This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 89

2014 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: positive integer , equation , number theory
Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers

2015 Greece JBMO TST, 3
Tags: number theory , perfect square , positive integer
Prove that there is not a positive integer $n$ such that numbers $(n+1)2^n, (n+3)2^{n+2}$ are both perfect squares.

2025 Alborz Mathematical Olympiad, P2
Tags: polynomial , positive integer , positive coefficients , algebra
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

2018 Lusophon Mathematical Olympiad, 4
Tags: diophantine equation , positive integer , number theory
Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.

2013 Balkan MO Shortlist, A7
Tags: algebra , composition , mapping , bijection , bijective function , positive integer
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]

2010 IMO Shortlist, 6
Tags: algebra , function , positive integer , functional equation
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]

2015 Kyiv Math Festival, P3
Tags: sum , number theory , positive integer
Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

2004 Junior Tuymaada Olympiad, 6
Tags: prime , sum , reciprocal sum , positive integer , number theory
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].

2013 Korea Junior Math Olympiad, 3
Tags: algebra , sequence , sum , positive integer , diophantine equation , recurrence relation
$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$). For positive integer $n$, de fine as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$ Prove that $b_n$ is positive integer.

2008 Indonesia TST, 2
Tags: factorial , positive integer , number theory
Find all positive integers $1 \le n \le 2008$ so that there exist a prime number $p \ge n$ such that $$\frac{2008^p + (n -1)!}{n}$$ is a positive integer.

2012 Greece JBMO TST, 4
Tags: combinatorics , positive integer , 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.

2018 Rioplatense Mathematical Olympiad, Level 3, 5
Tags: number theory , integer , positive integer , product
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 $d\ge 0$

2018 Bosnia and Herzegovina EGMO TST, 2
Tags: positive integer , divide , number theory
Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds $$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$

2006 Korea Junior Math Olympiad, 5
Tags: number theory , coprime , integer , positive integer
Find all positive integers that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also, $m,n$ are relatively prime positive integers.

2021 Argentina National Olympiad, 2
Tags: prime number , positive integer , perfect square , number theory
Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$

2024 Mozambican National MO Selection Test, P3
Tags: number theory , nt , diophantine equation , factorization , prime number , positive integer
Find all triples of positive integers $(a,b,c)$ such that: $a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$

1983 Austrian-Polish Competition, 4
Tags: set , positive integer , partition , number theory
The set $N$ has been partitioned into two sets A and $B$. Show that for every $n \in N$ there exist distinct integers $a, b > n$ such that $a, b, a + b$ either all belong to $A$ or all belong to $B$.

1983 Brazil National Olympiad, 5
Tags: positive integer , minimum , inequality , number theory
Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$. Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.

2007 Korea Junior Math Olympiad, 8
Tags: prime , positive integer , divisor , number theory
Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$). Prove that there are infi nite number of [i]Primes of the Year[/i].

2022 Pan-American Girls' Math Olympiad, 5
Tags: positive integer
Find all positive integers $k$ for which there exist $a$, $b$, and $c$ positive integers such that \[\lvert (a-b)^3+(b-c)^3+(c-a)^3\rvert=3\cdot2^k.\]

1991 Nordic, 3
Tags: induction , inequalities , positive integer
Show that $ \frac{1}{2^2} +\frac{1}{3^2} +...+\frac{1}{n^2} <\frac{2}{3}$ for all $n \ge 2 $.

2024 CAPS Match, 6
Tags: number theory , combinatorics , positive integer , combinatorial number theory , prime number , triplets
Determine whether there exist infinitely many triples $(a, b, c)$ of positive integers such that every prime $p$ divides \[\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.\]

2014 Korea Junior Math Olympiad, 4
Tags: greatest common divisor , positive integer , number theory
Positive integers $p, q, r$ satisfy $gcd(a,b,c) = 1$. Prove that there exists an integer $a$ such that $gcd(p,q+ar) = 1$.

2002 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: integer , positive integer , number theory
Determine all pairs $(a, b)$ of positive integers for which $\frac{a^2b+b}{ab^2+9}$ is an integer number.

2021 Polish Junior MO Finals, 1
Tags: algebra , positive integer , square root
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.