Olympiad problems

2018 Mexico National Olympiad, 4

Let $n\geq 2$ be an integer. For each $k$-tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$, consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$. Determine, in terms of $n$, the maximum possible value of the product $S_1S_2\cdots S_k$. [i]Proposed by Misael Pelayo[/i]

2012 India Regional Mathematical Olympiad, 5

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

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 $a \le b \le c$.

1997 Spain Mathematical Olympiad, 4

Tags: positive integer , number theory

Let $p$ be a prime number. Find all integers $k$ for which $\sqrt{k^2 -pk}$ is a positive integer.

2016 Peru IMO TST, 4

Tags: max , divide , min , positive integer , number theory

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

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 $b$.

1983 Brazil National Olympiad, 1

Tags: number theory , positive integer , diophantine equation

Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.

2017 JBMO Shortlist, NT3

Tags: positive integer , perfect square , number theory

Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.

1986 Brazil National Olympiad, 2

Tags: consecutive , positive integer , combinatorics , number theory

Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.

2011 Hanoi Open Mathematics Competitions, 9

Tags: two variables , positive integer , function

For every pair of positive integers $(x, y)$ we define $f(x,y)$ as follows: $f(x,1) = x$ $f(x,y) = 0$ if $y > x$ $f(x +1,y) = y[f(x,y)+ f(x, y-1)]$ Evaluate $f(5, 5)$.

2019 Saint Petersburg Mathematical Olympiad, 6

Tags: chessboard , infinite chessboard , positive integer , combinatorial geometry , combinatorics

Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each $n$ in each square $n \times n$ the sum of the numbers is a multiple of $n$?

2025 Alborz Mathematical Olympiad, P1

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

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

2006 Korea Junior Math Olympiad, 2

Tags: positive integer , integer , coprime , number theory

Find all positive integers that can be written in the following way $\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}$ . Also, $a,b, c$ are positive integers that are pairwise relatively prime.

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.

1993 Nordic, 3

Tags: digit , positive integer , system of equations , number theory

Find all solutions of the system of equations $\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$ where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.

2016 Peru IMO TST, 16

Tags: divisibility , positive integer , polynomial , divisible , algebra

Find all pairs $ (m, n)$ of positive integers that have the following property: For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.

2014 Korea Junior Math Olympiad, 5

Tags: multiple , positive integer , divisible , number theory

For positive integers $x,y$, find all pairs $(x,y)$ such that $x^2y + x$ is a multiple of $xy^2 + 7$.

2008 India Regional Mathematical Olympiad, 5

Tags: divisibility , positive integer , number theory

Let $N$ be a ten digit positive integer divisible by $7$. Suppose the first and the last digit of $N$ are interchanged and the resulting number (not necessarily ten digit) is also divisible by $7$ then we say that $N$ is a good integer. How many ten digit good integers are there?

2013 JBMO Shortlist, 5

Tags: number theory , positive integer , diophantine equation

Solve in positive integers: $\frac{1}{x^2}+\frac{y}{xz}+\frac{1}{z^2}=\frac{1}{2013}$ .

2024 Irish Math Olympiad, P3

Tags: modular arithmetic , number theory , positive integer , function

Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Determine all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ that satisfy: [list] [*]$f(mn)+1=f(m)+f(n)$ for all positive integers $m$ and $n$; [*]$f(2024)=1$; [*]$f(n)=1$ for all positive $n\equiv22\pmod{23}$. [/list]

2014 India Regional Mathematical Olympiad, 3

Tags: number theory , positive integer , divisibility

Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$.

1985 Spain Mathematical Olympiad, 4

Tags: positive integer , sum of powers , prime , number theory

Prove that for each positive integer $k $ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$. In all such cases prove that $a^3+b^3+c^3$ is not a prime.

2016 Saudi Arabia Pre-TST, 2.3

Tags: positive integer , number theory

Let $u$ and $v$ be positive rational numbers with $u \ne v$. Assume that there are infinitely many positive integers $n$ with the property that $u^n - v^n$ are integers. Prove that $u$ and $v$ are integers.

2007 JBMO Shortlist, 4

Tags: positive integer , algebra

Let $a$ and $ b$ be positive integers bigger than $2$. Prove that there exists a positive integer $k$ and a sequence $n_1, n_2, ..., n_k$ consisting of positive integers, such that $n_1 = a,n_k = b$, and $(n_i + n_{i+1}) | n_in_{i+1}$ for all $i = 1,2,..., k - 1$

2001 Bosnia and Herzegovina Team Selection Test, 2

Tags: equation , positive integer , number theory

For positive integers $x$, $y$ and $z$ holds $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$. Prove that $xyz\geq 3600$

2010 Belarus Team Selection Test, 1.1

Tags: number theory , sum , positive integer

Does there exist a subset $E$ of the set $N$ of all positive integers such that none of the elements in $E$ can be presented as a sum of at least two other (not necessarily distinct) elements from $E$ ? (E. Barabanov)