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

1999 German National Olympiad, 1
Tags: diophantine equation , diophantine , number theory
Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are a) natural numbers, b) integers

2014 NIMO Problems, 7
Tags: probability , geometry , geometric transformation , reflection , 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]

2004 VJIMC, Problem 4
Tags: diophantine equation , number theory
Find all pairs $(m,n)$ of positive integers such that $m+n$ and $mn+1$ are both powers of $2$.

2016 Postal Coaching, 4
Tags: number theory , diophantine equation , prime number
Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$

2010 Junior Balkan Team Selection Tests - Romania, 1
Tags: diophantine equation , diophantine , number theory
Determine the prime numbers $p, q, r$ with the property that: $p(p-7) + q (q-7) = r (r-7)$.

2017 Hanoi Open Mathematics Competitions, 3
Tags: diophantine equation , diophantine , number theory
The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is (A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.

PEN H Problems, 22
Tags: diophantine equation , symmetry
Find all integers $a,b,c,x,y,z$ such that \[a+b+c=xyz, \; x+y+z=abc, \; a \ge b \ge c \ge 1, \; x \ge y \ge z \ge 1.\]

2013 Estonia Team Selection Test, 1
Tags: number theory , system of equations , diophantine , diophantine equation
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

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

2021 OMpD, 3
Tags: diophantine equation , number theory
Determine all pairs of integer numbers $(x, y)$ such that: $$\frac{(x - y)^2}{x + y} = x - y + 6$$

2007 India IMO Training Camp, 2
Tags: modular arithmetic , diophantine equation , number theory
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$

2021 Canadian Mathematical Olympiad Qualification, 6
Tags: diophantine equation , algebra
Show that $(w, x, y, z)=(0,0,0,0)$ is the only integer solution to the equation $$w^{2}+11 x^{2}-8 y^{2}-12 y z-10 z^{2}=0$$

2024 Greece Junior Math Olympiad, 4
Tags: number theory , diophantine equation , diophantine
Prove that there are infinite triples of positive integers $(x,y,z)$ such that $$x^2+y^2+z^2+xy+yz+zx=6xyz.$$

2000 Singapore Senior Math Olympiad, 2
Tags: factorial , diophantine equation , diophantine , number theory
Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.

2009 Iran Team Selection Test, 2
Tags: pigeonhole principle , modular arithmetic , inequalities , prime number , diophantine equation , number theory
Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite

1998 Estonia National Olympiad, 4
Tags: factorial , diophantine , diophantine equation , number theory
Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .

1987 Czech and Slovak Olympiad III A, 2
Tags: number theory , diophantine equation , prime number , number of solutions
Given a prime $p>3$ and an odd integer $n>0$, show that the equation $$xyz=p^n(x+y+z)$$ has at least $3(n+1)$ different solutions up to symmetry. (That is, if $(x',y',z')$ is a solution and $(x'',y'',z'')$ is a permutation of the previous, they are considered to be the same solution.)

2003 Irish Math Olympiad, 1
Tags: diophantine equation , number theory
find all solutions, not necessarily positive integers for $(m^2+ n)(m+ n^2)= (m+ n)^3$

1982 IMO Longlists, 31
Tags: number theory , equation , diophantine equation , divisibility
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

2022 IMO Shortlist, N4
Tags: number theory , divisibility , lte lemma , diophantine equation
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

2001 Greece JBMO TST, 1
Tags: number theory , algebra , factorization , diophantine equation
a) Factorize $A= x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2$ b) Prove that there are no integers $x,y,z$ such that $x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 $

2018 Hanoi Open Mathematics Competitions, 15
Tags: diophantine equation , number theory
Find all pairs of prime numbers $(p,q)$ such that for each pair $(p,q)$, there is a positive integer m satisfying $\frac{pq}{p + q}=\frac{m^2 + 6}{m + 1}$.

1998 Korea Junior Math Olympiad, 1
Tags: number theory , diophantine equation
Show that there exist no integer solutions $(x, y, z)$ to the equation $$x^3+2y^3+4z^3=9$$

1997 India National Olympiad, 2
Tags: quadratic , modular arithmetic , diophantine equation , special factorizations , number theory
Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]

2013 Junior Balkan Team Selection Tests - Romania, 2
Tags: diophantine equation , diophantine , number theory
Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$