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: 436

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 $x^n + y^n = 3^m$.

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

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

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

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

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

1971 Polish MO Finals, 4
Tags: diophantine , number theory
Prove that if positive integers $x,y,z$ satisfy the equation $$x^n + y^n = z^n,$$ then $\min\, (x,y) \ge n$.

1998 Bundeswettbewerb Mathematik, 1
Tags: diophantine , number theory , diophantine equation
Find all integer solutions $(x,y,z)$ of the equation $xy+yz+zx-xyz = 2$.

2012 Dutch IMO TST, 3
Tags: number theory , diophantine , diophantine equation
Determine all pairs $(x, y)$ of positive integers satisfying $x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.

2014 Cuba MO, 1
Tags: diophantine , number theory , diophantine equation
Find all the integer solutions of the equation $ m^4 + 2n^2 = 9mn$.

1997 Tournament Of Towns, (558) 3
Tags: diophantine , number theory , diophantine equation
Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$. (N Vassiliev)

2018 India PRMO, 6
Tags: number theory , algebra , sum , diophantine , diophantine equation
Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?

2018 NZMOC Camp Selection Problems, 9
Tags: diophantine equation , power , diophantine , number theory
Let $x, y, p, n, k$ be positive integers such that $$x^n + y^n = p^k.$$ Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.

1993 Swedish Mathematical Competition, 3
Tags: diophantine , even , diophantine equation , number theory
Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.

VMEO III 2006, 12.2
Tags: diophantine , number theory , diophantine equation
Find all positive integers $(m, n)$ that satisfy $$m^2 =\sqrt{n} +\sqrt{2n + 1}.$$

2022-23 IOQM India, 4
Tags: diophantine , algebra
Starting with a positive integer $M$ written on the board , Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x+2$.Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he replaces it with $2x+27$.Given that Alice and Bob reach the same number after playing $4$ moves each, find the smallest value of $M+N$

2012 Czech-Polish-Slovak Junior Match, 5
Tags: diophantine , diophantine equation , number theory
Find all triplets $(a, k, m)$ of positive integers that satisfy the equation $k + a^k = m + 2a^m$.

2003 Abels Math Contest (Norwegian MO), 2a
Tags: diophantine , number theory , diophantine equation
Find all pairs $(x, y)$ of integers numbers such that $y^3+5=x(y^2+2)$

1997 Estonia National Olympiad, 1
Tags: diophantine , diophantine equation , number theory
Find: a) Any quadruple of positive integers $(a, k, l, m)$ such that $a^k = a^l + a^m,$ b) Any quintuple of positive integers $(a, k, l, m, n)$ for which $a^k = a^l + a^m+a^n$

2022 Dutch IMO TST, 1
Tags: number theory , diophantine , diophantine equation
Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.

2016 Hanoi Open Mathematics Competitions, 5
Tags: diophantine , diophantine equation , number theory
There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to (A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.

2013 Hanoi Open Mathematics Competitions, 5
Tags: diophantine , diophantine equation , number theory
The number of integer solutions $x$ of the equation below $(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above.

1993 Tournament Of Towns, (360) 3
Tags: diophantine , perfect square , diophantine equation , number theory
Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and $$\frac{ab}{a-b}=c$$ Prove that $a -b$ is a perfect square. (SL Berlov)

IV Soros Olympiad 1997 - 98 (Russia), 11.1
Tags: diophantine , diophantine equation , number theory
Solve the equation $xy =1997(x + y)$ in integers.

2001 Grosman Memorial Mathematical Olympiad, 6
Tags: diophantine , number theory , diophantine equation
(a) Find a pair of integers (x,y) such that $15x^2 +y^2 = 2^{2000}$ (b) Does there exist a pair of integers $(x,y)$ such that $15x^2 + y^2 = 2^{2000}$ and $x$ is odd?