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.

AND:
OR:
NO:

Found problems: 916

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 Costa Rica - Final Round, N1
Tags: number theory , diophantine equation , diophantine
Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.

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 Junior Macedonian Mathematical Olympiad, P1
Tags: number theory , diophantine equation , factorial
Determine all positive integers $a$, $b$ and $c$ which satisfy the equation $$a^2+b^2+1=c!.$$ [i]Proposed by Nikola Velov[/i]

2005 Austrian-Polish Competition, 4
Tags: fermat's little theorem , prime , number theory , diophantine equation , perfect square
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 \[\frac{a^p - a}{p}=b^2.\]

2017 OMMock - Mexico National Olympiad Mock Exam, 4
Tags: number theory , diophantine equation
Show that the equation $$a^2b=2017(a+b)$$ has no solutions for positive integers $a$ and $b$. [i]Proposed by Oriol Solé[/i]

2024 Kyiv City MO Round 1, Problem 5
Tags: number theory , diophantine equation
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$. [i]Proposed by Oleksii Masalitin[/i]

2021 Grand Duchy of Lithuania, 4
Tags: number theory , diophantine equation , diophantine
A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers. a) Find at least one primary triplet. b) Show that there are infinitely many primary triplets.

2005 iTest, 34
Tags: diophantine , diophantine equation , perfect square , number theory
If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.

1980 All Soviet Union Mathematical Olympiad, 288
Tags: diophantine , diophantine equation , number theory
Are there three integers $x,y,z$, such that $x^2 + y^3 = z^4$?

1996 Estonia National Olympiad, 1
Tags: sum , product , diophantine equation , diophantine , number theory
Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.

2008 Hanoi Open Mathematics Competitions, 2
Tags: number theory , diophantine equation , diophantine
Find all pairs $(m, n)$ of positive integers such that $m^2 + 2n^2 = 3(m + 2n)$

2010 NZMOC Camp Selection Problems, 3
Tags: diophantine , diophantine equation , number theory
Let $p$ be a prime number. Find all pairs $(x, y)$ of positive integers such that $x^3 + y^3 - 3xy = p -1$.

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.

2005 Thailand Mathematical Olympiad, 6
Tags: number theory , diophantine equation , diophantine
Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.

2002 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine , diophantine equation
Find all positive integers $a, b,c,d$ such that $a + b + c + d - 3 = ab + cd$.

2009 Korea Junior Math Olympiad, 8
Tags: prime number , diophantine equation , number theory
Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?

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: $p^3 + 107 = 2q (17q + 24)$ .

2009 Croatia Team Selection Test, 4
Tags: quadratic , diophantine equation , number theory
Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

2012 Greece JBMO TST, 2
Tags: number theory , least common multiple , coprime , diophantine equation
Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .

2016 Junior Balkan Team Selection Tests - Moldova, 6
Tags: diophantine , diophantine equation , number theory
Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.

2008 Greece JBMO TST, 4
Tags: number theory , diophantine equation , product , sum , integer
Product of two integers is $1$ less than three times of their sum. Find those integers.

2015 India PRMO, 11
Tags: number theory , algebra , diophantine equation , integer
$11.$ Let $a,$ $b,$ and $c$ be real numbers such that $a-7b+8c=4.$ and $8a+4b-c=7.$ What is the value of $a^2-b^2+c^2 ?$

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

2017 Thailand TSTST, 4
Tags: number theory , diophantine equation
Suppose that $m, n, k$ are positive integers satisfying $$3mk=(m+3)^n+1.$$ Prove that $k$ is odd.