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

1986 IMO Longlists, 76
Tags: number theory , geometry , diophantine equation , pythagorean theorem , algebra
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

PEN H Problems, 41
Tags: relatively prime , number theory , 3d geometry , geometry , diophantine equation
Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.

2002 Federal Math Competition of S&M, Problem 3
Tags: diophantine equation , number theory
Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.

1984 All Soviet Union Mathematical Olympiad, 379
Tags: exponential , algebra , diophantine equation , diophantine , number theory
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

PEN H Problems, 69
Tags: relatively prime , number theory , diophantine equation
Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.

KoMaL A Problems 2022/2023, A. 841
Tags: number theory , diophantine equation
Find all non-negative integer solutions of the equation $2^a+p^b=n^{p-1}$, where $p$ is a prime number. Proposed by [i]Máté Weisz[/i], Cambridge

2018 Korea Junior Math Olympiad, 4
Tags: combinatorics , diophantine equation , number theory
For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+2y+2z+3w=n$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that (i) $a+b+c+d=n$ (ii) $a \ge b \ge d$ (iii) $a \ge c \ge d$ Prove that for all $n$, $p(n) = q(n)$.

1998 Slovenia National Olympiad, Problem 1
Tags: number theory , diophantine equation
Find all integers $x,y$ which satisfy the equation $xy=20-3x+y$.

Revenge EL(S)MO 2024, 3
Tags: diophantine equation , square , algebra
Find all solutions to \[ (abcde)^2 = a^2+b^2+c^2+d^2+e^2+f^2. \] in integers. Proposed by [i]Seongjin Shim[/i]

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

2020 Switzerland - Final Round, 5
Tags: number theory , diophantine , diophantine equation , factorial
Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$

1991 Chile National Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Determine all nonnegative integer solutions of the equation $2^x-2^y = 1$

2013 German National Olympiad, 1
Tags: diophantine equation , number theory
Find all positive integers $n$ such that $n^{2}+2^{n}$ is square of an integer.

2013 Costa Rica - Final Round, N1
Tags: number theory , diophantine , diophantine equation
Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.

2007 Moldova Team Selection Test, 2
Tags: diophantine equation , search , polynomial , number theory , algebra
Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.

2016 Greece National Olympiad, 1
Tags: diophantine equation , number theory
Find all triplets of nonnegative integers $(x,y,z)$ and $x\leq y$ such that $x^2+y^2=3 \cdot 2016^z+77$

PEN H Problems, 89
Tags: diophantine equation
Prove that the number $99999+111111\sqrt{3}$ cannot be written in the form $(A+B\sqrt{3})^2$, where $A$ and $B$ are integers.

2012 Dutch BxMO/EGMO TST, 3
Tags: number theory , diophantine equation
Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

1987 Poland - Second Round, 5
Tags: diophantine , number theory , diophantine equation
Determine all prime numbers $ p $ and natural numbers $ x, y $ for which $ p^x-y^3 = 1 $.

2008 Bulgarian Autumn Math Competition, Problem 10.3
Tags: diophantine equation , number theory
Find all natural numbers $x,y,z$, such that $7^{x}+13^{y}=2^{z}$.

2022 Pan-American Girls' Math Olympiad, 2
Tags: diophantine equation , number theory
Find all ordered triplets $(p,q,r)$ of positive integers such that $p$ and $q$ are two (not necessarily distinct) primes, $r$ is even, and \[p^3+q^2=4r^2+45r+103.\]

2019 Silk Road, 3
Tags: number theory , eulers function , diophantine equation , relatively prime
Find all pairs of $ (a, n) $ natural numbers such that $ \varphi (a ^ n + n) = 2 ^ n. $ ($ \varphi (n) $ is the Euler function, that is, the number of integers from $1$ up to $ n $, relative prime to $ n $)

2005 Estonia Team Selection Test, 3
Tags: number theory , diophantine equation , diophantine
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.

2007 Argentina National Olympiad, 1
Tags: diophantine equation , diophantine , number theory
Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.

1979 IMO, 2
Tags: cauchy-schwarz inequality , algebra , system of equations , diophantine equation
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$