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 Shortlist, 3
Tags: geometry , number theory , algebra , diophantine equation , pythagorean theorem
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 $y.$

2013 Swedish Mathematical Competition, 3
Tags: number theory , diophantine equation , diophantine
Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$

2012 Dutch Mathematical Olympiad, 3
Tags: number theory , diophantine equation , diophantine
Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$, for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.

2014 Stars Of Mathematics, 1
Tags: number theory , diophantine equation
Prove there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^3+y \mid x+y^3$. ([i]Dan Schwarz[/i])

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

PEN H Problems, 55
Tags: floor function , diophantine equation
Given that \[34! = 95232799cd96041408476186096435ab000000_{(10)},\] determine the digits $a, b, c$, and $d$.

1966 Polish MO Finals, 1
Tags: number theory , diophantine , diophantine equation
Solve in integers the equation $$x^4 +4y^4 = 2(z^4 +4u^4)$$

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

2016 Indonesia MO, 2
Tags: number theory , diophantine equation
Determine all triples of natural numbers $(a,b, c)$ with $b> 1$ such that $2^c + 2^{2016} = a^b$.

2021 China Girls Math Olympiad, 5
Tags: number theory , diophantine equation
Proof that if $4$ numbers (not necessarily distinct) are picked from $\{1, 2, \cdots, 20\}$, one can pick $3$ numbers among them and can label these $3$ as $a, b, c$ such that $ax \equiv b \;(\bmod\; c)$ has integral solutions.

2018 Puerto Rico Team Selection Test, 1
Tags: number theory , diophantine , diophantine equation
Find all pairs $(a, b)$ of positive integers that satisfy the equation $a^2 -3 \cdot 2^b = 1$.

2017 Junior Balkan Team Selection Tests - Moldova, Problem 1
Tags: number theory , diophantine equation
Find all natural numbers $x,y$ such that $$x^5=y^5+10y^2+20y+1.$$

2023 Israel National Olympiad, P2
Tags: number theory , diophantine equation
The non-negative integers $x,y$ satisfy $\sqrt{x}+\sqrt{x+60}=\sqrt{y}$. Find the largest possible value for $x$.

2005 Austrian-Polish Competition, 4
Tags: number theory , perfect square , fermat's little theorem , prime , diophantine equation
Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

2021 OMpD, 3
Tags: number theory , diophantine equation
Determine all pairs of integer numbers $(x, y)$ such that: $$\frac{(x - y)^2}{x + y} = x - y + 6$$

1990 Austrian-Polish Competition, 2
Tags: number theory , diophantine , diophantine equation
Find all solutions in positive integers to $a^A = b^B = c^C = 1990^{1990}abc$, where $A = b^c, B = c^a, C = a^b$.

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

2018 JBMO Shortlist, NT2
Tags: number theory , diophantine equation
Find all ordered pairs of positive integers $(m,n)$ such that : $125*2^n-3^m=271$

2020 BMT Fall, 9
Tags: algebra , diophantine equation
There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$.

2009 VTRMC, Problem 6
Tags: diophantine equation , number theory
Let $n$ be a nonzero integer. Prove that $n^4-7n^2+1$ can never be a perfect square.

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

2014 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine , diophantine equation
Solve, in the positive integers, the equation $5^m + n^2 = 3^p$ .

PEN H Problems, 85
Tags: inequalities , diophantine equation
Find all integer solutions to $2(x^5 +y^5 +1)=5xy(x^2 +y^2 +1)$.

2015 Dutch IMO TST, 2
Tags: number theory , diophantine , system of equations , diophantine equation
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

2009 China Girls Math Olympiad, 1
Tags: inequalities , diophantine equation , number theory
Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc\equal{}2009(a\plus{}b\plus{}c).$