Olympiad problems

2021 Polish Junior MO Finals, 1

Positive integers $a$, $b$ an $n$ satisfy \[ \frac{a}{b}=\frac{a^2+n^2}{b^2+n^2}. \] Prove that $\sqrt{ab}$ is an integer.

2000 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: number theory , positive integers , Integer Part , Even

Let $a, b$ and $c$ be positive integers such that $a^2 + b^2 + 1 = c^2$ . Prove that $[a/2] + [c / 2]$ is even. Note: $[x]$ is the integer part of $x$.

2017 JBMO Shortlist, NT3

Tags: number theory , Perfect Square , positive integers

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

2016 Lusophon Mathematical Olympiad, 1

Tags: number theory , Prime factor , Product , coprime , positive integers

Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?

2014 India Regional Mathematical Olympiad, 3

Tags: number theory , Divisibility , positive integers

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

2010 IMO Shortlist, 6

Tags: function , algebra , functional equation , IMO Shortlist , positive integers , Hi

Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$ [i]Proposed by Alex Schreiber, Germany[/i]

2023 4th Memorial "Aleksandar Blazhevski-Cane", P6

Tags: number theory , functions , Divisibility , positive integers

Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: [b]•[/b] For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$. [b]•[/b] $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$. [i]Proposed by Nikola Velov[/i]

2024 Irish Math Olympiad, P3

Tags: function , positive integers , irmo , modular arithmetic , number theory

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]

2004 Junior Tuymaada Olympiad, 6

Tags: number theory , Sum , reciprocal sum , positive integers , prime

We call a positive integer [i] good[/i] if the sum of the reciprocals of all its natural divisors are integers. Prove that if $ m $ is a [i]good [/i] number, and $ p> m $ is a prime number, then $ pm $ is not [i]good[/i].

1986 Brazil National Olympiad, 2

Tags: number theory , combinatorics , consecutive , positive integers

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

1961 Czech and Slovak Olympiad III A, 1

Tags: Sequence , consecutive , positive integers

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.

2012 Greece JBMO TST, 4

Tags: combinatorics , positive integers , integer solutions

Numbers $x,y,z$ are positive integers and satisfy the equation $x+y+z=2013$. (E) a) Find the number of the triplets $(x,y,z)$ that are solutions of the equation (E). b) Find the number of the solutions of the equation (E) for which $x=y$. c) Find the solution $(x,y,z)$ of the equation (E) for which the product $xyz$ becomes maximum.

2025 Alborz Mathematical Olympiad, P2

Tags: algebra , polynomial , positive integers , Positive coefficients , AlborzMO

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

2008 India Regional Mathematical Olympiad, 3

Tags: inequalities , Inequality , positive integers , real number

Prove that for every positive integer $n$ and a non-negative real number $a$, the following inequality holds: $$n(n+1)a+2n \geqslant 4\sqrt{a}(\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}).$$

2018 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: number theory , Integer , Product , Products , positive integers

Let $n$ be a positive integer. Find all $n$- rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$ \frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$ is integer for every integer $d\ge 0$

2012 India Regional Mathematical Olympiad, 5

Tags: number theory , Diophantine equation , diophantine , prime , positive integers

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 $a \le b \le c$.

2019 Saint Petersburg Mathematical Olympiad, 6

Tags: combinatorial geometry , combinatorics , positive integers , Chessboard , infinite chessboard

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 $n$?

2014 Korea Junior Math Olympiad, 4

Tags: number theory , greatest common divisor , positive integers

Positive integers $p, q, r$ satisfy $gcd(a,b,c) = 1$. Prove that there exists an integer $a$ such that $gcd(p,q+ar) = 1$.

1987 Spain Mathematical Olympiad, 2

Tags: binomial coefficients , inequalities , positive integers

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

2011 Hanoi Open Mathematics Competitions, 9

Tags: function , two variables , positive integers

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 $f(5, 5)$.

2018 Mexico National Olympiad, 4

Tags: algebra , Inequality , maximum value , positive integers

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]

2016 Peru IMO TST, 16

Tags: polynomial , Divisibility , divisible , algebra , positive integers

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 $Q (x)$.

2018 Danube Mathematical Competition, 3

Tags: number theory , Sum , Product , rational , positive integers

Find all the positive integers $n$ with the property: there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$ such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.

2013 Balkan MO Shortlist, A7

Tags: composition , algebra , mapping , bijection , bijective function , positive integers

Suppose that $k$ is a positive integer. A bijective map $f : Z \to Z$ is said to be $k$-[i]jumpy [/i] if $|f(z) - z| \le k$ for all integers $z$. Is it that case that for every $k$, each $k$-jumpy map is a composition of $1$-jumpy maps? [i]It is well known that this is the case when the support of the map is finite.[/i]

2016 Saudi Arabia Pre-TST, 2.3

Tags: number theory , positive integers

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.