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

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

2013 USA Team Selection Test, 1
Tags: geometry , parallelogram , number theory , diophantine equation
Two incongruent triangles $ABC$ and $XYZ$ are called a pair of [i]pals[/i] if they satisfy the following conditions: (a) the two triangles have the same area; (b) let $M$ and $W$ be the respective midpoints of sides $BC$ and $YZ$. The two sets of lengths $\{AB, AM, AC\}$ and $\{XY, XW, XZ\}$ are identical $3$-element sets of pairwise relatively prime integers. Determine if there are infinitely many pairs of triangles that are pals of each other.

2022 Junior Balkan Mathematical Olympiad, 3
Tags: number theory , diophantine equation
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$

2022 Pan-American Girls' Math Olympiad, 2
Tags: number theory , diophantine equation
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 \[p^3+q^2=4r^2+45r+103.\]

2022 Turkey Team Selection Test, 1
Tags: modular arithmetic , diophantine equation , number theory
Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

1946 Moscow Mathematical Olympiad, 117
Tags: diophantine equation , number theory
Prove that for any integers $x$ and $y$ we have $x^5 + 3x^4y - 5x^3y^2 - 15x^2y^3 + 4xy^4 + 12y^5 \ne 33$.

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 $a^bb^cc^a = p$.

2004 Thailand Mathematical Olympiad, 11
Tags: diophantine , diophantine equation , number theory
Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$

2020 Abels Math Contest (Norwegian MO) Final, 2a
Tags: diophantine equation , diophantine , number theory
Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider $0$ to be a natural number.

2013 Irish Math Olympiad, 8
Tags: number theory , diophantine equation , divisor
Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satis ed by at least two pairs of integers $(x, y)$ with $1 < x \le y$.

2003 Federal Math Competition of S&M, Problem 1
Tags: number theory , diophantine equation
Find the number of solutions to the equation$$x_1^4+x_2^4+\ldots+x_{10}^4=2011$$in the set of positive integers.

PEN H Problems, 24
Tags: modular arithmetic , diophantine equation , 3d geometry , geometry
Prove that if $n$ is a positive integer such that the equation \[x^{3}-3xy^{2}+y^{3}=n.\] has a solution in integers $(x,y),$ then it has at least three such solutions. Show that the equation has no solutions in integers when $n=2891$.

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

2022 Nigerian Senior MO Round 2, Problem 6
Tags: number theory , diophantine equation
Let $k, l, m, n$ be positive integers. Given that $k+l+m+n=km=ln$, find all possible values of $k+l+m+n$.

1997 Tournament Of Towns, (548) 2
Tags: number theory , diophantine equation , diophantine
Prove that the equation $x^2 + y^2 - z^2 = 1997$ has infinitely many solutions in integers $x$, $y$ and $z$. (N Vassiliev)

1982 IMO Shortlist, 4
Tags: algebra , diophantine equation , polynomial , parametric equation , geometric progression
Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

2016 Belarus Team Selection Test, 3
Tags: number theory , diophantine equation
Solve the equation $2^a-5^b=3$ in positive integers $a,b$.

1965 Swedish Mathematical Competition, 2
Tags: number theory , diophantine equation , diophantine
Find all positive integers m, n such that $m^3 - n^3 = 999$.

2020 Tournament Of Towns, 4
Tags: diophantine equation , diophantine , number theory
For some integer n the equation $x^2 + y^2 + z^2 -xy -yz - zx = n$ has an integer solution $x, y, z$. Prove that the equation$ x^2 + y^2 - xy = n$ also has an integer solution $x, y$. Alexandr Yuran

1989 IMO Longlists, 50
Tags: number theory , system of equations , diophantine equation , additive number theory
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\] \[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$

2007 India IMO Training Camp, 2
Tags: modular arithmetic , diophantine equation , number theory
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$

2009 Swedish Mathematical Competition, 4
Tags: number theory , diophantine equation , diophantine
Determine all integers solutions of the equation $x + x^3 = 5y^2$.

1979 IMO Shortlist, 15
Tags: diophantine equation , algebra , system of equations , cauchy-schwarz inequality
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}.$

2010 IFYM, Sozopol, 5
Tags: greatest common divisor , diophantine equation , number theory
Let n is a natural number,for which $\sqrt{1+12n^2}$ is a whole number.Prove that $2+2\sqrt{1+12n^2}$ is perfect square.

2008 AIME Problems, 4
Tags: trigonometry , quadratic , special factorizations , diophantine equation
There exist unique positive integers $ x$ and $ y$ that satisfy the equation $ x^2 \plus{} 84x \plus{} 2008 \equal{} y^2$. Find $ x \plus{} y$.