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

2017 Danube Mathematical Olympiad, 4
Tags: number theory , diophantine equation
Determine all triples of positive integers $(x,y,z)$ such that $x^4+y^4 =2z^2$ and $x$ and $y$ are relatively prime.

PEN H Problems, 75
Tags: diophantine equation
Let $a,b$, and $x$ be positive integers such that $x^{a+b}=a^b{b}$. Prove that $a=x$ and $b=x^{x}$.

PEN H Problems, 13
Tags: diophantine equation , modular arithmetic , geometry , 3d geometry
Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]

2015 Greece National Olympiad, 1
Tags: diophantine equation , number theory , prime number
Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$

PEN H Problems, 21
Tags: diophantine equation , quadratic
Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.

2008 Hanoi Open Mathematics Competitions, 4
Tags: number theory , diophantine equation , diophantine
Find all pairs $(m,n)$ of positive integers such that $m^2 + n^2 = 3(m + n)$.

Revenge EL(S)MO 2024, 3
Tags: square , diophantine equation , 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]

1995 Israel Mathematical Olympiad, 4
Tags: diophantine , diophantine equation , number theory
Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$.

2019 Cono Sur Olympiad, 4
Tags: number theory , diophantine equation , prime
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.

PEN H Problems, 65
Tags: diophantine equation , modular arithmetic , number theory
Determine all pairs $(x, y)$ of integers such that \[(19a+b)^{18}+(a+b)^{18}+(19b+a)^{18}\] is a nonzero perfect square.

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]

2017 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , diophantine , diophantine equation
Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.

2017 QEDMO 15th, 1
Tags: diophantine , diophantine equation , number theory
Find all integers $x, y, z$ satisfy the $x^4-10y^4 + 3z^6 = 21$.

2014 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine equation , diophantine
Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$

2014 Saudi Arabia GMO TST, 1
Tags: number theory , diophantine equation , diophantine
Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$

2025 6th Memorial "Aleksandar Blazhevski-Cane", P1
Tags: number theory , prime number , diophantine equation
Determine all triples of prime numbers $(p, q, r)$ that satisfy \[p2^q + r^2 = 2025.\] Proposed by [i]Ilija Jovcevski[/i]

2004 Bosnia and Herzegovina Junior BMO TST, 1
Tags: diophantine equation , diophantine , number theory
In the set of integers solve the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}$, where $p$ is a prime number.

2020-IMOC, N2
Tags: diophantine equation , number theory
Find all positive integers $N$ such that the following holds: There exist pairwise coprime positive integers $a,b,c$ with $$\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.$$

2011 China Girls Math Olympiad, 1
Tags: diophantine equation , prime factorization , number theory
Find all positive integers $n$ such that the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ has exactly $2011$ positive integer solutions $(x,y)$ where $x \leq y$.

2019 Silk Road, 3
Tags: number theory , eulers function , relatively prime , diophantine equation
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 $)

2016 Hanoi Open Mathematics Competitions, 5
Tags: diophantine equation , diophantine , number theory
There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to (A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.

PEN H Problems, 50
Tags: diophantine equation , modular arithmetic
Show that the equation $y^{2}=x^{3}+2a^{3}-3b^2$ has no solution in integers if $ab \neq 0$, $a \not\equiv 1 \; \pmod{3}$, $3$ does not divide $b$, $a$ is odd if $b$ is even, and $p=t^2 +27u^2$ has a solution in integers $t,u$ if $p \vert a$ and $p \equiv 1 \; \pmod{3}$.

2019 District Olympiad, 1
Tags: diophantine equation , algebra , number theory
Determine the integers $a, b, c$ for which $$\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}$$

2017 Istmo Centroamericano MO, 4
Tags: diophantine equation , number theory , diophantine , system of equations , inequalities
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.

2014 Cuba MO, 7
Tags: diophantine equation , diophantine , number theory
Find all pairs of integers $(a, b)$ that satisfy the equation $$(a + 1)(b- 1) = a^2b^2.$$