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

2022 New Zealand MO, 1
Tags: number theory , diophantine equation , diophantine
Find all integers $a, b$ such that $$a^2 + b = b^{2022}.$$

2011 Grand Duchy of Lithuania, 3
Tags: diophantine , diophantine equation , number theory
Find all primes $p,q$ such that $p ^3-q^7=p-q$.

2019 Belarus Team Selection Test, 1.3
Tags: number theory , diophantine equation , perfect power
Given the equation $$ a^b\cdot b^c=c^a $$ in positive integers $a$, $b$, and $c$. [i](i)[/i] Prove that any prime divisor of $a$ divides $b$ as well. [i](ii)[/i] Solve the equation under the assumption $b\ge a$. [i](iii)[/i] Prove that the equation has infinitely many solutions. [i](I. Voronovich)[/i]

PEN A Problems, 31
Tags: diophantine equation , divisibility theory
Show that there exist infinitely many positive integers $n$ such that $n^{2}+1$ divides $n!$.

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

2016 Latvia Baltic Way TST, 18
Tags: diophantine equation , number theory , system of equations
Solve the system of equations in integers: $$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$

2006 Bosnia and Herzegovina Junior BMO TST, 1
Tags: number theory , diophantine , diophantine equation
. Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$

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

2022 Dutch IMO TST, 1
Tags: number theory , diophantine equation , diophantine
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 $c + d$.

PEN H Problems, 10
Tags: diophantine equation , modular arithmetic
Prove that there are unique positive integers $a$ and $n$ such that \[a^{n+1}-(a+1)^{n}= 2001.\]

2018 NZMOC Camp Selection Problems, 2
Tags: number theory , diophantine , diophantine equation
Find all pairs of integers $(a, b)$ such that $$a^2 + ab - b = 2018.$$

2021 Malaysia IMONST 1, 10
Tags: diophantine equation , diophantine , number theory
Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$

1980 IMO, 19
Tags: diophantine equation , algebra , equation , polynomial , number theory
Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

1977 Yugoslav Team Selection Test, Problem 2
Tags: diophantine equation , number theory
Determine all $6$-tuples $(p,q,r,x,y,z)$ where $p,q,r$ are prime, and $x,y,z$ natural numbers such that $p^{2x}=q^yr^z+1$.

2009 Thailand Mathematical Olympiad, 4
Tags: diophantine , diophantine equation , number theory
Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to $(m - n)^2 = kmn + m + n$.

1988 IMO Longlists, 14
Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

1988 Spain Mathematical Olympiad, 6
Tags: diophantine equation , parameterization , number theory
For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation $$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .

1954 Moscow Mathematical Olympiad, 262
Tags: diophantine equation , diophantine , number theory
Are there integers $m$ and $n$ such that $m^2 + 1954 = n^2$?

1981 Tournament Of Towns, (007) 1
Tags: number theory , diophantine equation , diophantine
Find all integer solutions to the equation $y^k = x^2 + x$, where $k$ is a natural number greater than $1$.

2019 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , diophantine , diophantine equation
Determine all positive integers $k$ for which there exist positive integers $n$ and $m, m\ge 2$, such that $3^k + 5^k = n^m$

2009 May Olympiad, 2
Tags: number theory , diophantine , diophantine equation
Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities. Remember that the number $1$ is not prime.

2017 Hanoi Open Mathematics Competitions, 2
Tags: diophantine equation , diophantine , number theory
How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

2022 Abelkonkurransen Finale, 1b
Tags: diophantine equation , number theory
Find all primes $p$ and positive integers $n$ satisfying \[n \cdot 5^{n-n/p} = p! (p^2+1) + n.\]

2013 Benelux, 4
Tags: modular arithmetic , quadratic , diophantine equation , number theory
a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers \[g^n - n\quad\text{ and }\quad g^{n+1} - (n + 1).\] b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers \[g^n - n^2\quad\text{ and }g^{n+1} - (n + 1)^2.\]

2019 Durer Math Competition Finals, 15
Tags: diophantine , diophantine equation , number theory , sum
The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?