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

2023 Taiwan TST Round 3, 4
Tags: inequalities , am-gm , diophantine equation , number theory
Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and \[a+b+c-3\sqrt[3]{abc}=1.\] [i]Proposed by usjl[/i]

2014 Polish MO Finals, 2
Tags: diophantine equation , number theory
Find all pairs $(x,y)$ of positive integers that satisfy $$2^x+17=y^4$$.

2013 Estonia Team Selection Test, 1
Tags: system of equations , diophantine equation , diophantine , number theory
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 $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

VI Soros Olympiad 1999 - 2000 (Russia), 9.1
Tags: diophantine equation , number theory
Prove that there is no natural number $k$ such that $k^{1999} - k^{1998} = 2k + 2$.

2020 Chile National Olympiad, 4
Tags: diophantine , diophantine equation , system of equations , number theory
Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$

2017 Simon Marais Mathematical Competition, B2
Tags: diophantine equation , number theory , modular arithmetic
Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

2006 Hanoi Open Mathematics Competitions, 3
Tags: system of equations , diophantine equation , number theory
Find the number of different positive integer triples $(x, y,z)$ satisfying the equations $x^2 + y -z = 100$ and $x + y^2 - z = 124$:

2024 Israel TST, P1
Tags: diophantine equation , exponential equation , number theory
Solve in positive integers: \[x^{y^2+1}+y^{x^2+1}=2^z\]

1998 Slovenia Team Selection Test, 4
Tags: diophantine , gcd , diophantine equation , number theory
Find all positive integers $x$ and $y$ such that $x+y^2+z^3 = xyz$, where $z$ is the greatest common divisor of $x$ and $y$

PEN H Problems, 23
Tags: diophantine equation , inequalities
Find all $(x,y,z) \in {\mathbb{Z}}^3$ such that $x^{3}+y^{3}+z^{3}=x+y+z=3$.

PEN H Problems, 47
Tags: diophantine equation
Show that the equation $x^4 +y^4 +4z^4 =1$ has infinitely many rational solutions.

1982 IMO Shortlist, 16
Tags: diophantine equation , divisibility , number theory , equation
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 $n=2891$.

2021 Grand Duchy of Lithuania, 4
Tags: diophantine equation , diophantine , number theory
A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers. a) Find at least one primary triplet. b) Show that there are infinitely many primary triplets.

2019 Azerbaijan Junior NMO, 5
Tags: diophantine equation , number theory
Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$

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

2014 Contests, 1
Tags: diophantine equation , modular arithmetic
Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

1966 Bulgaria National Olympiad, Problem 1
Tags: diophantine equation , number theory
Prove that the equation $$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.

1967 IMO Longlists, 38
Tags: modular arithmetic , perfect cube , diophantine equation , number theory
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

PEN H Problems, 86
Tags: function , euler , least common multiple , geometry , diophantine equation
A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?

2022 Israel National Olympiad, P4
Tags: diophantine equation , number theory
Find all triples $(a,b,c)$ of integers for which the equation \[x^3-a^2x^2+b^2x-ab+3c=0\] has three distinct integer roots $x_1,x_2,x_3$ which are pairwise coprime.

2010 Contests, 2
Tags: diophantine , diophantine equation , number theory
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

2023 Singapore Senior Math Olympiad, 2
Tags: diophantine equation , number theory
Find all positive integers $k$ such that there exists positive integers $a, b$ such that \[a^2+4=(k^2-4)b^2.\]

1980 IMO Shortlist, 12
Tags: diophantine equation , polynomial , equation , number theory , algebra
Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

2015 Saudi Arabia Pre-TST, 1.3
Tags: diophantine equation , diophantine , number theory
Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$. (Malik Talbi)

2012 India PRMO, 19
Tags: diophantine equation , number theory
How many integer pairs $(x,y)$ satisfy $x^2+4y^2-2xy-2x-4y-8=0$?