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: 916

2022 Azerbaijan National Mathematical Olympiad, 3
Tags: diophantine equation , diophantine , number theory
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

2011 Hanoi Open Mathematics Competitions, 6
Tags: diophantine , diophantine equation , number theory
Find all positive integers $(m,n)$ such that $m^2 + n^2 + 3 = 4(m + n)$

2021 China Team Selection Test, 3
Tags: algebra , number theory , diophantine equation
Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

2022 USAJMO, 5
Tags: diophantine equation , prime number , number theory
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

1999 Austrian-Polish Competition, 7
Tags: diophantine equation , diophantine , number theory
Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.

2012 Junior Balkan Team Selection Tests - Romania, 3
Tags: diophantine , number theory , diophantine equation
Let $m$ and $n$ be two positive integers, $m, n \ge 2$. Solve in the set of the positive integers the equation $x^n + y^n = 3^m$.

1988 Spain Mathematical Olympiad, 6
Tags: diophantine equation , parameterization , number theory
For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation $$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .

2015 Belarus Team Selection Test, 1
Tags: diophantine , diophantine equation , number theory
Solve the equation in nonnegative integers $a,b,c$: $3^a+2^b+2015=3c!$ I.Gorodnin

2019 Cono Sur Olympiad, 4
Tags: number theory , prime , diophantine equation
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.

2001 All-Russian Olympiad Regional Round, 11.1
Tags: diophantine , diophantine equation , number theory
Find all prime numbers $p$ and $q$ such that $p + q = (p -q)^3.$

2001 Moldova National Olympiad, Problem 6
Tags: diophantine equation , number theory
For a positive integer $n$, denote $A_n=\{(x,y)\in\mathbb Z^2|x^2+xy+y^2=n\}$. (a) Prove that the set $A_n$ is always finite. (b) Prove that the number of elements of $A_n$ is divisible by $6$ for all $n$. (c) For which $n$ is the number of elements of $A_n$ divisible by $12$?

2007 Alexandru Myller, 1
Tags: equation , perfect square , diophantine equation , divisibility , number theory
Solve $ x^3-y^3=2xy+7 $ in integers.

2016 Czech-Polish-Slovak Junior Match, 6
Tags: diophantine , system of equations , diophantine equation , number theory
Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that $a + b + c = 3k + 1$, $ab + bc + ca = 3k^2 + 2k$. Slovakia

2015 Middle European Mathematical Olympiad, 7
Tags: factorial , diophantine equation , exponential equation , number theory
Find all pairs of positive integers $(a,b)$ such that $$a!+b!=a^b + b^a.$$

2018 Korea Junior Math Olympiad, 7
Tags: diophantine equation , number theory
Find all integer pair $(m,n)$ such that $7^m=5^n+24$.

2012 Turkey Junior National Olympiad, 1
Tags: number theory , integration , quadratic , diophantine equation , calculus
Let $x, y$ be integers and $p$ be a prime for which \[ x^2-3xy+p^2y^2=12p \] Find all triples $(x,y,p)$.

2022 Dutch BxMO TST, 3
Tags: diophantine , diophantine equation , number theory
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$

2014 NIMO Problems, 7
Tags: geometric transformation , geometry , reflection , probability , diophantine equation
Find the sum of all integers $n$ with $2 \le n \le 999$ and the following property: if $x$ and $y$ are randomly selected without replacement from the set $\left\{ 1,2,\dots,n \right\}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number. [i]Proposed by Ivan Koswara[/i]

1999 Belarusian National Olympiad, 6
Tags: diophantine equation , number theory
Solve in integer:$x^6+x^2y=y^3+2y^2$

2017 India IMO Training Camp, 2
Tags: diophantine equation , number theory
Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$

2007 Ukraine Team Selection Test, 12
Tags: number theory , diophantine equation , polynomial
Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{2}\plus{}n\plus{}1$ are not more then $ \sqrt{n}$. [hide] Stronger one. Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{3}\minus{}1$ are not more then $ \sqrt{n}$.[/hide]

2013 Bulgaria National Olympiad, 6
Tags: number theory , diophantine equation
Given $m\in\mathbb{N}$ and a prime number $p$, $p>m$, let \[M=\{n\in\mathbb{N}\mid m^2+n^2+p^2-2mn-2mp-2np \,\,\, \text{is a perfect square} \} \] Prove that $|M|$ does not depend on $p$. [i]Proposed by Aleksandar Ivanov[/i]

2007 QEDMO 4th, 8
Tags: diophantine , number theory , diophantine equation
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$. Daniel Harrer

1979 IMO Shortlist, 15
Tags: algebra , cauchy-schwarz inequality , diophantine equation , system of equations
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

2014 Thailand TSTST, 1
Tags: diophantine equation , number theory
Find all triples of positive integers $(a, b, c)$ such that $$(2^a-1)(3^b-1)=c!.$$