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, 20
Tags: diophantine equation
Determine all positive integers $n$ for which the equation \[x^{n}+(2+x)^{n}+(2-x)^{n}= 0\] has an integer as a solution.

1989 IMO Longlists, 50
Tags: number theory , system of equations , diophantine equation , additive number theory
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\] \[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$

PEN H Problems, 61
Tags: diophantine equation
Solve the equation $2^x -5 =11^{y}$ in positive integers.

1983 Swedish Mathematical Competition, 3
Tags: diophantine , number theory , even , system of equations , system , diophantine equation
The systems of equations \[\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. \] has a solution in positive integers $x_i$. Show that $n$ must be even.

2016 Estonia Team Selection Test, 2
Tags: number theory , diophantine equation , diophantine
Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.

2022 Argentina National Olympiad, 5
Tags: diophantine , number theory , diophantine equation
Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$

2011 NZMOC Camp Selection Problems, 4
Tags: number theory , diophantine , diophantine equation
Find all pairs of positive integers $m$ and $n$ such that $$(m + 1)! + (n + 1)! = m^2n.$$

2004 Federal Math Competition of S&M, 1
Tags: number theory , diophantine equation
Find all pairs of positive integers $(a,b)$ such that $5a^b - b = 2004$.

2021 SG Originals, Q5
Tags: algebra , number theory , diophantine equation
Find all $a,b \in \mathbb{N}$ such that $$2049^ba^{2048}-2048^ab^{2049}=1.$$ [i]Proposed by fattypiggy123 and 61plus[/i]

1981 IMO, 3
Tags: number theory , algebra , polynomial , vieta , diophantine equation
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

1982 IMO Longlists, 31
Tags: equation , number theory , 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 $n=2891$.

2015 China Northern MO, 1
Tags: number theory , diophantine equation , diophantine
Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$

2019 Hanoi Open Mathematics Competitions, 11
Tags: number theory , diophantine , diophantine equation
Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.

1999 Swedish Mathematical Competition, 3
Tags: number theory , diophantine equation , diophantine
Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.

2011 NZMOC Camp Selection Problems, 1
Tags: number theory , diophantine , diophantine equation
Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$ .

2010 Germany Team Selection Test, 3
Tags: number theory , induction , diophantine equation
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

2011 Dutch IMO TST, 1
Tags: number theory , diophantine equation
Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.

2016 District Olympiad, 1
Tags: diophantine equation , equation , algebra
Solve in $ \mathbb{N}^2: $ $$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$

1998 Singapore MO Open, 3
Tags: number theory , diophantine , diophantine equation
Do there exist integers $x$ and $y$ such that $19^{19} = x^3 +y^4$ ? Justify your answer.

2012 France Team Selection Test, 3
Tags: number theory , modular arithmetic , diophantine equation
Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that: \[a^p+b^p=p^c.\]

PEN H Problems, 3
Tags: diophantine equation , quadratic
Does there exist a solution to the equation \[x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65\] in integers with $x, y, z, u, v$ greater than $1998$?

2012 Vietnam Team Selection Test, 1
Tags: algebra , number theory , polynomial , diophantine equation
Consider the sequence $(x_n)_{n\ge 1}$ where $x_1=1,x_2=2011$ and $x_{n+2}=4022x_{n+1}-x_n$ for all $n\in\mathbb{N}$. Prove that $\frac{x_{2012}+1}{2012}$ is a perfect square.

2009 Korea National Olympiad, 3
Tags: number theory , diophantine equation
Let $n$ be a positive integer. Suppose that the diophantine equation \[z^n = 8 x^{2009} + 23 y^{2009} \] uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.

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 $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

2014 Contests, 1
Tags: number theory , diophantine equation , diophantine
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.