Olympiad problems

2025 Alborz Mathematical Olympiad, P2

Suppose that for polynomials $ P, Q, R $ with positive integer coefficients, the following two conditions hold: $\bullet$ The constant terms of $ P, Q, R $ are equal. $\bullet$ For all real numbers $ x $, the following relations hold: \[ P(Q(R(x))) = Q(R(P(x))) = R(P(Q(x))) = P(R(Q(x))) = Q(P(R(x))) = R(Q(P(x))). \] Prove that for every real number $ x $, $ P(x) = Q(x) = R(x) $. Proposed by Soroush Behroozifar & Ali Nazarboland

1961 Czech and Slovak Olympiad III A, 1

Tags: sequence , consecutive , positive integer

Consider an infinite sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, \ldots, \underbrace{n,\ldots,n}_{n\text{ times}},\ldots.$$ Find the 1000th term of the sequence.

1987 Spain Mathematical Olympiad, 2

Tags: binomial coefficient , positive integer , inequalities

Show that for each natural number $n > 1$ $1 \cdot \sqrt{{n \choose 1}}+ 2 \cdot \sqrt{{n \choose 2}}+...+n \cdot \sqrt{{n \choose n}} <\sqrt{2^{n-1}n^3}$

2018 Lusophon Mathematical Olympiad, 4

Tags: positive integer , diophantine equation , number theory

Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.

2015 JBMO Shortlist, C3

Tags: positive integer , combinatorics , array , table

Positive integers are put into the following table. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{tabular} Find the number of the line and column where the number $2015$ stays.

2019 Poland - Second Round, 4

Tags: positive integer , number theory , divisibility

Let $a_1, a_2, \ldots, a_n$ ($n\ge 3$) be positive integers such that $gcd(a_1, a_2, \ldots, a_n)=1$ and for each $i\in \lbrace 1,2,\ldots, n \rbrace$ we have $a_i|a_1+a_2+\ldots+a_n$. Prove that $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

2025 Alborz Mathematical Olympiad, P1

Tags: positive integer , functional equation , number theory , algebra

Let $ \mathbb{Z^{+}} $ denote the set of all positive integers. Find all functions $ f: \mathbb{Z^{+}} \rightarrow \mathbb{Z^{+}} $ such that for every pair of positive integers $ a $ and $ b $, there exists a positive integer $ c $ satisfying: $$ f(a)f(b) - ab = 2^{c-1} - 1. $$ Proposed by Matin Yousefi

2002 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: integer , number theory , positive integer

Determine all pairs $(a, b)$ of positive integers for which $\frac{a^2b+b}{ab^2+9}$ is an integer number.

2012 India Regional Mathematical Olympiad, 5

Tags: positive integer , number theory , diophantine equation , diophantine , prime

Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.

2024 Irish Math Olympiad, P1

Tags: positive integer , recursion

The [i]runcible[/i] positive integers are defined recursively as follows: [list] [*]$1$ and $2$ are runcible [*]If $a$ and $b$ are runcible (where $a$ and $b$ are not necessarily distinct) then $2a + 3b$ is runcible. [/list] Is $2024$ runcible?

2015 Kyiv Math Festival, P3

Tags: positive integer , number theory , sum

Is it true that every positive integer greater than $50$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

2003 Bosnia and Herzegovina Team Selection Test, 3

Tags: inequality , algebra , positive integer

Prove that for every positive integer $n$ holds: $(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$

2018 Baltic Way, 19

Tags: number theory , positive integer

An infinite set $B$ consisting of positive integers has the following property. For each $a,b \in B$ with $a>b$ the number $\frac{a-b}{(a,b)}$ belongs to $B$. Prove that $B$ contains all positive integers. Here, $(a,b)$ is the greatest common divisor of numbers $a$ and $b$.

2013 Korea Junior Math Olympiad, 3

Tags: algebra , sequence , recurrence relation , sum , positive integer , diophantine equation

$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$). For positive integer $n$, define as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$ Prove that $b_n$ is positive integer.