Olympiad problems

1997 Pre-Preparation Course Examination, 4

Let $f : \mathbb N \to \mathbb N$ be an injective function such that there exists a positive integer $k$ for which $f(n) \leq n^k$. Prove that there exist infinitely many primes $q$ such that the equation $f(x) \equiv 0 \pmod q$ has a solution in prime numbers.

2011 AIME Problems, 9

Tags: trigonometry , algebra , polynomial , rational root theorem , logarithm

Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$. Find $24\cot^2{x}$.

2001 Saint Petersburg Mathematical Olympiad, 10.7

Tags: graph theory , combinatorics , processes

On the parliament of Sikinia, for any two deputies, there is third deputy, which knows exactly one of the two. Every deputy belongs to one of the two ruling parties. Every day, he president tells a certain group of deputies to change the party that they belong, and all the deputies which which know at least one of the deputies of the group has to change their party too. Prove that, the president can reach any configuration of deputies between two parties.(The president himself isn't a member of the parliament of Sikinia). [I]Proposed by S. Berlov[/i]

2018 Czech and Slovak Olympiad III A, 6

Tags: combinatorial number theory , combinatorics , number theory

Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square.

2020 HMNT (HMMO), 4

Tags: combinatorics

Marisa has two identical cubical dice labeled with the numbers $\{1, 2, 3, 4, 5, 6\}$. However, the two dice are not fair, meaning that they can land on each face with different probability. Marisa rolls the two dice and calculates their sum. Given that the sum is $2$ with probability $0.04$, and $12$ with probability $0.01$, the maximum possible probability of the sum being $7$ is $p$. Compute $\lfloor 100p \rfloor$.

1978 Bulgaria National Olympiad, Problem 4

Tags: inequalities

Find the greatest possible real value of $S$ and smallest possible value of $T$ such that for every triangle with sides $a,b,c$ $(a\le b\le c)$ to be true the inequalities: $$S\le\frac{(a+b+c)^2}{bc}\le T.$$

2025 Kyiv City MO Round 2, Problem 2

Tags: algebra

Mykhailo chose three distinct positive real numbers $ a, b, c $ and wrote the following numbers on the board: \[ a + b, \quad b + c, \quad c + a, \quad ab, \quad bc, \quad ca. \] What is the minimum possible number of distinct numbers that can be written on the board? [i]Proposed by Anton Trygub[/i]

2016 Balkan MO Shortlist, C1

Tags: palindrome , number theory , number base

Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_1,b_2,..., b_K$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_1,b_2,..., b_K$.

1995 Vietnam National Olympiad, 1

Tags: algebra

Find all real solutions to $ x^3 \minus{} 3x^2 \minus{} 8x \plus{} 40 \minus{} 8\sqrt[4]{4x \plus{} 4} \equal{} 0$

2008 ISI B.Math Entrance Exam, 6

Tags: algebra

Let $\dbinom{n}{k}$ denote the binomial coefficient $\frac{n!}{k!(n-k)!}$ , and $F_m$ be the $m^{th}$ Fibonacci number given by $F_1=F_2=1$ and $F_{m+2}=F_m+F_{m+1}$ for all $m\geq 1$. Show that $\sum \dbinom{n}{k}=F_{m+1}$ for all $m\geq 1$ . Here the above sum is over all pairs of integers $n\geq k\geq 0$ with $n+k=m$ .

2018 Sharygin Geometry Olympiad, 23

Tags: geometry , combinatorial geometry

The plane is divided into convex heptagons with diameters less than 1. Prove that an arbitrary disc with radius 200 intersects most than a billion of them.

1998 Greece JBMO TST, 2

Tags: trapezoid , geometry

Let $ABCD$ be a trapezoid with parallel sides $AB, CD$. $M,N$ lie on lines $AD, BC$ respectively such that $MN || AB$. Prove that $DC \cdot MA + AB \cdot MD = MN \cdot AD$.

2017 Dutch IMO TST, 4

Tags: geometry , combinatorics

Let $n \geq 2$ be an integer. Find the smallest positive integer $m$ for which the following holds: given $n$ points in the plane, no three on a line, there are $m$ lines such that no line passes through any of the given points, and for all points $X \neq Y$ there is a line with respect to which $X$ and $Y$ lie on opposite sides

2000 VJIMC, Problem 2

Tags: combinatorics

If we write the sequence $\text{AAABABBB}$ along the perimeter of a circle, then every word of the length $3$ consisting of letters $A$ and $B$ (i.e. $\text{AAA}$, $\text{AAB}$, $\text{ABA}$, $\text{BAB}$, $\text{ABB}$, $\text{BBB}$, $\text{BBA}$, $\text{BAA}$) occurs exactly once on the perimeter. Decide whether it is possible to write a sequence of letters from a $k$-element alphabet along the perimeter of a circle in such a way that every word of the length $l$ (i.e. an ordered $l$-tuple of letters) occurs exactly once on the perimeter.

2013 Turkey Junior National Olympiad, 1

Tags: inequalities , algebra , polynomial , vieta , analytic geometry , function , calculus

Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of \[ |(x-y)(y-z)(z-x) | \]

2019 Puerto Rico Team Selection Test, 3

Tags: algebra , inequalities

Find the largest value that the expression can take $a^3b + b^3a$ where $a, b$ are non-negative real numbers, with $a + b = 3$.

2012 Online Math Open Problems, 27

Tags: geometry , circumcircle , incenter , trigonometry , angle bisector , cyclic quadrilateral

Let $ABC$ be a triangle with circumcircle $\omega$. Let the bisector of $\angle ABC$ meet segment $AC$ at $D$ and circle $\omega$ at $M\ne B$. The circumcircle of $\triangle BDC$ meets line $AB$ at $E\ne B$, and $CE$ meets $\omega$ at $P\ne C$. The bisector of $\angle PMC$ meets segment $AC$ at $Q\ne C$. Given that $PQ = MC$, determine the degree measure of $\angle ABC$. [i]Ray Li.[/i]

2002 Moldova National Olympiad, 3

Tags: pigeonhole principle

There are $ 16$ persons in a company, each of which likes exactly $ 8$ other persons. Show that there exist two persons who like each other.

2018 MIG, 17

Tags:

Two standard six sided dice labeled with the numbers $1$-$6$ are rolled, and the numbers that come up are multiplied. What is the probability that their product is a multiple of five? $\textbf{(A) } \dfrac14\qquad\textbf{(B) } \dfrac5{18}\qquad\textbf{(C) } \dfrac{11}{36}\qquad\textbf{(D) } \dfrac13\qquad\textbf{(E) } \dfrac49$

1979 Kurschak Competition, 3

Tags: combinatorics

An $n \times n$ array of letters is such that no two rows are the same. Show that it must be possible to omit a column, so that the remaining table has no two rows the same.

2015 ISI Entrance Examination, 7

Tags: circles , triangle , geometry

Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$

2014 Contests, 1

Tags: induction , modular arithmetic , quadratic , number theory

Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$ a) Prove that the sequence consists only of natural numbers. b) Check if there are terms of the sequence divisible by $2011$.

2014 Greece Junior Math Olympiad, 4

Tags: combinatorics

We color the numbers $1, 2, 3,....,20$ with two colors white and black in such a way that both colors are used. Find the number of ways, we can perform this coloring if the product of white numbers and the product of black numbers have greatest common divisor equal to $1$.

2011 Baltic Way, 2

Tags: algebra , functional equation , functional equation in n

Let $f:\mathbb{Z}\to\mathbb{Z}$ be a function such that for all integers $x$ and $y$, the following holds: \[f(f(x)-y)=f(y)-f(f(x)).\] Show that $f$ is bounded.

1977 Vietnam National Olympiad, 3

Tags: combinatorial geometry , circles , geometry

Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?