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

2016 NZMOC Camp Selection Problems, 7
Tags: diophantine , diophantine equation , number theory
Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.

2014 Contests, 3
Tags: diophantine equation , number theory
Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

2020 LIMIT Category 2, 15
Tags: diophantine equation , diophantine , limit , number theory
How many integer pairs $(x,y)$ satisfies $x^2+y^2=9999(x-y)$?

2019 Hanoi Open Mathematics Competitions, 7
Tags: diophantine equation , diophantine , number theory
Let $p$ and $q$ be odd prime numbers. Assume that there exists a positive integer $n$ such that $pq-1= n^3$. Express $p+q$ in terms of $n$

2016 Mathematical Talent Reward Programme, MCQ: P 6
Tags: diophantine equation , algebra
Number of solutions of the equation $3^x+4^x=8^x$ in reals is [list=1] [*] 0 [*] 1 [*] 2 [*] $\infty$ [/list]

2016 Regional Olympiad of Mexico Northeast, 1
Tags: diophantine , diophantine equation , number theory
Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that: $$a^3 + b^3 + c^3 = 2016$$

2022 Lusophon Mathematical Olympiad, 4
Tags: diophantine equation , number theory
How many integer solutions exist that satisfy this equation? $$x+4y-343\sqrt{x}-686\sqrt{y}+4\sqrt{xy}+2022=0$$.

2010 Federal Competition For Advanced Students, P2, 2
Tags: diophantine equation , number theory , diophantine
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

2021 Ecuador NMO (OMEC), 6
Tags: diophantine equation , number theory
Find all positive integers $a, b, c$ such that $ab+1$ and $c$ are coprimes and: $$a(ba+1)(ca^2+ba+1)=2021^{2021}$$

PEN H Problems, 6
Tags: number theory , diophantine equation
Show that there are infinitely many pairs $(x, y)$ of rational numbers such that $x^3 +y^3 =9$.

1989 IMO Shortlist, 4
Tags: polynomial , diophantine equation , rational roots , equation , algebra
Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

2022 Austrian Junior Regional Competition, 4
Tags: diophantine equation , diophantine , number theory
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]

2006 IMO, 4
Tags: diophantine equation , number theory
Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]

1989 IMO Longlists, 94
Tags: algebra , diophantine equation , linear equation , counting
Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation \[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\] where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]

2017 USA TSTST, 4
Tags: number theory , diophantine equation , factorial
Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$. [i]Proposed by Mark Sellke[/i]

1996 Poland - Second Round, 5
Tags: diophantine equation , diophantine , number theory
Find all integers $x,y$ such that $x^2(y-1)+y^2(x-1) = 1$.

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

2003 Chile National Olympiad, 2
Tags: diophantine , diophantine equation , number theory
Find all primes $p, q$ such that $p + q = (p-q)^3$.

2014 Peru IMO TST, 9
Tags: diophantine equation , number theory
Prove that for every positive integer $n$ there exist integers $a$ and $b,$ both greater than $1,$ such that $a ^ 2 + 1 = 2b ^ 2$ and $a - b$ is a multiple of $n.$

2013 Benelux, 4
Tags: quadratic , diophantine equation , modular arithmetic , 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 \[g^n - n^2\quad\text{ and }g^{n+1} - (n + 1)^2.\]

2024 Bangladesh Mathematical Olympiad, P1
Tags: number theory , diophantine equation , prime number , modular arithmetic
Find all prime numbers $p$ and $q$ such that\[p^3-3^q=10.\] [i]Proposed by Md. Fuad Al Alam[/i]

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

2010 ELMO Shortlist, 3
Tags: quadratic , diophantine equation , number theory
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

2017 Swedish Mathematical Competition, 2
Tags: diophantine , diophantine equation , number theory
Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$ \frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$

2010 ELMO Shortlist, 3
Tags: quadratic , diophantine equation , number theory
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]