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

PEN H Problems, 29
Tags: diophantine equation , quadratic , algebra , quadratic formula
Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]

1917 Eotvos Mathematical Competition, 1
Tags: number theory , system of equations , diophantine , diophantine equation
If $a$ and $b$ are integers and if the solutions of the system of equations $$y - 2x - a = 0$$ $$y^2 - xy + x^2 - b = 0$$ are rational, prove that the solutions are integers.

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 $12$?

2012 NZMOC Camp Selection Problems, 6
Tags: diophantine equation , diophantine , number theory
Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.

2020 Thailand TSTST, 3
Tags: diophantine equation , number theory
Find all pairs of positive integers $(m, n)$ satisfying the equation $$m!+n!=m^n+1.$$

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

PEN H Problems, 28
Tags: diophantine equation , modular arithmetic , number theory , relatively prime
Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]

1963 Poland - Second Round, 3
Tags: number theory , system of equations , diophantine , diophantine equation
Solve the system of equations in integers $$x + y + z = 3$$ $$x^3 + y^3 + z^3 = 3$$

2017 Israel National Olympiad, 4
Tags: algebra , diophantine equation , number theory
Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.

1955 Moscow Mathematical Olympiad, 302
Tags: diophantine , diophantine equation , number theory
Find integer solutions of the equation $x^3 - 2y^3 - 4z^3 = 0$.

VMEO IV 2015, 10.3
Tags: number theory , diophantine equation , diophantine
Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$

1969 IMO Longlists, 7
Tags: modular arithmetic , number theory , diophantine equation , additive number theory , perfect cube
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

2017 Regional Olympiad of Mexico Northeast, 3
Tags: number theory , diophantine equation , diophantine
Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation $$a^3 + 2017a = b^3 -2017b.$$

2016 Mathematical Talent Reward Programme, MCQ: P 6
Tags: algebra , diophantine equation
Number of solutions of the equation $3^x+4^x=8^x$ in reals is [list=1] [*] 0 [*] 1 [*] 2 [*] $\infty$ [/list]

2001 Grosman Memorial Mathematical Olympiad, 6
Tags: diophantine equation , diophantine , number theory
(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?

1950 Poland - Second Round, 6
Tags: number theory , diophantine equation , diophantine
Solve the equation in integer numbers $$y^3-x^3=91$$

PEN H Problems, 40
Tags: diophantine equation
Determine all pairs of rational numbers $(x, y)$ such that \[x^{3}+y^{3}= x^{2}+y^{2}.\]

1964 Swedish Mathematical Competition, 2
Tags: number theory , diophantine equation , diophantine
Find all positive integers $m, n$ such that $n + (n+1) + (n+2) + ...+ (n+m) = 1000$.

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

2022 Austrian Junior Regional Competition, 4
Tags: number theory , diophantine equation , diophantine
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]

1984 Brazil National Olympiad, 1
Tags: number theory , positive integer , factorial equation , factorial , diophantine equation
Find all solutions in positive integers to $(n+1)^k -1 = n!$

1979 Chisinau City MO, 179
Tags: diophantine , diophantine equation , number theory
Prove that the equation $x^2 + y^2 = 1979$ has no integer solutions.

2014 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine equation , diophantine
Determine the prime numbers $p$ and $q$ that satisfy the equality: $p^3 + 107 = 2q (17q + 24)$ .

2020 China Team Selection Test, 4
Tags: number theory , diophantine equation
Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers $$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$

2018 Ecuador NMO (OMEC), 1
Tags: number theory , diophantine equation , diophantine
Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.