problem 226 Let   be a function defined from  all positive real numbers such that  for all   for all   for all  Prove that  for all   for all 

 real, 

 to the set of

INMO 2019

Problem 227 Let   and   be two circles of unequal radii, with centers   and   respectively, intersecting in two distinct points   and  . Assume that the center of each circle is outside the other circle. The tangent to   at   intersects   again in  , different from  ; the tangent to   at   intersects   again at  , different from  . The bisectors of   and   meet   and   again in   and  , respectively. Let   and   be the circumcenter of triangles   and  , respectively. Prove that   is the perpendicular bisector of the line segment  . INMO 2020

96

Problem 228 Suppose   is a polynomial with real coefficients, satisfying the condition  , for every real  . Prove that   can be expressed in the form for some real numbers 

 and non-negative integer  . INMO 2020

Problem 229 Let   be a subset of  . Suppose there is a positive integer   such that for any integer  , one can find positive integers   so that   and all the digits in the decimal representations of   (expressed without leading zeros) are in  . Find the smallest possible value of  . INMO 2020

Problem 230 Let  numbers such that 

 be an integer and let 

 be   real . Prove that INMO 2020

Problem 231 Infinitely many equidistant parallel lines are drawn in the plane. A positive integer   is called frameable if it is possible to draw a regular polygon with   sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon. (a) Show that   are frameable. (b) Show that any integer   is not frameable. (c) Determine whether   is frameable. INMO 2020

97

Problem 232 A stromino is a   rectangle. Show that a   board divided into twenty-five   squares cannot be covered by   strominos such that each stromino covers exactly three squares of the board, and every square is covered by one or two strominos. (A stromino can be placed either horizontally or vertically on the board.) INMO 2020

Problem 233 Suppose 

 is an integer, and let 

 be 

 integers such that for any two integers   and   satisfying  . Determine the maximum possible value of  . INMO 2021

Problem 234 Find all pairs of integers   so that each of the two cubic polynomials has all the roots to be integers. INMO 2021

Problem 235 Betal marks   points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any   of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using only a straightedge. Can Vikram always complete this challenge? Note. A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points. INMO 2021

98

Problem 236 A Magician and a Detective play a game. The Magician lays down cards numbered from   to   face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise. Prove that the Detective can guarantee a win if and only if she is allowed to ask at least   questions. INMO 2021

Problem 237 In a convex quadrilateral   and  . Extend   at  . Prove that  .

,  ,  ,   to meet the circumcircle of triangle  INMO 2021

Problem 238 Let  functions 

 be the set of all polynomials with real coefficients. Find all  satisfying the following conditions:

a.  maps the zero polynomial to itself, b. for any non-zero polynomial  ,  c. for any two polynomials  , the polynomials   have the same set of real roots.

, and  and  INMO 2021

99

Problem 239 Let 

 be an interior point on the side 

. Let the circumcircle of triangle  circumcircle of triangle   intersect 

 intersect   again at 

 intersect the circumcircle of triangle  , respectively. Let   and  , respectively. Prove that 

 of an acute-angled triangle   again at  . Let 

 again at 

 be the incentres of triangles   are concyclic.

 and the , and 

, 

, 

 and   and  INMO 2022

Problem 240 Find all natural numbers   for which there is a permutation   of   that satisfies:

INMO 2022

Problem 241 For a positive integer  , let   denote the number of arrangements of the integers   into a sequence   such that   for all  ,   and   for all  ,  . For example,   is  , since the possible arrangements are 

 and 

(a) Find  (b) If 

 is the largest non-negative integer so that  .

(c) Find the largest non-negative integer 

 so that 

 divides 

, show that 

 divides  INMO 2022

100

Indian Team Selection Test (TST) Problems (2001 - 2019)

101

Problem 1

Let   ,   , 

. Prove that if 

, then 

. TST 2001

Problem 2

Two symbols   and   obey the rule  . Given a word   consisting of   letters   and   letters  , show that there is a unique cyclic permutation of this word which reduces to  . TST 2001

Problem 3 In a triangle   with incircle   and incenter   , the segments   ,   ,   cut   at   ,   ,   , respectively. Rays   ,   ,   meet the sides   ,   ,   at   ,   ,   respectively. Prove that: When does equality occur? TST 2001

Problem 4

For any positive integer  , show that there exists a polynomial 

degree   with integer coefficients such that  powers of  .

 of

 are all distinct TST 2001

102

Problem 5

Let 

 be a cubic polynomial with integer coefficients. Suppose that a

prime   divides 

 for 

 ,  ,  ,  , where 

integers from the set 

 are distinct

. Prove that   divides all the coefficients of 

. TST 2001

Problem 6

Find the number of all unordered pairs 

-element set, such that 

 and 

 of subsets of an  . TST 2001

Problem 7

If on 

such that  , 

, triangles  ,  .

 and  .

COnstructed externally on   is triangle  . Prove that 1.   is perpendicular to  . 2. If 

 is the projection of 

 are constructed externally

 on 

 with 

, then prove that 

 , 

. TST 2001

Problem 8

Find all functions 

 satisfying :

for all 

. TST 2001

103

Problem 9

Points 

triangle 

 are chosen on side 

 in that order. Let 

 be the inradius of triangle 

 , and   be the inradius of   independent of   such that :

 of a  for 

. Show that there is a constant 

TST 2001

Problem 10 Complex numbers   ,   ,   have the property that  integer for every natural number  . Prove that the polynomial

 is an

has integer coefficients. TST 2001

Problem 11 Let 

 be a prime. For each 

the unique integer in 

 such that 

, define 

 to be

 and set 

. Prove that :

TST 2001

Problem 12 Each vertex of an  grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if:  all the three colors occur at the vertices of the square, and  one side of the square has the endpoints of the same color. Show that the number of properly colored squares is even. TST 2001

104

Problem 13 Let   be a rectangle, and let   be a circular arc passing through the points   and  . Let   be the circle tangent to the lines   and   and to the circle  , and lying completely inside the rectangle  . Similarly let   be the circle tangent to the lines   and   and to the circle  , and lying completely inside the rectangle  . Denote by   and   the radii of the circles   and  , respectively, and by   the inradius of triangle  . (a) Prove that  . (b) Prove that one of the two common internal tangents of the two circles   and   is parallel to the line 

 and has the length 

. TST 2001

Problem 14 A strictly increasing sequence   for all  for which there exist positive integers   and 

 has the property that  . Suppose   is the least positive integer  such that 

. Prove that 

. TST 2001

Problem 15 Let   be a polynomial of degree   with real coefficients and let  . Prove that

TST 2001

105

Problem 16 Let 

 and 

 be three points on a line with 

 between 

 and 

. Let   be semicircles, all on the same side of   and with   as diameters, respectively. Let   be the line perpendicular to   through  . Let   be the circle which is tangent to the line  , tangent to   internally, and tangent to   externally. Let   be the point of contact of   and  . The diameter of   through   meets   in  . Show that  . TST 2002

Problem 17 Show that there is a set of   consecutive positive integers containing exactly   primes. (You may use the fact that there are   primes less than 1000) TST 2002

Problem 18 Let  form  of  ?

. How many quadratics are there of the , with equal roots, and such that 

 are distinct elements TST 2002

Problem 19 Let   be the circumcenter and   the orthocenter of an acute triangle  . Show that there exist points  ,  , and   on sides  ,  , and   respectively such that and the lines  ,  , and   are concurrent. TST 2002

Problem 20 Let 

 be positive reals such that 

. Prove that

TST 2002

106

Problem 21 Determine the number of  -tuples of integers  that 

 for each 

 and 

 such

 for 

. TST 2002

Problem 22 Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle. TST 2002

Problem 23 Let 

be the sum of positive divisors of an integer  Show that 

 for positive integers 

.

 and   with  TST 2002

Problem 24 Find all positive integers   such that 

 is a power of  . TST 2002

107

Problem 25 On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first   days, apples for the next   days, followed by oranges for the next   days, and so on. Srinath has oranges for the first   days, apples for the next   days, followed by oranges for the next   days, and so on. If  , and if the tour lasted for  eat the same kind of fruit?

 days, on how many days did they TST 2002

Problem 26 Let 

 denote the set of all ordered triples 

integers. Find all functions 

 of nonnegative

 satisfying

for all nonnegative integers  ,  ,  . TST 2002

Problem 27 Let   be a triangle and   an exterior point in the plane of the triangle. Suppose the lines  ,  ,   meet the sides  ,  ,   (or extensions thereof) in  ,  ,  , respectively. Suppose further that the areas of triangles  ,  ,   are all equal. Prove that each of these areas is equal to the area of triangle   itself. TST 2002

108

Problem 28 Let 

 be integers with 

. A set 

 of non-negative

integers is olympic if   and if  . Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets. TST 2002

Problem 29 Let   and 

 and 

 be two triangles such that

 is the midpoint of 

Prove that 

.

 bisects 

 is the midpoint of 

 and 

 bisects 

. TST 2002

Problem 30 Let   be an odd prime and let   be an integer not divisible by  . Show that there are 

 triples of integers 

 with 

 and such

that  TST 2002

Problem 31 Let 

 be arbitrary real numbers. Prove the inequality

TST 2002

Problem 32 Is it possible to find   positive integers not exceeding  that all pairwise sums of them are different?

, such TST 2002

109

Problem 33 Let   be a positive integer and let   is the square root of 

 where 

, and   and   are polynomials with real coefficients. Show

that for any real number   the equation 

 has only real roots. TST 2002

Problem 34 Consider the square grid with   and   at its diagonal ends. Paths from   to   are composed of moves one unit to the right or one unit up. Let   (n-th catalan number) be the number of paths from   to   which stay on or below the diagonal  . Show that the number of paths from   to   which cross 

 from below at most twice is equal to  TST 2002

Problem 35 Let  triangles exterior to  that

 be an acute triangle. Let  , with 

the intersection of lines   and  , and let   be the intersection of  sum

, and  , and 

, let   and 

 be isosceles , such

Let   be  be the intersection of   and  . Find, with proof, the value of the

TST 2002

Problem 36 Let 

 be positive real numbers. Prove that

TST 2002 110

Problem 37 Given a prime  , show that there exists a positive integer   such that the decimal representation of 

 has a block of 

 consecutive zeros. TST 2002

Problem 38 Let 

 be the midpoints of the sides 

of an acute non-isosceles triangle 

, and let 

, respectively,  be the feet of the

altitudes through the vertices   on these sides respectively. Consider the arc   of the nine point circle of triangle   lying outside the triangle. Let the point of trisection of this arc closer to   be  . Define analogously the points   (on arc  ) and  (on arc  ). Show that triangle   is equilateral. TST 2003

Problem 39 Find all triples  (i)  ; (ii)  (iii) 

 of positive integers such that

; and  is divisible by each of the numbers 

. TST 2003

Problem 40 Find all functions 

 such that for all reals   and  , TST 2003

111

Problem 41 There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines. TST 2003

Problem 42 On the real number line, paint red all points that correspond to integers of the form  , where   and   are positive integers. Paint the remaining integer point blue. Find a point   on the line such that, for every integer point  , the reflection of   with respect to   is an integer point of a different color than  . TST 2003

Problem 43 A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find  , the maximum number of regions into which   zig-zags can divide the plane. For example,  (see the diagram). Of these   regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in   of degree not exceeding  .

TST 2003

112

Problem 44

 is a polynomial with integer coefficients and for every natural   we

have 

. 

 is divisible by 

 is a sequence that: 

 for every 

 one of 

 Prove that  TST 2003

Problem 45 Let 

 be a triangle, and let 

exradii opposite the vertices 

 denoted its inradius and the

, respectively. Suppose 

. Prove that  is acute,

(a) triangle  (b) 

. TST 2003

Problem 46 Let   be a positive integer and   such that   such that one of 

 a partition of 

. Prove that there exist   is the sum of the other two.

, 

,  TST 2003

Problem 47 Let   be a positive integer greater than  , and let   be a prime such that   divides 

 and   divides 

. Prove that 

 is a square. TST 2003

Problem 48 Let   be a cyclic quadrilateral. Let  ,  ,   be the feet of the perpendiculars from   to the lines  ,  ,  , respectively. Show that   if and only if the bisectors of  with 

 and 

 are concurrent

. TST 2004 113

Problem 49 Prove that for every positive integer   there exists an  -digit number divisible by   all of whose digits are odd. TST 2004

Problem 50 For 

 positive reals find the minimum value of

TST 2004

Problem 51 Given a permutation  pair 

 of 

 is called an inversion of   if 

 , an ordered  and 

. Let 

 denote the no. of inversions of the permutation  . Find the average of   as   varies over all permutations. TST 2004

Problem 52 Prove that in any triangle 

,

TST 2004

Problem 53 Find all triples 

 of positive integers such that TST 2004

114

Problem 54 Suppose the polynomial 

 has only real zeros

and let 

. Assume that 

 has no real roots.

Prove that  TST 2004

Problem 55 Let   be a bijection of the set of all natural numbers onto itself. Prove that there exists positive integers   such that  TST 2004

Problem 56 A set   of 4 points in the plane is said to be Athenian set if there is a point   of the plane satisfying (*) 

 does not lie on any of the lines 

(**) the line joining  joining 

 for 

 to the midpoint of the line 

 to the midpoint of 

, 

;  is perpendicular to the line

 being distinct.

(a) Find all Athenian sets in the plane. (b) For a given Athenian set, find the set of all points  and (**)

 in the plane satisfying (*) TST 2004

Problem 57 Determine all integers   such that  some 

 is divisible by 

 for TST 2004

115

Problem 58 The game of 

 is played on an infinite board of lattice points 

. Initially there is a  point 

 at 

and placing a 

. A move consists of removing a   at each of the points 

 from

 and 

 provided both are vacant. Show that at any stage of the game there is a  some lattice point 

 at

 with  TST 2004

Problem 59 Let   be a triangle and let   be a point in its interior. Denote by  ,   the feet of the perpendiculars from   to the lines  ,  ,  , respectively. Suppose that by  ,  ,   the excenters of the triangle  of the triangle  .

. Prove that 

, 

Denote  is the circumcenter TST 2004

Problem 60 Show that the only solutions of the equation integers 

, in positive

 and prime   are

(i)  (ii) 

and   is a prime of the form 

,  TST 2004

Problem 61 Determine all functions 

 such that for all reals 

 where 

 is a given

constant. TST 2004

116

Problem 62 Let   be a triangle and   its incenter. Let  triangles   and   respectively. (a) Show that there exists a function   and 

 and 

 be the inradii of

 such that

where 

(b) Prove that TST 2004

Problem 63 Define a function  (a)   is nondecreasing

 by the following rule:

(b) for each  , 

 i sthe number of times   appears in the range of  ,

Prove that 

 and 

 for all  TST 2004

Problem 64 Two runners start running along a circular track of unit length from the same starting point and in the same sense, with constant speeds   and   respectively, where   and   are two distinct relatively prime natural numbers. They continue running till they simultaneously reach the starting point. Prove that (a) at any given time  , at least one of the runners is at a distance not more than   units from the starting point. (b) there is a time   such that both the runners are at least  the starting point. (All distances are measured along the track).  integer function.

 units away from  is the greatest TST 2004

117

Problem 65 Let 

 be   real numbers such that 

. Prove

that TST 2004

Problem 66 Find all primes  , the number

 with the following property: for any prime 

is squarefree (i.e. is not divisible by the square of a prime). TST 2004

Problem 67 Every point with integer coordinates in the plane is the center of a disk with radius 

.

(1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than  . TST 2004

118

Problem 68 Let 

 be an acute-angled triangle and   be a circle with 

 as

diameter intersecting   and   at   and   respectively. Tangents are drawn at   and   to   intersect at  . Show that the ratio of the circumcentre of triangle   to that if   is a rational number. TST 2004

Problem 69 Let 

 and 

two real polynomials. Suppose that there exists an interval  than   SUCH THAT BOTH   AND   ARE nEGATIVE FOR  both are positive for   and  . Show that there is a real 

be  of length greater  and  such that  TST 2004

Problem 70 An integer   is said to be good if   is not the square of an integer. Determine all integers   with the following property:   can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. TST 2004

Problem 71 Let  point of triangle 

 be a triangle with all angles  . Let   be the Fermat , that is, the interior point of   such that  . For each one of the three triangles  ,  , draw its Euler line - that is, the line connecting its circumcenter

 and  and its centroid. Prove that these three Euler lines pass through one common point. Remark. The Fermat point   is also known as the first Fermat point or the first Torricelli point of triangle  . TST 2005

119

Problem 72 Prove that one can find a  positive integers  ,   ,   such that (i)  (ii) 

 such that 

, there exist three

;  is the cube of an integer. TST 2005

Problem 73 If  ,   ,  are three positive real numbers such that  , prove that TST 2005

Problem 74 Consider a  -sided polygon inscribed in a circle ( ). Partition the polygon into   triangles using non-intersecting diagonals. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant. TST 2005

Problem 75 Let   denote the number of positive divisors of the positive integer  . Prove that there exist infinitely many positive integers   such that the equation   does not have a positive integer solution  . TST 2005

120

Problem 76 There are   students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of   societies. Suppose that the following conditions hold: i.) Each pair of students are in exactly one club. ii.) For each student and each society, the student is in exactly one club of the society. iii.) Each club has an odd number of students. In addition, a club with   students (  is a positive integer) is in exactly   societies. Find all possible values of  . TST 2005

Problem 77 Let   be two rational numbers. Let  numbers with the properties: (i) 

 and 

(ii) if    Let  that 

 and 

 be a set of positive real

; , then 

.

denote the set of all irrational numbers in  . prove that every   such ,   contains an element   with property  TST 2005

121

Problem 78 Find all functions  for any two positive integers  Remark. The abbreviation 

 satisfying  and  .  stands for the set of all positive integers:

. By 

, we mean 

 (and not 

). TST 2005

Problem 79 A merida path of order  - plane joining 

 to 

 is a lattice path in the first quadrant of 

 using three kinds of steps 

 and 

, i.e. 

 joins 

 to 

,   etc... An ascent

in a merida path is a maximal string of consecutive steps of the form  . If   denotes the number of merida paths of order   with exactly   ascents, compute   and 

. TST 2005

Problem 80 Let   be a convex quadrilateral. The lines parallel to   and   through the orthocentre   of   intersect   and   Respectively at   and  . prove that the perpendicular through  orthocentre of triangle 

 to the line 

 passes through the TST 2005

122

Problem 81 Given real numbers  that there exist integers   and   s.t.

 s.t. 

 and 

, prove

TST 2005

Problem 82 Consider a matrix of size   whose entries are real numbers of absolute value not exceeding  . The sum of all entries of the matrix is  . Let   be an even positive integer. Determine the least number   such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding   in absolute value. TST 2005

Problem 83 For a given triangle ABC, let X be a variable point on the line BC such that the point C lies between the points B and X. Prove that the radical axis of the incircles of the triangles ABX and ACX passes through a point independent of X. TST 2005

Problem 84 Determine all positive integers 

 , such that TST 2005

123

Problem 85 For real numbers 

 not all equal to   , define a real function  . Suppose 

 for some real 

. prove that there exist a real number   s.t.  TST 2005

Problem 86 Let   be a positive integer divisible by  . Find the number of permutations   of 

 which satisfy the condition   for all 

. TST 2006

Problem 87 Let   be a parallelogram. A variable line   through the vertex   intersects the rays   and   at the points   and  , respectively. Let   and   be the  -excenters of the triangles   and  . Show that the angle   is independent of the line  . TST 2006

Problem 88 There are   markers, each with one side white and the other side black. In the beginning, these   markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if   is not divisible by  . TST 2006

124

Problem 89 Let   be a triangle and let   be a point in the plane of  inside the region of the angle   but outside triangle  .

 that is

(a) Prove that any two of the following statements imply the third. (i) the circumcentre of triangle 

 lies on the ray 

(ii) the circumcentre of triangle 

 lies on the ray 

(iii) the circumcentre of triangle 

 lies on the ray 

. . .

(b) Prove that if the conditions in (a) hold, then the circumcenter of triangles   and 

 lie on the circumcircle of triangle  TST 2006

Problem 90 Let   be a prime number and let   be a finite set containing at least   elements. A collection of pairwise mutually disjoint  -element subsets of   is called a  -family. (In particular, the empty collection is a  -family.) Let  (respectively,  ) denote the number of  -families having an even (respectively, odd) number of  -element subsets of  Prove that   and   differ by a multiple of  . TST 2006

Problem 91 Let   and  . Prove that if 

 be an equilateral triangle, and let   respectively. Let 

 and 

 be points on 

 and 

, then the union of the triangular regions   covers the triangle 

. TST 2006

125

Problem 92 Let 

 be a triangle with inradius  , circumradius 

, and with sides 

. Prove that

TST 2006

Problem 93 the positive divisors  Suppose 

 of a positive integer   are ordered

. Find all possible values of 

. TST 2006

Problem 94 Let   be arithmetic progressions of integers, each of   terms, such that any two of these arithmetic progressions have at least two common elements. Suppose   of these arithmetic progressions have common difference   and the remaining arithmetic progressions have common difference   where  . Prove that

TST 2006

Problem 95 Find all triples 

 such that   satisfying 

 are integers in the set   and 

. TST 2006

126

Problem 96 Let   be a real number for each   be an integer such that

Let 

 and each 

 and   be positive integers such that 

integers 

 and let 

. Prove that there exist

 not all zero, such that

TST 2006

Problem 97 Let 

 be subsets of a finite set   such that 

each  . For a subset   of   let  for each subset   of   at least one of the following conditions holds , Prove that 

, 

 for . Suppose

.

. TST 2006

Problem 98 Show that in a non-equilateral triangle, the following statements are equivalent:  The angles of the triangle are in arithmetic progression.  The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line. TST 2007

127

Problem 99 Find all integer solutions of the equation TST 2007

Problem 100 Let   be the set of all bijective functions from the set   to itself. For each 

 define

Determine  (Here 

 for all 

) TST 2007

Problem 101 Let   be a trapezoid with parallel sides   and   lie on the line segments   and  , respectively, so that  . Suppose that there are points  segment 

 and 

 satisfying

that the points 

, 

, 

. Points 

 on the line Prove

 and 

 are concyclic.| TST 2007

Problem 102 Let 

 be non-negative real numbers such that   and 

 Show that TST 2007

128

Problem 103 Given a finite string   of symbols   and  , we denote  number of s in   minus the number of  s (For example,  ). We call a string 

 as the

 balanced if every substring 

 of

(consecutive symbols)   has the property   (Thus   is not balanced, since it contains the substring   whose   value is   Find, with proof, the number of balanced strings of length  TST 2007

Problem 104 A sequence of real numbers  here   denotes the greatest integer not exceeding   for   sufficiently large.

 is defined by the formula  is an arbitrary real number,  , and 

. Prove that  TST 2007

Problem 105 Let 

 be a finite set of points in the plane such that no three of them are

on a line. For each convex polygon 

 whose vertices are in  , let 

 be the

number of vertices of  , and let   be the number of points of   which are outside  . A line segment, a point, and the empty set are considered as convex polygons of  ,  , and   vertices respectively. Prove that for every real number  where the sum is taken over all convex polygons with vertices in  . TST 2007

129

Problem 106 Circles   and   with centers   and   are externally tangent at point   and internally tangent to a circle   at points   and   respectively. Line   is the common tangent of   and   at  . Let   be the diameter of   perpendicular to  , so that   are on the same side of  . Prove that lines  ,  ,   and   are concurrent. TST 2007

Problem 107 Find all integer solutions 

 of the equation 

 where 

 is a prime such that  TST 2007

Problem 108 Find all function(s) 

 satisfying the equation

For all  TST 2007

Problem 109 Let   be a triangle with  contact of Incircle And Nine-Point Circle, Then   being inradius.

.Prove that if  ,

 is point of

TST 2009

130

Problem 110 Let us consider a simple graph with vertex set   of integers with 

, are elements of V.

 is connected to  for all integers k. Prove that for all 

Problem 111 Let 

. All ordered pair 

 by an edge and to  , there exists a path fromm 

 by another edge  to 

. TST 2009

 be two distinct odd natural numbers.Define a Sequence 

 like following:

. Prove that there exists a natural number 

 such that 

Problem 112 Let   be circumcircle of  &  internally.Define  Prove That 

.Let 

. TST 2009

 be radius of circle touching 

 similarly. . TST 2009

131

Problem 113 Let  and  complex coefficients.

 be two monic polynomials of degree=  having

We know that there exist complex numbers 

, such that

. Prove that there exists 

 such that . TST 2009

Problem 114 Prove The Following identity:

. The Second term on the left hand side is to be regarded as zero for j=0. TST 2009

Problem 115 Let 

 be any point in the interior of a 

.Prove That

. TST 2009

Problem 116 Let   be a natural number 

 which divides 

.Prove That 

. TST 2009

132

Problem 117 Let

 and 

 be two polynomials with real

coefficients. Let g(x) have  than  .

 as two of its roots. Prove That 

 has a positive root less TST 2009

Problem 118 For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle. TST 2009

Problem 119 Find all integers 

 with the following property:

There exists three distinct primes   such that whenever   are   distinct positive integers with the property that at least one of   divides  one of   divides all of these differences.

, TST 2009

133

Problem 120 Let  and

 be a simple graph with vertex set 

 are connected by an edge for 

. Let 

 be the induced subgraph associated with  components of  Let

 .  be a subset of 

. Let 

 and 

 be number of

 having an odd number of vertices.  for 

Prove That 

.

. TST 2009

Problem 121 Let  of 

, 

 be a triangle in which 

 be the altitude from 

. Suppose that  orthocenter of 

 on 

, and 

. Let 

 be the midpoint

 be the altitude from 

 produced meets   (extended) at  , prove that   is perpendicular to 

. If  .

 onto 

 is the TST 2010

Problem 122 Two polynomials 

 and 

 have real coefficients, and   is an interval on the real line of length greater than  . Suppose   and   take negative values on  , and they take non-negative values outside  . Prove that there exists a real number   such that 

. TST 2010

134

Problem 123 For any integer 

, let 

 be the maximum number of triples 

 consisting of non-negative integers   (not necessarily distinct) such that the following two conditions are satisfied: (a) 

 for all 

(b) 

, then 

Determine 

;

,   for all 

 and 

.

. TST 2010

Problem 124 Let  . Prove that

 be positive real numbers such that 

TST 2010

Problem 125 Given an integer  distinct positive integers 

 and  (Here  prime to 

, show that there exist an integer an   and , all greater than  , such that the sums 

 are both  -the powers of some integers.  denotes the number of positive integers less than  .)

 and relatively TST 2010

135

Problem 126 Let   be a given integer. Show that the number of strings of length   consisting of  s and  s such that there are equal number of   and   blocks in each string is equal to

TST 2010

Problem 127 Let   be a cyclic quadrilateral and let   be the point of intersection of its diagonals   and  . Suppose   and   meet in  . Let the midpoints of   and   be   and   respectively. If   is the circumcircle of triangle  , prove that   is tangent to  . TST 2010

Problem 128 Call a positive integer good if either   or   can be written as product of Even number of prime numbers, not necessarily distinct. Let 

 where 

 are positive integers.

(a) Show that there exist distinct positive integers 

 such that 

 are all good numbers. (b) Suppose  . Prove that 

 are such that  .

 is a good number for all positive integers  TST 2010

136

Problem 129 Let   be a   array of positive real numbers such that the sum of numbers in row as well as in each column is  . Show that there exists 

 and 

 such that

TST 2010

Problem 130 Let 

 be a triangle. Let 

 be the brocard point. Prove that 

TST 2010

Problem 131 Find all functions   for all reals 

 such that  TST 2010

Problem 132 Prove that there are infinitely many positive integers  exists consecutive odd positive integers   and 

 for which there

 such that 

 are both perfect squares. If 

 are two positive

integers satisfying this condition, then we have  TST 2010

137

Problem 133 Let 

 be a triangle each of whose angles is greater than 

. Suppose a circle centered with   in 

 cuts segments 

 in 

 in 

 and 

 such that they are on a circle in counterclockwise direction in that

order.Suppose further 

 are equilateral. Prove that:

 The radius of the circle is 

 where 

 is an area.

TST 2011

Problem 134 Let the real numbers   and 

 satisfy the relations 

 Prove that TST 2011

Problem 135 A set of   distinct integer weights 

 is said to

be balanced if after removing any one of weights, the remaining   weights can be split into two subcollections (not necessarily with equal size)with equal sum.  Prove that if there exist balanced sets of sizes  size 

 then also a balanced set of

.

 Prove that for all odd 

 there exist a balanced set of size  . TST 2011

138

Problem 136 Find all positive integer   satisfying the conditions

 is a perfect square. TST 2011

Problem 137 Suppose 

 are non-integral real numbers for 

 is an integer for all integers   is rational.

 such that 

. Prove that none of  TST 2011

Problem 138 Let   be a non-empty finite subset of positive integers  . A subset   of   is called good if for every integer   there exists an   in   such that  . Let

Prove that :  If 

 is not good then the number of pairs 

 the number of good subsets of 

 in 

 is even.

 is odd. TST 2011

139

Problem 139 Let 

 be a convex pentagon such that   and 

 Let 

let   be the circumcenter of triangle  that 

 

 be the midpoint of 

 Given that 

 and  prove

TST 2011

Problem 140 Prove that for no integer   is 

 a perfect square. TST 2011

Problem 141 Consider a   square grid which is divided into   unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an  -staircase. Find the number of ways in which an  -stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas. TST 2011

Problem 142 Let  bisectors with 

 be an acute-angled triangle. Let   on 

 be internal

 respectively. Prove that

TST 2011

Problem 143 Find all pairs 

 of nonnegative integers for which TST 2011 140

Problem 144 Let  such that

 and 

for all integers 

 be two infinite sequences of integers

. Prove that there exists a positive integer   such that TST 2011

Problem 145 Let  on the segment   such that 

 be an isosceles triangle with  . Let   be a point  such that  . Let   be a point on the segment  . Prove that  . TST 2012

Problem 146 Let 

 and 

 be real numbers. Prove that the equation has real roots. TST 2012

Problem 147 How many  -tuples  which  simultaneously true?

 of natural numbers are there for  and 

 are TST 2012

141

Problem 148 Let 

 be a trapezium with 

 such that   is between   and   respectively. Let   intersect  that 

. Let 

 be a point on 

; and let   be the midpoints of   in   and   intersect   in 

. Prove

. TST 2012

Problem 149 Let   be integers where   is a prime. Prove that the following statements are equivalent:

TST 2012

Problem 150 Let 

 be a function such that   for all   for all 

. Prove that   satisfies 

. TST 2012

Problem 151 The circumcentre of the cyclic quadrilateral  intersection point of the circles   and  , other than 

 is  , is 

. The second , which lies in

the interior of the triangle  . Choose a point   on the extension of   beyond  , and a point   on the extension of   beyond  . Prove that   if and only if 

. TST 2012

142

Problem 152 Let  complex coefficients such that 

 be a polynomial with  and 

. Prove that

TST 2012

Problem 153 Determine the greatest positive integer   that satisfies the following property: The set of positive integers can be partitioned into   subsets   such that for all integers  exist two distinct elements of   whose sum is 

 and all 

 there TST 2012

Problem 154 Determine all sequences   of positive integers, such that for every positive integer   there exists an integer   with

TST 2012

Problem 155 Show that there exist infinitely many pairs   of positive integers with the property that   divides  ,   divides  ,   and  TST 2012

143

Problem 156 Suppose that   students are standing in a circle. Prove that there exists an integer   with   such that in this circle there exists a contiguous group of   students, for which the first half contains the same number of girls as the second half. TST 2012

Problem 157 Let 

 be a triangle with 

. The angle bisector of 

 and let 

 be the midpoint of 

 intersects the circle through 

 and 

 at the

point   inside the triangle  . The line   intersects the circle through   and   in two points   and  . The lines   and   meet at a point  , and the lines   and   meet at a point  . Show that   is the incentre of triangle  . TST 2012

Problem 158 Let 

 be a nonempty set of primes satisfying the property that for each

proper subset   of  , all the prime factors of the number  in  . Determine all possible such sets  .

 are also TST 2012

Problem 159 In a  array we have positive reals s.t. the sum of the numbers in each of the   columns is  . Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most 

. TST 2012

144

Problem 160 A quadrilateral  a circle with center  . Prove that  quadrilateral 

 without parallel sides is circumscribed around  is a point of intersection of middle lines of

 (i.e. barycentre of points  .

) iff  TST 2012

Problem 161 Find the least positive integer that cannot be represented as   for some positive integers 

. TST 2012

Problem 162 Let 

 denote the set of all positive real numbers. Find all functions 

 satisfying

for all 

. TST 2012

Problem 163 For a prime  , a natural number   and an integer  , we let   denote the exponent of   in the prime factorisation of   and 

. Find all pairs 

. For example, 

 such that 

. TST 2013

145

Problem 164 Let 

 by a cyclic quadrilateral with circumcenter 

the point of intersection of the diagonals  circumcenters of triangles 

 and 

, and 

, 

. Let 

 be

 the

, respectively. Prove that  TST 2013

Problem 165 We define an operation 

 on the set 

 by

For two natural numbers   and  , which are written in base   as   and 

 (possibly with leading 0's), we define 

 where   written in base   is  . For example, we have  For a natural number  , let 

 with   since  , where 

, for   and 

.

 denotes the largest

integer less than or equal to  . Prove that   is a bijection on the set of natural numbers. TST 2013

Problem 166 Let   be positive real numbers such that  positive integer then prove that

. If   is a

TST 2013

146

Problem 167 In a triangle 

 with 

 such that the inradii of triangles  prove that 

,   and 

 is a point on the segment   are equal. If 

 then

. TST 2013

Problem 168 A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the folLowing game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers  , neither of which was chosen earlier by any player and move the marker by   units in the horizontal direction and   units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning. Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well). TST 2013

Problem 169 Let   be an integer. There are   beads numbered  . Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed). For example, with  , the necklace with four beads 

 in the clockwise order is same as the one with 

 in the clockwise order, but is different from the one with  clockwise order.

 in the

We denote by   (respectively  ) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least  . Prove that 

 divides 

. TST 2013

147

Problem 170 In a triangle  , with  , let   and   denote its circumcenter and orthocenter, respectively. Let   be the reflection of   with respect to 

. Prove that 

 and 

 are collinear if and only if 

. TST 2013

Problem 171 For a positive integer  , a cubic polynomial   is said to be  -good if there exist  distinct integers   such that all the roots of the polynomial   are integers for   prove that there exists an  -good cubic polynomial.

. Given a positive integer  TST 2013

Problem 172 Find all functions   from the set of real numbers to itself satisfying for all real numbers 

. TST 2013

Problem 173 An integer   is called friendly if the equation   has a solution over the positive integers. a) Prove that there are at least   friendly integers in the set  b) Decide whether   is friendly.

. TST 2013

148

Problem 174 Players   and   play a game with   coins and   boxes arranged around a circle. Initially   distributes the coins among the boxes so that there is at least   coin in each box. Then the two of them make moves in the order   by the following rules: (a) On every move of his   passes   coin from every box to an adjacent box. (b) On every move of hers   chooses several coins that were not involved in  's previous move and are in different boxes. She passes every coin to an adjacent box. Player  's goal is to ensure at least   coin in each box after every move of hers, regardless of how   plays and how many moves are made. Find the least   that enables her to succeed. TST 2013

Problem 175 For a positive integer  , a sum-friendly odd partition of   is a sequence 

 of odd positive integers with   and   such that for all positive integers  ,   can be uniquely written as a subsum  . (Two subsums   and   with   and   are considered the same if 

 and 

 for 

.) For example,   is a sum-friendly odd partition of  . Find the number of sum-friendly odd partitions of  . TST 2013

Problem 176 In a triangle 

, let   denote its incenter. Points 

chosen on the segments   and 

 are

, respectively, such that  . The circumcircles of triangles 

 intersect lines 

, respectively, at points 

), respectively. Prove that 

 (different from 

 are concyclic. TST 2013

149

Problem 177 Let   be an integer and   the set of all positive integers that are greater than or equal to  . Let   be a nonempty subset of   such that the following two conditions hold: I. II.

if 

 with 

if   with  Prove that 

, then  , then 

.

. TST 2013

Problem 178 A positive integer   is called a double number if it has an even number of digits (in base 10) and its base 10 representation has the form   with   for  , and  . For example,   is a double number. Determine whether or not there are infinitely many double numbers   such that   is a square and   is not a power of  . TST 2013

Problem 179 Let 

 be an integer and 

 a sequence of

polynomials with integer coefficients. One is allowed to make moves  follows: in the  -th move 

 one chooses an element 

 as

 of the sequence with

degree of   at least   and replaces it with  stops when all the elements of the sequence are of degree  . If 

. The process

, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of   identical polynomials of degree 1. TST 2013

150

Problem 180 In a triangle  , with  ,   is a point on the line   such that   is perpendicular to  . A circle passing through   and touching the line  . Let 

 at a point 

 intersects the line 

 be a point on the line 

 different from 

 be the point of intersection of the lines 

 and 

 are concyclic if and only if 

 for the second time at   such that 

. Let 

. Prove that the points 

 is perpendicular to 

. TST 2013

Problem 181 Let   be an odd prime and   an odd natural number.Show that   does not divide  TST 2014

Problem 182 Let 

 be positive real numbers.Prove that  . TST 2014

Problem 183 In a triangle  , points   and   are on   and   respectively such that  ,  is not perpendicular to   and   is not perpendicular to  .Let   be the circle with   as center and   as its radius.Find the angles of triangle   given that the orthocenter of triangles   and   lie on  . TST 2014

151

Problem 184 Let   and   be rational numbers, such that  that   is the square of a rational number.

. Prove TST 2014

Problem 185 Let   be a natural number.A triangulation of a convex n-gon is a division of the polygon into   triangles by drawing   diagonals no two of which intersect at an interior point of the polygon.Let   denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine 

 in terms of  . TST 2014

Problem 186 For integers 

 we define   if 

Given a natural number 

 if 

 and 

.  show that there exist natural numbers 

 such that 

 with 

,where ,  being composed with itself   times. TST 2014

Problem 187 Find all polynomials 

 with integer coefficients such that 

 and 

 are co-prime for all natural numbers  . TST 2014

152

Problem 188 Let   be a positive integer. Find the smallest integer   with the following property; Given any real numbers   such that   and   for  , it is possible to partition these numbers into   groups (some of which may be empty) such that the sum of the numbers in each group is at most  . TST 2014

Problem 189 Starting with the triple  sequence of triples 

, define a

 by

for  .Show that each of the sequences  to a limit and finds these limits.

 converges TST 2014

Problem 190 In a triangle  incircle touches the line  triangle 

, let   be its incenter;   the point at which the ;   the midpoint of   and   the orthocenter of

. Prove that the line 

 is perpendicular to the line 

. TST 2014

153

Problem 191 For   let   be non-zero real numbers, and let  .Suppose that the following statements hold:

 satisfy triangle inequality  also satisfy triangle inequality. Prove that exactly one of 

 is negative. TST 2014

Problem 192 Let   be a positive integer, and let   be an infinite sequence of real numbers. Assume that for all nonnegative integers   and   there exists a positive integer 

 such that

Prove that the sequence is periodic, i.e. there exists some   for all  .

 such that  TST 2014

Problem 193 In a triangle  , with   and  ,   is a point on line   different from  . Suppose that the circumcenter and orthocenter of triangles   and   lie on a circle. Prove that  . TST 2014

154

Problem 194 Determine whether there exists an infinite sequence of nonzero digits   and a positive integer   such that for every integer  , the number   is a perfect square. TST 2014

Problem 195 In how many ways rooks can be placed on a   by   chess board such that every row and every column has at least one rook? (Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them) TST 2014

Problem 196 Prove that in any set of  pairs 

 and 

 with 

 distinct real numbers there exist two  or 

, such that TST 2014

Problem 197 Find all positive integers   and   such that 

. TST 2014

Problem 198 Let 

 be a triangle with 

. Let 

 and 

 be two different

 and 

 is located

points on line 

 such that 

between 

. Suppose that there exists an interior point 

 and 

 for which  that 

. Let the ray 

 intersect the circle 

 of segment   at 

. Prove

. TST 2014 155

Problem 199 Find all positive integers  integers.

 such that 

 and 

 are also TST 2015

Problem 200 A 

-digit number is called a 

 number if its digits belong to the set 

 and the difference of every pair of consecutive digits is  . a) Find the total number of cute numbers. b) Prove that the sum of all cute numbers is divisible by  . TST 2015

Problem 201 Prove that for any triangle 

, the inequality 

 holds. TST 2015

Problem 202 Let   be a triangle in which  . Let   be its orthocentre and   its circumcentre. Let   and   be respectively the midpoints of the arc   not containing   and arc   not containing  . Let   and   be respectively the reflections of  on a circle if and only if 

 in 

 and 

 in 

. Prove that 

 lie

 are collinear. TST 2015

156

Problem 203 For a composite number  , let 

 denote its largest proper divisor.

Show that there are infinitely many   for which 

 is a perfect square. TST 2015

Problem 204 Every cell of a  board is coloured either by red or blue. Find the number of all colorings in which there are no   squares in which all cells are red. TST 2015

Problem 205 Let   intersect at 

 be a convex quadrilateral and let the diagonals  . Let 

 and 

 be respectively the incentre of triangles  . Let 

triangles 

 be respectively the excentres of  opposite 

circle if and only if 

. Show that 

 lie on a

 lie on a circle. TST 2015

Problem 206 Let   and   be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose  the sets  an integer   such that 

 and 

 is odd and

 are the same. Prove that there exists . TST 2015

157

Problem 207 Let   points be given inside a rectangle   such that no two of them lie on a line parallel to one of the sides of  . The rectangle   is to be dissected into smaller rectangles with sides parallel to the sides of   in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect   into at least   smaller rectangles. TST 2015

Problem 208 Let 

 be an integer, and let 

 be the set

Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of   . TST 2015

Problem 209 Find all functions from  , for all 

 such that  . TST 2015

Problem 210 Let 

 be a simple graph on the infinite vertex set 

. Suppose every subgraph of  -colorable, Prove that   itself is  -colorable.

 on a finite vertex subset is  TST 2015

158

Problem 211 In a triangle  the incentres of triangles 

, a point   and 

 intersect the circumcircle of triangle 

 is on the segment  , Let   and   respectively. The lines   and   at 

 and 

 be

, respectively.

Let   be the point of intersection of lines   and  . Suppose   is also the reflection of   in   where   is the incentre of triangle  . Prove that  . TST 2015

Problem 212 Find all triples  positive integers   and   such that 

 consisting of a prime number   and two  and 

 are both powers of  . TST 2015

Problem 213 There are   lamps, each with two states:   or  . For each non-empty subset   of the set of these lamps, there is a   which operates on the lamps in  ; that is, upon   this button each of the lamps in   changes its state(on to off and off to on). The buttons are identical and it is not known which button corresponds to which subset of lamps. Suppose all the lamps are off initially. Show that one can always switch all the lamps on by performing at most 

 operations. TST 2015

Problem 214 Consider a fixed circle   with three fixed points  Also, let us fix a real number 

. For a variable point 

, let   be the point on the segment   such that  second point of intersection of the circumcircles of the triangles  . Prove that as 

 varies, the point 

 and 

 on it.  on 

 . Let   be the  and 

 lies on a fixed circle. TST 2015 159

Problem 215 Let   in 

 we have 

necessarily from  all 

 be a finite set of pairs of real numbers such that for any pairs   be a pair of real numbers(not

). We define 

, if   in 

. Let 

 for which 

there exists an integer 

 for all   we let 

; otherwise we choose a pair 

 and set   such that 

 as follows: for . Show that

. TST 2015

Problem 216 Let 

 be a given integer. Prove that infinitely many terms of the

sequence  , defined by are odd. (For a real number  ,   denotes the largest integer not exceeding  .) TST 2015

Problem 217 An acute-angled   is inscribed into a circle  . Let   be the centroid of  , and let   be the altitude of this triangle. A ray   meets   at  . Prove that the circumcircle of the triangle   is tangent to  TST 2016

Problem 218 Given that   is a natural number such that the leftmost digits in the decimal representations of   and   are the same, find all possible values of the leftmost digit. TST 2016

160

Problem 219 Let a,b,c,d be real numbers satisfying  . Prove that 

 and 

TST 2016

Problem 220 We say a natural number   is perfect if the sum of all the positive divisors of   is equal to  . For example,   is perfect since its positive divisors   add up to   distinct prime divisors.

. Show that an odd perfect number has at least 

Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result. TST 2016

Problem 221 Find all functions  for all reals  .

 such that TST 2016

161

Problem 222 An equilateral triangle with side length   is divided into   congruent triangular cells as shown in the figure below. Initially all the cells contain  . A move consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by   simultaneously. Determine all positive integers   such that after performing several such moves one can obtain   consecutive numbers  some order.

 in

TST 2016

Problem 223 Let   be an acute triangle with orthocenter  . Let   be the point such that the quadrilateral   is a parallelogram. Let   be the point on the line   such that   bisects  . Suppose that the line   intersects the circumcircle of the triangle   at   and  . Prove that  . TST 2016

Problem 224 Suppose that a sequence 

 of positive real numbers satisfies

for every positive integer  . Prove that   for every  . TST 2016

162

Problem 225 Let   be a natural number. A sequence  is called 

 if each 

 of 

 is an element of the set 

 numbers

 and the

ordered pairs   are all different for   (here we consider the subscripts modulo  ). Two  good sequences   and   are called   if there exists an integer   such that   for all   (again taking subscripts modulo  ). Suppose that there exists a non-trivial permutation (i.e., a permutation which is different from the identity permutation)   of 

 and an 

 good sequence 

 which is similar to 

. Show that 

. TST 2016

Problem 226 Suppose  positive integers  each of 

 are two positive rational numbers. Assume for some , it is known that 

 and 

 is a rational number. Prove that

 is a rational number. TST 2016

Problem 227 Let 

 and   be positive integers such that 

 for   are integers, then 

. Define 

. Prove that if all the numbers   is divisible by an odd prime. TST 2016

163

Problem 228 For a finite set   of positive integers, a partition of   into two disjoint nonempty subsets   and   is   if the least common multiple of the elements in   is equal to the greatest common divisor of the elements in  . Determine the minimum value of   such that there exists a set of   positive integers with exactly   good partitions. TST 2016

Problem 229 Let   be a natural number. We define sequences   and  integers as follows. We let   and  . For  , we let

 of

Given that   is a power of two. TST 2016

 for some natural number  , prove that 

Problem 230 Let   be an acute triangle and let   be the midpoint of  . A circle   passing through   and   meets the sides   and   at points   and   respectively. Let  that 

 be the point such that 

 lies on the circumcircle of 

 is a parallelogram. Suppose

. Determine all possible values of  . TST 2016

Problem 231 Let   be an odd natural number. We consider an 

 grid which is

made up of   unit squares and   edges. We color each of these edges either   or  . If there are at most     edges, then show that there exists a unit square at least three of whose edges are  . TST 2016

164

Problem 232 Let 

 be an acute triangle with circumcircle  . Let 

 and 

 be respectively the midpoints of the arcs   and   of  . Show that the inradius of triangle   is not less than the inradius of triangle  . TST 2016

Problem 233 Find all functions   such that for all   (Here   denotes the set of all real numbers.) TST 2016

Problem 234 Let 

 denote the set of all natural numbers. Show that there exists two

nonempty subsets   and   of   such that every number in   can be expressed as the product of a number in   and a number in  ; each prime number is a divisor of some number in   and also some number in  ; one of the sets   and   has the following property: if the numbers in this set are written as  , then for any given positive integer   there exists   such that 

. Each set has infinitely many composite numbers. TST 2016

165

Problem 235 Let 

 and   with   an integer and 

 real numbers,  . Define 

 and 

. (a) Find the number of unordered pairs of polynomials  two common roots. (b) For any 

 with exactly

, find the sum of the elements of 

. TST 2017

Problem 236 Find all positive integers 

 such that TST 2017

Problem 237 Let 

 be a cyclic quadrilateral inscribed in circle   with 

. Let  lines 

 and 

 be the projections of 

 respectively. Let 

 on the

 be the mid-points of sides 

 respectively. (a) Prove that 

 are concyclic.

(b) If   is the radius of   and   is the distance between its center and  , then find the radius of the circle in (a) in terms of   and  . TST 2017

166

Problem 238 In an acute triangle  . Let 

, points 

 and 

 lie on side 

 with 

 be the circumcenters of triangles  , respectively. Prove that 

 are con-cyclic if and only if 

 are collinear. TST 2017

Problem 239 Let 

 be pairwise distinct positive integers such that is an integer. Prove that 

 is not a

prime number.” TST 2017

Problem 240 There are   lamps   arranged in a circle in that order. At any given time, each lamp is either on or off. Every second, each lamp undergoes a change according to the following rule: (a) For each lamp  , if   have the same state in the previous second, then   is off right now. (Indices taken mod  .) (b) Otherwise, 

 is on right now.

Initially, all the lamps are off, except for   which is on. Prove that for infinitely many integers   all the lamps will be off eventually, after a finite amount of time. TST 2017

167

Problem 241 Let 

 be distinct positive real numbers with 

. Prove that

TST 2017

Problem 242 Define a sequence of integers   for all 

 and 

. Suppose 

 is a prime with 

. Prove that it is possible to choose 

 such that 

 for any 

. TST 2017

Problem 243 Let   be a positive integer. An   matrix is called good if each entry is a non-negative integer, the sum of entries in each row and each column is equal. A permutation matrix is an   matrix consisting of   ones and   zeroes such that each row and each column has exactly one non-zero entry. Prove that any good matrix is a sum of finitely many permutation matrices. TST 2017

Problem 244 Suppose 

 are non constant polynomials. Suppose neither of 

 is the square of a real polynomial but  square of a real polynomial.

 is. Prove that 

 is not the TST 2017

168

Problem 245 Let   be a positive integer relatively prime to  . We paint the vertices of a regular  -gon with three colors so that there is an odd number of vertices of each color. Show that there exists an isosceles triangle whose three vertices are of different colors. TST 2017

Problem 246 Let   and  plane. A nonempty, bounded subset   there is a point  entirely in  ; and  for any triangle   of the indices  similar.

 in 

 be fixed points on the coordinate  of the plane is said to be nice if

 such that for every point 

 in  , the segment 

, there exists a unique point   for which triangles 

points in  .

, then the product 

 and 

 and a permutation 

 and 

Prove that there exist two distinct nice subsets   and   such that if 

 in 

 lies

 are

 of the set   are the unique choices of

 is a constant independent of the triangle  TST 2017

Problem 247 Find all positive integers   for which all positive divisors of   can be put into the cells of a rectangular table under the following constraints: each cell contains a distinct divisor; the sums of all rows are equal; and the sums of all columns are equal. TST 2017

169

Problem 248 Let  incenter. The line   meets   at  of triangle  .

 be a triangle with   and let   be its  meets   at  , and the line through   perpendicular to  . Prove that the reflection of   in   lies on the circumcircle TST 2017

Problem 249 Prove that for any positive integers   and   we have

TST 2017

Problem 250 Let 

 be an acute angled triangle with incenter  . Line

perpendicular to 

 at   meets 

 be the incenters of   lie on a circle. Prove that 

 and   and  .

 at points 

 and 

 respectively. Let 

 respectively. Suppose  TST 2017

Problem 251 For each 

 define the polynomial Prove that

(a) For each  (b) 

, 

 has a unique positive real root 

;

 is a strictly increasing sequence;

(c)  TST 2017

170

Problem 252 Let   be a positive integer which is not a perfect square, and consider the equation

Let 

equation admits a solution in 

 be the set of positive integers   for which the  with 

, and let 

integers for which the equation admits a solution in  that  .

 be the set of positive

 with 

. Show TST 2017

Problem 253 Let 

 be an acute triangle. 

 are the touch points of

incircle with 

 respectively. 

 intersect incircle at 

 respectively. If, prove that 

Then

. Also prove that there exists integers 

, 

 such that, 

. TST 2018

Problem 254 A  divisible by 

Problem 255 Let 

 digit number is called interesting if its digits are distinct and is . Then find the number of interesting numbers. TST 2018

 be sequences of positive reals such that, for all 

Prove that, 

. . TST 2018

171

Problem 256 Let   be a convex quadrilateral inscribed in a circle with center   which does not lie on either diagonal. If the circumcentre of triangle   lies on the line  , prove that the circumcentre of triangle   lies on the line  . TST 2018

Problem 257 For an integer 

 find all 

 so that

(a) 

(b) 

(c)  TST 2018

Problem 258 A convex polygon has the property that its vertices are coloured by three colors, each color occurring at least once and any two adjacent vertices having different colors. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the interior of the polygon, in such a way that all the resulting triangles have vertices of all three colors. TST 2018

Problem 259 A rectangle   with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of   are either all odd or all even. TST 2018

172

Problem 260 Let   be three points in that order on a line   in the plane, and suppose  . Draw semicircles   and   respectively with   and   as diameters, both on the same side of  . Let the common tangent to   and   touch them respectively at  segment   and   in 

 and 

, 

 such that the semicircle  .

Prove that 

. Let 

 and 

 with 

 be points on the

 as diameter touches 

 in 

 are concyclic.

Prove that 

 are concyclic. TST 2018

Problem 261 Find all functions 

 such that for all 

. TST 2018

Problem 262 For a natural number 

, define 

 to be the set of all triplets 

 of natural numbers, with   odd and   and   divides  . Find all values of   for which 

Problem 263 In triangle 

, let   be the excircle opposite to 

 be the points where   is tangent to   intersects line  the circle 

, such that   is finite. TST 2018

 at 

 and 

, and  . Let 

. Let 

 and 

, respectively. The circle 

 be the midpoint of 

. Prove that

 is tangent to  . TST 2018

173

Problem 264 Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: Choose any number of the form  , where   is a non-negative integer, and put it into an empty cell. Choose two (not necessarily adjacent) cells with the same number in them; denote that number by  . Replace the number in one of the cells with   and erase the number in the other cell. At the end of the game, one cell contains  , where   is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of  . TST 2018

Problem 265 Let   be a positive integer. Define a chameleon to be any sequence of   letters, with exactly   occurrences of each of the letters   and  . Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon   , there exists a chameleon   such that   cannot be changed to   using fewer than 

 swaps. TST 2018

Problem 266 Let 

 be a finite set, and let 

. Let   be an element of 

, and let 

that 

 for every   in 

 be the set of all functions from   to   be the image of   with 

 under  . Suppose

. Show that 

. TST 2018

174

Problem 267 Find the smallest positive integer   or show no such   exists, with the following property: there are infinitely many distinct  -tuples of positive rational numbers 

 such that both are integers. TST 2018

Problem 268 Let 

 be a triangle and 

point  . Suppose each of the quadrilaterals  both circumcircle and incircle. Prove that  the center of the triangle.

 be cevians concurrent at a  and   has  is equilateral and   coincides with TST 2018

Problem 269 Let  numbers such that 

 be a natural number. Let   and 

 be real  If 

smallest integer larger than 

, the

, then show that  TST 2018

Problem 270 Determine all integers   having the following property: for any integers   whose sum is not divisible by  , there exists an index   such that none of the numbers is divisible by  . Here, we let   when  . TST 2018

175

Problem 271 In an acute angled triangle   with  , let   denote the incenter and   the midpoint of side  . The line through   perpendicular to   intersects the tangent from   to the incircle (different from line  ) at a point  > Show that   is tangent to the circumcircle of triangle  . TST 2019

Problem 272 Show that there do not exist natural numbers  that the numbers powers of 

 such are all TST 2019

Problem 273 Let   be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of   squares in a row, numbered   to   from left to right. Initially,   stones are put into square  , and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with   stones, takes one of these stones and moves it to the right by at most   squares (the stone should say within the board). Sisyphus' aim is to move all   stones to square  . Prove that Sisyphus cannot reach the aim in less than turns. (As usual, 

 stands for the least integer

not smaller than  . ) TST 2019

Problem 274 Let  functions 

 denote the set of all positive rational numbers. Determine all  satisfying

for all  TST 2019

176

Problem 275 Let   be a natural number. A tiling of a   board is a placing of   dominos (of size   or  ) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two separate tilings of a   board: one with red dominos and the other with blue dominos. We say two squares are red neighbors if they are covered by the same red domino in the red tiling; similarly define blue neighbors. Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on its red and blue neighbors i.e the number on its red neighbor minus the number on its blue neighbor. Show that   is divisible by  TST 2019

Problem 276 Let 

 be a function such that   for all pairs 

exists a positive integer 

 of positive integers. Prove that there

 which divides all values of  . TST 2019

Problem 277 Given any set   of positive integers, show that at least one of the following two assertions holds:

(1) There exist distinct finite subsets  ;

 and 

(2) There exists a positive rational number  finite subsets   of  .

 of   such that 

 such that 

 for all TST 2019

177

Problem 278 Let  circumcenter 

 be an acute-angled scalene triangle with circumcircle   and . Suppose 

incenter of triangle  of triangle  Let 

. Let 

, containing 

 be a point on the arc  . Let 

lines 

. Let 

 and 

 be the orthocenter and   be the

 be the midpoint of the arc 

 of the circumcircle

.  of   not containing 

, such that 

 be the circumcenter of triangle 

. Prove that the

 meet on  . TST 2019

Problem 279 Let   be a positive integer. The organizing committee of a tennis tournament is to schedule the matches for   players so that every two players play once, each day exactly one match is played, and each player arrives at the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay   coin to the hotel. The organizers want to design the schedule so as to minimize the total cost of all players' stays. Determine this minimum cost. TST 2019

Problem 280 Determine all non-constant monic polynomials   with integer coefficients for which there exists a natural number   such that for all 

, 

 divides  TST 2019

178

Problem 281 Determine all functions 

 satisfying for all 

. TST 2019

Problem 282 Let   be the circumcentre, and   be the circumcircle of an acute-angled triangle  . Let   be an arbitrary point on  , distinct from  ,  ,  , and their antipodes in  . Denote the circumcenter of the triangles  ,  , and   by  ,  , and  , respectively. The lines  ,  ,   perpendicular to  ,  , and   pass through  ,  , and  , respectively. Prove that the circumcircle of triangle formed by  ,  , and   is tangent to the line  . TST 2019

Problem 283 Let 

 be a set of 

 distinct positive even numbers and 

 be a set of   distinct positive odd numbers such that Prove that TST 2019

Problem 284 Let 

 be a triangle with 

chosen on the sides   respectively so that   Let   and   be the perimeters of the triangles   and  Prove that 

 Points 

 are

, respectively. TST 2019

179

Problem 285 Let 

 be an integer. Solve in reals: TST 2019

Problem 286 Let the points   and   be the circumcenter and orthocenter of an acute angled triangle   Let   be the midpoint of   Let   be the point on the angle bisector of   such that   Let   be the point such that   is a rectangle. Prove that 

 are collinear. TST 2019

Problem 287 Determine all positive integers   satisfying the condition that there exists a unique positive integer   such that there exists a rectangle which can be decomposed into   congruent squares and can also be decomposed into   congruent squares. TST 2019

Problem 288 There are   coins on a table. Some are placed head up and others tail up. A group of   persons perform the following operations: the first person chooses any one coin and then turns it over, the second person chooses any two coins and turns them over and so on and the  -th person turns over all the coins. Prove that no matter which sides the coins are up initially, the   persons can come up with a procedure for turning the coins such that all the coins have the same side up at the end of the operations. TST 2019

180

Solutions for Solved Problems (Page 10 - 19)

Solutions Solution. 1. Note that g(1) = f (1) = 0, so 1 is a root of both f ( x ) and g( x ). Let p and q be the other two roots of f(x), so p2 and q2 are the other two roots of g( x ). We then get pq = −c andp2 .q2 = − a, so a = −c2 . Also, (− a)2 = ( p + q + 1)2 = p2 + q2 + 1 + 2( pq + p + q) = −b + 2b = b.Therefore b = c4 . Since f (1) = 0we therefore get 1 + c − c2 + c4 = 0 . Factorising, we get (c + 1)(c3 − c2 + 1) = 0. Note that c3 − c2 + 1 = 0 has no integer root and hence c = −1, b = 1, a = −1. Therefore a2023 + b2023 + c2023 = −1

Solution. 2. denote Sn = αn + βn + γn also ,Sn ∈ Z+ claim:- S10,000+n+k = Sk now from netwon sums clearly Sn can be recursively written in terms of Sn−1 + Sn−2 + · · · + S1 α35005 + β35005 + γ35005 ≡ S35005 ≡ S5005 ( mod 10000) S13 ≡ Sn ( mod 16) S5 ≡ Sn ( mod 625) by CRT Sn ≡ −7( mod 625) Sn ≡ 9( mod 16) Sn ≡ −7( mod 10, 000) Sn ≡ 9993( mod 10, 000), where n = 35005 so S35005 ≡ 9993( mod 10, 000)

Solution. 3. Let D = (0, 0), A = (0, a), B = (−b, 0), C = (c, 0). If you construct squares circumscribing ABWX and CAYZ whose sides are parallel to the axes, it becomes clear that: W X Y Z

= (−( a + b), b) = (− a, a + b) = ( a, a + c) = ( a + c, c)

So we need to show that M and P lie on the y-axis. Since X and Y are equidistant from the y-axis, M lies on the y-axis. The equations for lines CW and BZ are: b ( x − c) a+b+c c y= ( x + b) a+b+c   We now see that the intersection, P, is 0, a+bcb+c , so we’re done. y=−

182

Solutions

√

Solution. 4. 2x + 1 = 4x + 2x+1 + 1 which can be written in another form as p √ x + 4x +1 + 2x +2 + 1, now this can be further expanded so on to get g ( x ) = 4 q p √ 4x + 4x+1 + 4x+2 + · · · so we have g( x ) = 2x + 1. so g(2021) = 22021 + 1 which is 8609( mod 10000) so we get 23 as our ans

Solution. 5. as we are given xyzw = 1 , so from this we get second equation as y z

+ yx + wx + wz = 2. so say a = yz , b = yx , wx = c, wz = d. so we get a + b + c + d = 2. from fourth equation we get 1a + 1b + 1c + 1d = −1. so we get abc + abd + acd + bcd = −1. also from third equation we get ab + bc + cd + ad + w2 y2 + x2 z2 = −3. notice we want ac and bd. so ac = x21z2 . so this gives ab + bc + cd + ad + ac + bd = −3. and abcd = 1. so we get a equation α4 − 2α3 − 3α2 + α + 1 = 0 whose roots are a, b, c, d. so we get (α + 1)(α3 − 3α2 + 1) = 0. this gives α = −1. and three distinct complex ( not necessarily non real) solutions. so as α = −1. we get any one pair say yx = −1. so x = −y = k for some k ∈ C. so as z, w, will be distinct we will get 4 quadruples from −k, k, w, z solution so we can have such 4 · 4 = 16 quadruples.

√ Solution. 6. so we construct a triangle ABC such that ∠ ACB = π2 and AB = 3, BC = √ 2, AC = 1. now choose a point P inside the triangle such√that ∠ APC = ∠ APB√= 3 2 ∠ BPC = 2π 3 . so we get [ APC ] + [ APB ] + [ BPC ] = [ ABC ]. so 4 · ( xy + yz + zx ) = 2 . √ 2√ 2 . 3

now adding the given three equations we get x2 + p √ y2 + z2 = 9−3 6 . and hence we get x + y + z = 3 + 6. now we are provided that p √ √ x2 + xz√+ z2 = 3. so we get ( x + z)2 − xz = 3 or y2 + 3 + 6 − 2y 3 + 6 − zx = 3 2√ and 9−3 6 − y2 + zx = 3. adding these two equations we get y2 = . solving on so we get xy + yz + zx = √

we get

y2

=

√ 6−2 6 9 .

3(3+ 6)

so we get m + n + p + q = 23 .

183

Solutions p Solution. 7. Squaring the given equation, we arrive at the conclusion that 2p q2 + r √ p p is an integer, or therefore q2 + r is an integer. Let q2 + r = u and s2 + t = v. Therefore, we arrive at u2 − q2 = r and v2 − s2 = t. This gives us u − q = 1 from difference of squares and r being prime, and similarly v − s = 1. Therefore, we get u + q = r and v + s = t. This gives us the following four vital equations which we will constantly use throughout the solution: 2q + 1 = r (1) 2s + 1 = t(2) u = q + 1(3) v = s + 1(4) √ p Go back to the original equation now. Since q2 + r = u = q + 1 and s2 + t = v = s + 1, plugging this back into the original equation, we arrive at p + (q + 1) = (s + 1) =⇒ p+q = s . Taking the equation modulo 2, we arrive at p = 2, q = 2, or , s = 2. Assume p p = 2, which p gives us s = q + 2. Going back to the original equation, 2 2+r = we have 2 + q (q + 2)2 + t. Therefore, p psquaring this, we get 4 + (q + r ) + 4 q2 + r = q2 + 4q + 4 + t. Therefore, r + 4 q2 + r = 4q + t =⇒ (3) =⇒ r + 4(q + 1) = 4q + t =⇒ r + 4 = t. Therefore, t = 2q + 5 using (1). Since t is prime, 2q + 5 is prime, since s is prime q + 2 is prime, and q is prime, and since r is prime 2q + 1 is prime. Therefore, we have an obvious contradiction mod 3 unless one of the values is 3. The only value that gives q being prime is 3, which gives ( p, q, r, s, t) = (2, 3, 7, 5, p11). √ If q = 2, then from q2 + r = q + 1 we have 4 + r = 3 =⇒ r = 5. Plugging this back into the original equation, we obtain p + 3 = s + 1 =⇒ p + 2 = s. We also have t = 2s + 1 from (2), so therefore from p being prime, we must have s − 2 being prime, from t being prime we must have 2s + 1 being prime, and s is prime. This is again a contradiction mod 3 unless one of the values is 3. If s = 3, then we get p = 1 absurd. If s − 2 = 3, then s = 5 and p = 3 and t = 11. There are no other solutions for this case. Therefore, we arrive at the solution ( p, q, r, s, t)√ = (3, 2, 5, 5, 11). 2 Lastly, p we have to consider s = 2. Since s + t = s + 1 = 3, we must have p + q2 + r = 3, which is clearly absurd. The solutions are whence ( p, q, r, s, t) = (2, 3, 7, 5, 11), (3, 2, 5, 5, 11) .

Solution. 8. Let P( x, y) be the assertion that f ( x f (y) + f ( x )) = 2 f ( x ) + xy. P(1, y) shows that f is surjective. Hence there exists z such that f (z) = 0. P(z, z) implies f (0) = z2 . Assume z = 0. Then P( x, 0) implies f ( f ( x )) = 2 f ( x ) and because of surjectivity we get f ( x ) = 2x for all x but this does not satisfy the original equation. Hence z ̸= 0. Now, P(z, y) implies f (z f (y)) = zy and therefore f must also be 184

Solutions injective hence bijective. P(z, 0) implies f (z3 ) = 0 = f (z) and because of injectivity we get z = z3 and thus z = ±1. In any case f (0) = z2 = 1. Now, P(−1, −1) implies f (−1) = 0 and thus z = −1. Now, P( x, −1) yields f ( f ( x )) = 2 f ( x ) − x meaning we can prove easily by induction f (n) = n + 1 for every positive integer n. Also, P(−1, y) yields f (− f ( x )) = − x so that we can extend this proof to the negative integers setting x = n. Assume f ( a) = 2a for some a. Then P( a, −4) implies f (− a) = 0 and thus a = 1 meaning this a is unique. Now, P( x, −2) implies f ( f ( x ) − x ) = 2( f ( x ) − x ) and therefore f ( x ) − x = a = 1 and hence f ( x ) = x + 1 for every real x which is indeed a solution.

Solution. 9. The question is asking, how many permutations of this set have no orbit with length greater than 2? There can be 1, 3, 5, or 7 orbits of length 1. One orbit of length 1: pick a number, then count how many pairings are possible. 7 · 5 · 3 · 1 = 105 Three orbits of length 1: choose three, then count the pairings. (73) · 3 · 1 = 105 Five orbits of length 1: choose five, then just one pairing is possible. (75) · 1 = 21 Seven orbits of length 1: that is, f ( a) = a 1 105 + 105 + 21 + 1 = 232

Solution. 10. Lemma 1: Let

jp

q



f n ( x ) ̸=

q



f n+1 ( x ) =⇒

q



f n +1 ( x ) =

q

f n +2 ( x )



k f n ( x ) = a.

a2 ≤ f n ( x ) ≤ a2 + 2a and a2 + 2a + 1 ≤ f n+1 ( x ). a2 + j2a + 1 ≤ kf n+1 ( x ) ≤ a2 + 3a + 1 = a2 + 3a + 1 < ( a + 2)2 p =⇒ f n ( x ) = a + 1. Calculating f n+2 ( x ) for the largest value of f n+1 ( x ): 2 2 f n+j2 ( x ) ≤ a2 +k 3a + 1+1+a+ jp k 1 = a + 4a + 3 < ( a + 2) p f n +1 ( x ) = f n+2 ( x ) = a + 1. ■ So q  q  q  q  Lemma 2: f n (x) = f n+1 ( x ) = a =⇒ f n +2 ( x ) = f n +3 ( x ) = a + 1 a2 ≤ f n ( x ) < f n+1 ( x ) ≤ a2 + 2a. Calculating f n+2 ( x ) for the smallest value of f n ( x ): a2 + a + 1 ≤ f n+1 ( x ) and f n+2 ( x ) ≥ a2 + a + 1 + 1 + a = a2 + 2a + 2 > ( a + 1)2 . 185

Solutions Calculating f n+2 ( x ) for the largest value of f n+1 ( x ): f n+2 ( x ) ≤ a2 + 2a + 1 + a = a2j+ 3a + 1 0, α ∈ R such that z = ρ(cos α + i sin α).  α α √ The equation w3 = z has the roots: w0 = 3 ρ cos + i sin ; 3     3 α + 2π α + 4π α + 2π α + 4π √ √ 3 3 w1 = ρ cos + i sin ; w2 = ρ cos + i sin . Hence: 3 3 3 3 {z1 + z2 , z1 + z3 , z2 + z3 } ⊂ {w0 , w1 , w2 }. Using the property P results: {z1 + z2 , z1 + z3 , z2 + z3 } = {w0 , w1 , w2 }. WLOG, we can consider: z1 +z2 = w0; z1 + z3= w1 ; z 2+ √ α α √ + π − i sin +π ; z3 = w2 . Results: z1 − z2 = w1 − w2 = 3 ρ · 3 sin 3  3 α+π α+π √ √ z2 − z3 = w0 − w1 = 3 ρ · 3 sin − i cos ; z 3 − z 1 = w2 − w0 = 3 3   √ α + 2π α + 2π √ √ √ 3 ρ· 3 sin − i cos . | z1 − z2 | = | z2 − z3 | = | z3 − z1 | = 3 ρ · 3 3 3

Solution. 15. First note that

R π/2 0

sin(sin x )dx =

R1 0

√sin x dx. 1− x 2

Now we also know that

187

Solutions x−

x3 3!

≤ sin x ≤ x for all x ∈ R. So 3 Z 1 x − x3!

√

0

=⇒

Z 1

√

0

x 1 − x2

1−

x2

dx −

dx ≤

1 3!

Z 1

Z 1 0

√

0

=⇒ 1 − =⇒

sin x √ dx ≤ 1 − x2

x3 1 − x2

1 ≤ 9

8 ≤ 9

dx ≤

Z 1 0

Z π/2 0

Z 1 0

Z 1 0

√

x 1 − x2

dx

sin x √ dx ≤ 1 − x2

Z 1

√

0

x 1 − x2

dx

sin x √ dx ≤ 1 1 − x2

sin(sin x )dx ≤ 1

Hence, proved.

Solution. 16. Let L := limx→∞ x f ( x ). So there exists N > 1 such that | x f ( x ) − L| ≤ 1 for x ≥ N. Now write Z ∞ f (x)

x

1

dx =

Z N f (x)

x

1

Z ∞ f (x)

dx +

N

x

(1)

dx.

The first integral on the right-hand side of (1) is a proper integral and so convergent. For the second integral, write

| x f ( x ) − L| | L| 1 + | L| | f ( x )| | x f ( x )| ≤ + 2 ≤ . (2) = 2 2 x x x x x2 R ∞ f (x) dx is convergent, 2 N x dx is (absolutely) convergent too, by the comx

R∞ So, since N parison test. Next, put x t = y. Then t

Z a 1

f ( x t ) dx =

and so lim t

t→∞

Z a 1

Z at Z at Z at f (y) 1/t f (y) 1/t f (y) y dy = (y − 1) dy + dy 1

y

1

t

f ( x ) dx = lim

Z at f (y)

t→∞ 1

y

y

(y

1/t

y

1

− 1) dy +

Z ∞ f (y) 1

y

dy.

So we are done if we show that lim

Z at f (y)

t→∞ 1

y

(y1/t − 1) dy = 0.

(3)

To prove (3), we use (2) to write 0≤

Z at | f (y)| 1

y

(y1/t − 1) dy ≤

Z N | f (y)| 1

y

(y1/t − 1) dy + (1 + | L|)

Z at 1/t y −1 N

y2

dy. 188

Solutions So if M := maxy∈[1,N ] | f (y)|, then 0≤

Z at | f (y)| 1

y

(y1/t − 1) dy ≤ M

Z N 1/t y −1

y

1

dy + (1 + | L|)

Z at 1/t y −1 N

y2

dy.

An easy calculation shows that lim

Z N 1/t y −1

t→∞ 1

y

dy = lim

Z at 1/t y −1

t→∞ N

y2

dy = 0

and that completes the proof of (3) and the first part of your problem. For the second part of your problem, apply the first part to the function f ( x ) = which satisfies all the conditions required, to get lim t

t→∞

Z a 1

dx = 1 + xt

Z ∞ 1

1 1+ x ,

dx = ln 2. x (1 + x )

Solution. 17. Note that since RHS ∈ Z, then all solutions must be integers.   Let n = 2 and the equation is x = n2 and so : x = 2k implies 2k = k and so k = 0 and so x = 0 which indeed is a solution. x = 2k + 1 implies 2k + 1 = k and so k = −1 and so x = −1 which indeed is a solution.   SoA2 = {−1, 0} Let n = 3 and the equation is x = n2 + n3 and so : x = 6k implies 6k = 5k and so k = 0 and so x = 0 which indeed is a solution. x = 6k + 1 implies 6k + 1 = 5k and so k = −1 and so x = −5 which indeed is a solution. x = 6k + 2 implies 6k + 2 = 5k + 1 and so k = −1 and so x = −4 which indeed is a solution. x = 6k + 3 implies 6k + 3 = 5k + 2 and so k = −1 and so x = −3 which indeed is a solution. x = 6k + 4 implies 6k + 4 = 5k + 3 and so k = −1 and so x = −2 which indeed is a solution. x = 6k + 5 implies 6k + 5 = 5k + 3 and so k = −2 and so x = −7 which indeed is a solution. So A2 = {−7, −5, −4, −3, −2, 0} And so A2 ∪ A3 = {−7, −5, −4, −3, −2, −1, 0}

Solution. 18. Substitute x := y + 1 to the preposition to get | f (y + 1) − f (y)| ≤

|y + 1 − y| = 1. Because f is injective, | f (y + 1) − f (y)| = 1. Now set f (0) = a. Because | f (1) − f (0)| = 1, either f (1) = a + 1 or f (1) = a − 1. Suppose f (1) = a + 1. We induct that f (n) = a + n. For n = 0, 1, this is correct. Suppose for n = k, k + 1, this claim is correct, then: | f (k + 2) − f (k + 1)| = 1 f (k + 2) = f (k + 1) + 1 ∨ f (k + 1) − 1 f (k + 2) = a + k + 189

Solutions 2 ∨ a + k But f (k ) = a + k, so since f is injective, f (k + 2) = a + k + 2, and the claim is proven for n ≥ 0. For n = 1, 0, this is correct. Suppose for n = k + 1, k, this claim is correct, then: | f (k − 1) − f (k)| = 1 f (k − 1) = f (k) + 1 ∨ f (k) − 1 f (k − 1) = a + k + 1 ∨ a + k − 1 But f (k + 1) = a + k + 1, so f (k − 1) = a + k − 1, and the claim is proven for n ≤ 0. Hence f (n) = a + n for all integer n. The same argument can be applied to where f (0) = a, f (1) = a − 1 to get f (n) = a − n. Hence we have our solutions: f ( x ) = c + x for all x, or f ( x ) = c − x for all x, for any c.

Solution. 19. Let P( a, b) be the assertion a2 + ab + b2


Rb

f ( x ) dx = 3

a

t ∈ R, we have P(0, t) =⇒ t

2

Zt

f ( x ) dx = 3

0

Zt

Rb

x2 f ( x ) dx, and

a

x2 f ( x ) dx

0

since f is continuous here, we know that f ′ exists, and differentiating with respect to t on both sides gives 2

t f (t) + 2t

Zt

2

f ( x ) dx = 3t f (t) =⇒

0

Zt

f ( x ) dx = t f (t)

0

and differentiating with respect to t again gives f (t) = f (t) + t f ′ ( x ) =⇒ f ′ (t) = 0 and so f is a constant function. It is easy to check that f ( x ) = C where C ∈ R works. ■

Rt Rt Rt Solution. 20. t = 0 dx ≤ 0 e x dx ≤ 0 edx = et ⇒ 1 + t ≤ et ≤ 1 + et, (∀) t ∈ [0, 1] . R1 R 1 xn R1 1 e n n ⇒ 1 + n+ 1 = 0 (1 + x ) dx ≤ 0 e dx ≤ 0 (1 + ex ) dx = 1 + n+1 , (∀) n ∈ N R1 n ⇒ lim 0 e x dx = 1. n→∞

190

Solutions Solution. 21. Lemma: If P, Q, R are three 2 × 2 matrices, then det( P + Q + R) = det( P + Q) + det( Q + R) + det( P + R) − det( P) − det( Q) − det( R). Apply the above lemma to get that det( A + 2B) − det( B + 2A) = det(2B) − det(2A) − (det( B) − det( A)). Now use that det( A) = 0 and det( B) = 2 to get the answer as 6.

Solution. 22. Since 346346 ≡ 4 (mod 9), we need at least four perfect cubes. On the other hand, (7 · 346115 )3 + (346115 )3 + (346115 )3 + (346115 )3 = (73 + 13 + 13 + 13 ) · 346345 = 346346 . So the minimum number is 4 .

Solution. 23. Suppose , that x = c is a integer solution to this equation . Then, 2abc4 − a2 c2 − b2 − 1 = 0 =⇒ ac2 (2bc2 − a) = b2 + 1 . Note that b2 + 1 prime factors are either 2 or of form 4k + 1. Also we see that if , a is even then , we get 4|b2 + 1 a contradiction . similarly we see that c is also odd. And indeed if b is odd , then we again get a similar contradiction. Now , if a ≡ 3 (mod 4) then , we it means there is prime factor of a which is of form 4k + 3 a condtradiction. Also c must also be of form 4k + 1 . And b is even . But then we have 2bc2 − a2 ≡ 3 (mod 4) a contradiction .

Solution. 24. ∠ AUC ≡ ∠ AXB ≡ ∠ DXC ≡ ∠ AVD ⇒ ∠ AUT + ∠CUT = ∠ AVT +

∠TVP ⇒ ∠CUT ≡ ∠TVA. Denote { R} = (VPU ) ∩ AV. Using the fact that ∠CUT ≡ ∠TVR, we get that R, C and U are colinear. RP AP RP CP Now AR · AV = AP · AU ⇒ △ ARP ∼ △ AUV ⇒ UV = AV , but UV = CU , AP CP thus AV = CU ⇒ △CPU ∼ △ ACV. Similarly we prove that △CRV ∼ △ ACU ⇒ AC AV △ ACV ∼ △ ACU ⇒ ∠VAC ≡ ∠CAU ⇒ AC = AU ⇒ AV = AU ⇒ A, C and M are colinear. ∠ ATP ≡ ∠ ACP ≡ ∠VCO ≡ ∠OCU, where {O} = AC ∩ UV ∠ MVT + ∠ MAT = ∠ MVT + π − ∠TAC = ∠VUT + π − ∠TPV = π, hence MVTA is cyclic⇒ ∠ MVA ≡ ∠ MTA 191

Solutions

∠ MVA = ∠VUR =

π 2

− ∠OCU ⇒ ∠ MTA + ∠ ATP =

π 2

⇒ ∠ PTM = π2 .

Solution. 25. We prove both parts. Define MB , MC as the arc midpoints of AC and AB, O as the circumcenter, and TB , TC as the B-extouch point with AC and the C-extouch point with AB. Part 1: If AB + AC = 3BC, then P, I, Q are collinear. Proof: Observe that s = 2BC, so BTC = s − BC = BC, and similarly CTB = BC. Therefore, we have 1 ∠QCA = ∠TC CB = 90 − B 2 This implies Q is the arc midpoint of ABC. Similarly, P is the arc midpoint of APB, so P, O, MC and Q, O, MB are collinear. Observe that MB MC is the perpendicular bisector of AI. Then, 1 BC 1 BC QMC = 2R sin A = sin A = 2 sin A 2 2 cos 12 A If QMC = 12 AI, we’re done. Observe that the length of the tangent from A to the incircle is s − BC = BC, so 1 1 r sin 12 A 1 r BC tan A = = ⇒ QMC = AI 2 BC 2 = 2 2 cos 12 A and we are done. Part 2: If P, I, Q are collinear, then AB + AC = 3BC. If we let SB , SC be the touch-points of ωC , ω B with AC, AB respectively, then by homothety, P, SC , MC and Q, SB , MB are collinear. Furthermore, it is well known that ( QISB A) and ( PISC A) are cyclic. If O′ = QMB ∩ PMC , then

∠ ASC O′ + ∠ ASB O′ = ∠ AIQ + ∠ AIP = 180 which means ( ASC O′ SB ) is cyclic. Therefore, ∠ MC O′ MB = 180 − ∠ A. However, since MC MB = 90 − 12 A, this means PQ = 90 − 12 A, so PQ = MC MB . Define S = CQ ∩ PB. By pascals on PBMB QCMC , we have I, S, O′ are collinear. Since S is the exsimilicenter of the incircle and circumcircle, we have I, O, S are collinear. Now, I claim the only way for O′ , O, I to be collinear is when O = O′ . First of all, this is possible by setting P, Q as the reflection of MC , MB over O. Next, for any other placement of P, Q is fixed (since PQ = MC MB . Inverting with the circumcenter, O′ goes to the intersection of MC MB and PQ, so the locus of O′ is ( MC OMB ). However, since MC MB ∩ PQ lies outside of MC MB , this means the locus of O′ must also be within ( ABC ), so the only intersection of ( MC OMB ) with OI is O. Therefore, O′ = O. Finally, this means 1 ∠ BCTC = ∠QCA = 90 − ∠ ACMB = 90 − B 2 192

Solutions so BTC = BC, and s − BC = BC so AB + AC = 3BC.

4 x −cos2 x

Solution. 26. Proof 2sin 4 x +sin2 x

4 x −sin2 x

− 2cos

4 x +cos2 x

4 x +1−cos2 x

= cos 2x ⇔ 2sin

4 x +sin2 x



4 x +1−sin2 x

− 2cos 4

2

=

4

2x

1 − 2cos x+cos x−sin x−sin

4 2 2 cos 2x 2sin x+sin x 1 − 22 cos 2x = 2 cos 2x ⇒ 2 cos 2x 1 − 22 cos 2x ≥ 0 ⇔ cos 2x =
 4 2 0 ⇒ 2sin x+sin x 1 − 22 cos 2x = 2 cos 2x ⇔ cos 2x = 0 ⇔ x ∈ ± π4 + πk, k ∈ Z 2 cos 2x ⇔ 2sin

− 2cos

= 2 cos 2x ⇔ 2sin

Solution. 27. Let f ( x ) = x2 − ( a2 + b2 + c2 + d2 + 1) x + ab + bc + cd + da, let roots of f be x = r1 , r2 . Since f ∈ Z[ X ] is monic and has an integer root, we see that it must necessarily have two integer roots. We see that r1 + r2 = a2 + b2 + c2 + d2 + 1 > 0 and r1 r2 = ab + bc + cd + da > 0, therefore r1 , r2 > 0. 1 Now, a2 + b2 + c2 + d2 = ab + bc + cd + da + [( a − b)2 + (b − c)2 + (c − d)2 + (d − 2 1 2 a) ] and so this re-arranges to (1 − r1 )(r2 + 1) = [( a − b)2 + (b − c)2 + (c − d)2 + 2 (d − a)2 ] ≥ 0 . . . (♣), however r1 , r2 ̸= 0 as ( a, b, c, d) ∈ N and so r1 , r2 ≥ 1, implying that (1 − r1 )(r2 − 1) ≤ 0 and using (♣), we have that (1 − r1 )(r2 − 1) = 0 or one of the roots r1 or r2 is 1, let r1 = 1, then a2 + b2 + c2 + d2 = r1 + r2 = r1 + 1 = ab + bc + cd + da + 1 =⇒ a2 + b2 + c2 + d2 = ab + bc + cd + da and so a = b = c = d must be true, implying that x = r1 or r2 = ab + bc + cd + da = 4a2 = (2a)2 or 1 = 12 which is as desired

Solution. 28. If x > 1 then LHS > RHS, if 0 < x < 1 then LHS < RHS. so x = 1 is the only solution

193



=

Solutions Solution. 29. a + b − c = 2 and 2ab − c2 = 4 =⇒ ( a + b − c)2 = 2ab − c2 =⇒ a2 + b2 + 2c2 − 2c( a + b) = 0 a2 + b2 + 2c(c − a − b) = 0 =⇒ 2ab − 4c ≤ a2 + b2 − 4c = 0 =⇒ 2ab − 4c ≤ 0 2ab = 4 + c2 =⇒ 2ab − 4c = 4 + c2 − 4c = (c − 2)2 ≤ 0. So c − 2 = 0 =⇒ c = 2. Therefore, a + b = 4 and 2ab = 8, ( a + b)2 = 16 and 4ab = 16 =⇒ ( a + b)2 = 4ab =⇒ a = b Then we deduce that a = b = c = 2 So △ ABC is an equilateral triangle

Solution. 30. Let ab = 1010 and suppose a and b have no zeroes in their decimal representation. One of a, b must be divisible by 2 and one must be divisible by 5, and neither can be divisible by both. Therefore a = 210 and b = 510 . But then a = 1024, which has a zero, contradiction.

Solution. 31.

n n −1 −1 ( n −1)2

=

(n−1)(nn−2 +nn−3 +···+1) ( n −1)2

=

nn−2 +nn−3 +···+1 . n −1

I now show nn−2 +

nn−3 + · · · + 1 ≡ 0 (mod n − 1). We know n ≡ 1 (mod n − 1), so nn−2 + nn−3 + · · · + 1 ≡ n − 1 ≡ 0 (mod n − 1)■

Solution. 32. The equality is obvious at x = 2 Now using a is the longest side, For x > 2 a x = a2 a x−2 = (b2 + c2 )( a x−2 ) > b x + c x

194

Solutions Solution. 33. (a) x n + x n+1 is strictly increasing and continuous on R+ .And x n + x n+1 goes from 0 to +∞ on R+ .Then by intermediate value theorem, ∃!xn > 0 s.t. xnn + xnn+1 = 1■ +1 n +2 (b) Obviously 0 < xn < 1.If xn+1 < xn ,then 1 = xnn + xnn+1 > xnn+ 1 + xn+1 which is absurd.Thus xn+1 ≥ xn .Then { xn } is increasing and upper-bounded.Therefore ∃α s.t. limn→+∞ xn = α.0 < α ≤ 1.If 0 < α < 1,then limn→+∞ ln xnn = limn→+∞ n ln xn = −∞.Hence limn→+∞ xnn = 0.And 0 < xnn+1 < xnn ,by squeeze theorem, limn→+∞ xnn+1 = 0.Thus limn→+∞ xnn + xnn+1 = 0 which is absurd.Therefore α = 1■


p Solution. 34. 1)Using power mean inequality (k p + (n − k ) p ) ≥ 2 n2 so 1 p + 2 p +
p ... + (n − 1) p + n p + (n − 1) p + ... + 2 p + 1 p ≥ 2n · n2 −→ ∞ as n −→ ∞ 2)For the second case it is easy to calculate

1 p +2 p +...+(n−1) p +n p +(n−1) p +...+2 p +1 p n2

=1

Solution. 35. ∠ DFE = ∠ DAE by angles in the same segment property. ∠ ADF = 90◦ − ∠ DAF =⇒ ∠ MDF = 90◦ − ∠OAF = B using MD || AO ∠OAF = 90◦ − B hence △ FMD ∼ △ ADB by angle equalities

n −r +1 r −1 n Solution. 36. Rewrite the identity as ∑∞ d=1 ( d )(r −d) = ( r ) (1). Arbitrarily split n

objects into a group of n − r + 1 (Group 1) and r − 1 (Group 2) objects. In order to choose r objects from the n total, we can either choose 1 object from Group 1 and r − 1 from Group 2 or 2 from Group 1 and r − 2 from Group 2, etc. Thusly we can count the choosing of r objects from n in two ways, proving the equality of the L.H.S. and R.H.S. of (1). Note that d cannot be 0 because we cannot choose r objects from Group 2, which only has r − 1 objects. Also when d > n − r + 1 or d − 1 > r − 1, the binomial product vanishes (the sum is equally valid from d = 1 to min({n − r + 1, r })).

195

Solutions 1

1

1

1

ln(2)

Solution. 37. let P = 2 2 · 4 4 · 8 8 · (2n ) 2n from here we get ln( P) = 2   n ln(2) 3 3 1 2 3 n ln ( 2 ) + · · · + ln ( 2 ) + · · · + ln ( P ) = ln ( 2 ) + + + · · · + 8 2n 2n 8   2 4 8   n +1 n +1 ln( P) = ln(2) −2 21 (n + 1) − 2 12 + 2 < 2 ln(2)

+

ln(2) 2

+

p p √ √ Solution. 38. Let N = 3 13 + x + 3 13 − x p√ p √ note that N > 3 x + 3 − x = 0 p √ √ p √ p √ 3 3 3 = (13 + cubing both sides yeilds:N x ) + 3 ( 13 + x )( 13 − x )( 3 13 + x + p √ √ 3 13 − x ) + (13 − x )

√ 3

N 3 − 26 169 − x = 3N

3

−26 3 x = 169 − ( N 3N ) >0 It is easy to see that if N ≥ 5,then x < 0 therefore,N ∈ {1, 2, 3, 4} put the values N to get x = 196 as the only integer for N = 2

Solution. 39. Let P( x, y) be the assertion f (max( x, y) + min( f ( x ), f (y))) = x + y Let a = f (0) P( x, x ) =⇒ f ( x + f ( x )) = 2x Let x > y. If f ( x ) ≤ f (y), then P( x, y) =⇒ f ( x + f ( x )) = x + y = 2x and so x = y, impossible. So x > y =⇒ f ( x ) > f (y) and P( x, y =⇒ : f ( x + f (y)) = x + y ∀ x > y (I) So f ( x ) = x + y − f (y) ∀ x > f (y) Setting there y = 0, we get f ( x ) = x − a ∀ x > a (II) Let then y ∈ R and x > max(y, a − f (y)) : x > y implies f ( x + f (y)) = x + y (see I above) x + f (y) > a implies f ( x + f (y)) = x + f (y) − a (see II above) And so f (y) = y + a ∀y Plugging this back in original equation, we get a = 0 and so f ( x ) = x ∀ x

196

Solutions Solution. 40. We have ( a + d)(b + c) = ( ab + cd) + ( ac + bd) = 9 and ( a + d) + (b + c) = 6. Therefore, we must have a + d = b + c = 3. Similarly, ( a + b)(c + d) = 5 and ( a + b) + (c + d) = 6. Therefore, { a + b, c + d} = {1, 5}. Also similarly, ( a + c)(b + d) = ( ab + cd) + ( ad + bc) = 8 and ( a + c) + (b + d) = 6. Therefore, we must have { a + c, b + d} = {2, 4}. Note that 2a = ( a + b) + ( a + c) − (b + c) ∈ {0, 2, 4, 6}. Therefore, a ∈ {0, 1, 2, 3}. However, each of these gives us a unique solution (because each value of a corresponds to a unique value of a + b and a + c), and therefore all the solutions are ( a, b, c, d) = (0, 1, 2, 3), (1, 0, 3, 2), (2, 3, 0, 1), (3, 2, 1, 0)

Solution. 41. .Given tan( x + y + z) =

sin 3x +sin 3y+sin 3z cos 3x +cos 3y+cos 3z ,

then sin( x + y + z)(cos 3x +

cos 3y + cos 3z) = cos( x + y + z)(sin 3x + sin 3y + sin 3z), sin(4x + y + z) + sin(−2x + y + z) + sin( x + 4y + z) + sin( x − 2y + z) + sin( x + y + 4z) + sin( x + y − 2z) = sin(4x + y + z) + sin(2x − y − z) + sin( x + 4y + z) + sin(− x + 2y − z) + sin( x + y + 4z) + sin(− x − y + 2z), simplifying: sin(2x − y − z) + sin(− x + 2y − z) + sin(2z − x − y) = 0, [sin(2x − y y − z) + sin(− x + 2y − z)] + sin 2(z − 2x − 2 ) = 0, y 3y y y y x x x 2 sin( 2x + 2 − z) cos( 3x 2 − 2 ) + 2 sin( z − 2 − 2 ) cos( z − 2 − 2 ) = 0, sin( 2 + 2 − 3y y x z)[cos( 3x 2 − 2 ) − cos( z − 2 − 2 )] = 0. Solutions: x + y = 2z or 2x = y + z R = 1 + 2 cos answer

3( x − y ) 2

which gives 4 as desired

Solution. 42. Let x = cos θ and y = sin θ. Then the second condition becomes 4 sin(θ ) cos(θ )cos(2θ ) = 1, which becomes 2 sin(2θ ) cos(2θ ) = 1, becomes √ which √
2 sin(4θ ) = 1. Thus, 4θ = π2 , so θ = π8 . Thus, x = cos π8 = 2+ , so the an2

2 2 swer is 2 + 2 = 4 S2 :16x2 y2 2x2 − 1 = 1 =⇒ 16x2 (1 − x2 ) 2x2 − 1 = 1 =⇒ 64x8 − 128x6 + 80x4 − 16x2 + 1 = 0
2 =⇒ 64x8√− 128x6 + 64x4 + 16x4 − 16x2 + 1 = 0 =⇒ 8x4 − 8x2 + 1 = 0 =⇒ √ −(−8)± (−8)2 −4·8·1 2± 2 2 x = = 4 2·8 √ √ 2 So the largest root is x = 2+ 2 And again, a + b = 2 + 2 = 4.

197

Solutions Solution. 43. We can rewrite f ( x ) as −( x − 2p)2 + 4p2 − p + 1. The area of this triangle is equal to half the base times the height. Taking the base as the side of the triangle on the x-axis, the base is the difference between the roots and the height is the y-coordinate of the vertex. The y-coordinate of the vertex is, from the above 4p2 − p + 1. Let this be q. √ form, p −4p± 16p2 −4p+4 From the quadratic formula, the roots are 2p ± 4p2 − p + 1, so = −2 p 2 − p + 1 = 2√ q. the base of the triangle, or the difference in the roots, is 2 4p   √ √ Therefore, the area of the triangle is 12 (q)(2 q) = q q. This is an integer if and only if q is the square of an integer. Now let 4p2 − p + 1√= x2 , so that 4p2 − p + 1 − x2 = 0. From the quadratic formula, √

2

1±

(4x )2 −15

−15 = . p = 1± 16x 8 8 2 2 Now let (4x ) − 15 = y , or (4x − y)(4x + y) = 15. Therefore, either 4x − y = 1, 4x + y = 15, or 4x − y = 3, 4x + y = 5. Both of these give integral values for x, so there are two integral values for x. Each integral value for x gives two rational values for p, so we have 4 rational values. Namely, p = − 43 , 0, 14 , 1.

Solution. 44. Using L’hospital’s rule and differentiation under integral sign, we have: R 1 lim A → ∞ A1 1 A A x dx = L’Hospital = 1 −1 1 R R 1 lim A → ∞A A + 1 A A xx dx = 1 + lim A → ∞ A1 1 A Axx dx = L’Hospital 1 1 −1 1 R R =1+ lim A → ∞ AAA + 1 A A xx2 dx = 1 + lim A → ∞ A1 1 A Ax2x dx = 1 + lim A→∞ A1 · 1

A− A A ln A

=1

Solution. 45. By adding all equations we get a3 + b3 + c3 + d3 = 0 which implies that at least one of them is positive and at least one of them is negative. Suppose a is positive. If b was positive this will imply c is positive which will imply d being positive which can’t happen. Then b is negative. From d3 + a = b we get that d is negative too. Since c3 = a − d we have that c must be positive. The other cases are the same thing, basically a and c share signs and b and d also share signs, opposite to the sign of a and c. Therefore, suppose a, c are positive and b, d are negative. For simplicity, we will set a = x 2 , b = − y2 , c = z2 , d = − w2 . 198

Solutions Our equations then become  6 x − y2 = z2     − y6 + z2 = − w2  z6 − w2 = x 2    − w6 + x 2 = − y2 Which after rearranging become  6 x = y2 + z2     y6 = z2 + w2  z6 = w2 + x 2    6 w = x 2 + y2 Clearly, x6 + z6 = y6 + w6 Suppose x2 > z2 . From this we have x6 > z6 → y2 + z2 > w2 + x2 → y2 > w2 → y6 > w6 However, x2 + y2 > z2 + w2 → w6 > y6 which is a contradiction. Similarly, asumming z2 > x2 will give us a contradiction in the same way. Therefore x 2 = z2 . Using the exact same method we can prove y2 = w2 Since x6 + z6 = y6 + w6 we have x6 = y6 I will switch back to a, b, c, d keeping in mind a = c , b = d and a3 = −b3 We have the two equations ( a3 + b = a b3 + a = b Since a3 = −b3 we must have a = −b √ √ Therefore we must solve a3 = 2a. equation √ The√solutions √ √to this √ √ are √ 0,√ 2, − 2. Hence ( a, b, c, d) = (0, 0, 0, 0), ( 2, − 2, 2, − 2), (− 2, 2, − 2, 2)

Solution. 46. Let ( x, y) := ( g(0), f ( g(0))): g( f ( g(0)) − f ( g(0))) = f ( g( f ( g(0)))) + g(0) f ( g( f ( g(0)))) = 0 Let ( x, y) := ( g( f ( g(0))), 0): g( f ( g( f ( g(0))))) = f ( g(0)) + g( f ( g(0))) (1) g(0) = f ( g(0)) + g( f ( g(0))) Let ( x, y) := ( g(0), 0): (2) g( f ( g(0))) = f ( g(0)) + g(0) From (1) and then (2) we have: g(0) = f ( g(0)) + g( f ( g(0))) = f ( g(0)) + f ( g(0)) + g(0) = 2 f ( g(0)) + g(0), so f ( g(0)) = 0. Let ( x, y) := (0, 0): g( f (0)) = f ( g(0)) = 0 Let y := 0: g( f ( x )) = f ( g(0)) + x = x Let y := f ( x ): g(0) = f ( g( f ( x ))) + x, but g( f ( x )) = x, so g(0) = f ( x ) + x, so f ( x ) = − x + g(0). We use f ( x ) = − x + g(0) in g( f ( x )) = x and we get: g(− x + g(0)) = x Let x := − x + g(0) in g(− x + g(0)) = x: g( x ) = − x + g(0) So f ( x ) = g( x ) = − x + g(0). Check: LHS = g( f ( x ) − y) = g(− x + g(0) − y) = x + y − g(0) + g(0) = x + y RHS = f ( g(y)) + x = f (−y + g(0)) + x = y − g(0) + g(0) + x = x + y. So functions f ( x ) = g( x ) = − x + c for any c are solutions of this equation. 199

Solutions

Solution. 47. Case I: x = y then ( x, y) = (50, 50), (1, 1). Case II: x < y then ( x, y) = (2, 225 ), (5, 510 ), (10, 105 ), (25, 252 ). These pairs are √ 50 x found by using equation y = x50 = x x . Clearly, x must be a divisor of 50. Case III: x > y then this means x50 > y50 so y x = x50 if x > 50. But from the 50 equation in Case II, if x > 50 then x x is not an integer so there is no solutions in this condition. ∴ The all pairs of solutions are (1, 1), (2, 225 ), (5, 510 ), (10, 105 ), (25, 252 ), (50, 50).

Solution. 48. .Let the midpoint of BC be M, let AM and BE meet at H, and let BE meet AC at N. Since AB = AC, we have that AM ⊥ BD and we already know that BE ⊥ AD, so H is the orthocenter of △ ABD, so DH ⊥ AB. Yet, AC ⊥ AB, so DH ∥ AC. Thus, HM MD MC − DC = = = AH DC DC

BC 2

− BC 3

BC 3

=

BC 6 BC 3

=

1 2

It follows that since H is on median AM, we have that H is the centroid of △ ABC. Thus, N is the midpoint of AC. Now, notice that ∠ AEN = 90 = ∠ BAN, so △ AEN ∼ EN 2 2 = AN △ BAN, which means that AN BN , so BN · EN = AN = CN . This gives that EN NC NC = BN , so △ ENC ∼ △CNB, so ∠ NCE = ∠ NBC. Furthermore, from similar triangles AEN and BAN, we have that ∠EAN = ∠ ABN, so

∠ AEC = 180 − ∠EAC − ∠ECA = 180 − ∠ ABN − ∠CBN = 180 − 45 = 135 from which we conclude that ∠ DEC = 180 − ∠ AEC = 45 .

Solution. 49. We start by noting that equality holds for {n, n + 1, n + 2}. This tells us that the inequality may be hard to prove, and it’s logical to try our hand at a simpler analogue. We may construct a 2-variable version, where equality holds for {n, n + 1}. Namely, this is    1 1 a− b− ≤ ab − 2. (1) a b 200

Solutions Proof. After multiplying by ab and expanding, we arrive at a2 b2 − a2 − b2 + 1 ≤ a2 b2 − 2ab, which rearranges to 1 ≤ ( a − b)2 , which is true. Equality holds only for | a − b| = 1. ■ Using this inequality, we may easily establish the problem’s claim:       1 1 1 1 a− b− c− ≤ a− (bc − 2) ≤ abc − ( a + b + c), a b c a where the last estimate is equivalent to abc − 2a −

bc 2 + ≤ abc − ( a + b + c), a a

bc ( a − b)( a − c) 2 ≤ a−b−c+ = , a a a which is clearly true if ( a − b)( a − c) ≥ 2. This holds if a = max{ a, b, c}, which we may assume WLOG. ■

Solution. 50. WLOG a ≥ b and c ≥ d and b ≥ d abcd = 4( a + b)(c + d) ≤ 4( a + a)(c + c) = 16ac ⇒ bd ≤ 16 ⇒ d2 ≤ bd ≤ 16 ⇒ d ≤ 4 2(c + d) = ab ⇒ 2c = ab − 2d 2( a + b) = cd ⇒ 4( a + b) = 2cd ⇒ 4( a + b) = ( ab − 2d)d ⇒ abd2 − 4ad − 4bd − 2d3 = 0 ⇒ ( ad − 4)(bd − 4) = 2d3 + 16 and obviously we can put d = 1, 2, 3, 4 in the last equality,and find solutions... d = 1: ( a − 4)(b − 4) = 18 ⇒ b = 5, a = 22, c = 54 or b = 6, a = 13, c = 38 or b = 7, a = 10, c = 34 d = 2: (2a − 4)(2b − 4) = 32 ⇒ ( a − 2)(b − 2) = 8 ⇒ b = 3, a = 10, c = 13 or b = 4, a = 6, c = 10 d = 3: (3a − 4)(3b − 4) = 70 ⇒ b = 3, a = 6, c = 6 d = 4: (4a − 4)(4b − 4) = 144 ⇒ ( a − 1)(b − 1) = 9 ⇒ b = 4, a = 4, c = 4 therefore all solutions are: {{ a, b}, {c, d}} = {{22, 5}, {54, 1}}, {{13, 6}, {38, 1}}, {{7, 10}, {34, 1}} , {{10, 3}, {13, 2}}, {{6, 4}, {10, 2}}, {{6, 3}, {6, 3}}, {{4, 4}, {4, 4}}

201

About the Authors Pranav M. Sawant Pranav Milind Sawant is a Grade 12 student from Army Public School, Pune. He has been studying Olympiad mathematics for the past four years and has done exceedingly well, alongside consistently bagging single-digit ranks, in several prestigious Olympiads such as the IOQM (Indian Olympiad Qualifier in Mathematics), INMO (Indian National Mathematics Olympiad), SouthEast Asian Maths Olympiad, Asia International Math Olympiad, Purple Comet and many others. With a background in Competitive programming as well, Pranav loves to solve Combinatorics, Game Theory and Number Theory problems. His research interests include the likes of Number Theory, Stochastic Analysis, Mathematical Modelling, Game Theory and Astronomy. He has also authored two research papers and has a provisional patent to his name. You can find more about him in the links provided. Personal Website: https://pranavsawant.in Linkedin: https://www.linkedin.com/in/pranavsawant-2005/ ResearchGate: https://www.researchgate.net/profile/Pranav-Sawant-6 GoogleScholar: https://scholar.google.com/citations?user=Io6UOIQAAAAJ&hl=en&authuser=2 Academia: https://independentresearcher.academia.edu/PranavSawant

Piyush K. Jha Piyush Kumar Jha is a mathlete with multiple accolades in Mathematical Olympiads. He is currently in grade 12 at Rishabh Public School, Delhi. He has been preparing for math Olympiads for the past 3 years and coining math problems since he was in grade 10. With his commitment and hard work, he qualified for the Indian National Math Olympiad (INMO) in 2020-21. He also has excellent performances in exams such as Limit, Purple Comet and Sharygin Olympiad. To share the knowledge he acquired over these years and give back to the community he runs "Mathematical Society", an educational platform. His research interests include real analysis, complex analysis, geometry, algebra, number theory and combinatorics. To connect with Piyush or get hints and solutions for the Unsolved Challenges, he can be contacted via his email [email protected] or his AOPS id #lifeismathematics.

Anshuman Shukla’s Links Personal Website: https://anshuman.email Linkedin: https://www.linkedin.com/in/anshuman-shukla-a90a93227/ ResearchGate: https://www.researchgate.net/profile/Anshuman-Shukla-17 Medium: https://medium.com/@anshuman.shukla07

Recommended Resources for Olympiad Mathematics 1) Handouts ● Evan Chen Handouts: https://web.evanchen.cc/olympiad.html ● Yufei Zhao Handouts: https://yufeizhao.com/olympiad.html ● Alexander Remorov Handouts: https://alexanderrem.weebly.com/math-competitions.html ● Po-Shen Loh Handouts: https://math.cmu.edu/~ploh/olympiad.shtml 2) Books ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

Mathematical Circles (Russian Experience) by Fomin, Genkin and Itenberg Excursion in Mathematics by Bhaskaracharya Pratishthana, Pune Inequalities (Little Mathematical Library) by Pavel Korovkin Functional Equation by B.J. Venkatchala The Art and Craft of Problem Solving by Paul Zeitz Mathematical Olympiad Challenges by Titu Andreescu Mathematical Olympiad Treasures by Titu Andreescu Putnam and Beyond by Gelca and Andreescu Lecture Notes on Mathematical Olympiad Courses by Xu Jiagu Euclidean Geometry in Mathematical Olympiads by Evan Chen Game Theory and Strategy by Philip D. Straffin The USSR Olympiad Problem Book by Shklarsky, Chentzov and Yaglom Lemmas in Olympiad Geometry by Adreescu, Korsky and Pohoata Inequalities: Theorems, Techniques and Selected Problems by Zdravko Cvetkovski Problems in Plane Geometry by Igor Sharygin The Mathematical Olympiad Handbook by A. Gardiner Problem Solving Strategies by Arthur Engel Principles and Techniques in Combinatorics by Chi Chuan Elementary Number Theory by David Burton Polynomials by E.J. Barbeau Elementary Number Theory by Waclaw Sierpinski Graph Theory by Frank Harary Introductory Combinatorics by Richard Brualdi Secrets in Inequalities Vil. I and II by Pham Kim Hung Functional Equations and How to Solve Them - (Springer) - Christopher G. Small.