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: 436

2004 Austria Beginners' Competition, 2
Tags: number theory , inequalities , diophantine
For what pairs of integers $(x,y)$ does the inequality $x^2+5y^2-6\leq \sqrt{(x^2-2)(y^2-0.04)}$ hold?

2021 Malaysia IMONST 1, 10
Tags: diophantine equation , diophantine , number theory
Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$

2021 Junior Balkan Team Selection Tests - Moldova, 7
Tags: diophantine , number theory
Determine all pairs of integer numbers $(a, b)$ that satisfy the relation: $$(a + b)(a + b + 6) = 34 \cdot 3^{|2a-b|}- 7$$

2009 Thailand Mathematical Olympiad, 4
Tags: diophantine , diophantine equation , number theory
Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to $(m - n)^2 = kmn + m + n$.

1977 Swedish Mathematical Competition, 3
Tags: diophantine , system of equations , system , number theory
Show that the only integral solution to \[\left\{ \begin{array}{l} xy + yz + zx = 3n^2 - 1\\ x + y + z = 3n \\ \end{array} \right. \] with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.

1954 Moscow Mathematical Olympiad, 262
Tags: diophantine equation , diophantine , number theory
Are there integers $m$ and $n$ such that $m^2 + 1954 = n^2$?

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

2019 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , diophantine , diophantine equation
Determine all positive integers $k$ for which there exist positive integers $n$ and $m, m\ge 2$, such that $3^k + 5^k = n^m$

2009 May Olympiad, 2
Tags: number theory , diophantine , diophantine equation
Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities. Remember that the number $1$ is not prime.

2017 Hanoi Open Mathematics Competitions, 2
Tags: diophantine equation , diophantine , number theory
How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

2019 Durer Math Competition Finals, 15
Tags: diophantine , diophantine equation , number theory , sum
The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?

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

1972 Swedish Mathematical Competition, 1
Tags: diophantine , system of equations , system , number theory
Find the largest real number $a$ such that \[\left\{ \begin{array}{l} x - 4y = 1 \\ ax + 3y = 1\\ \end{array} \right. \] has an integer solution.

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.

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)

1948 Moscow Mathematical Olympiad, 148
Tags: diophantine , exponential , algebra , number theory
a) Find all positive integer solutions of the equation $x^y = y^x$ ($x \ne y$). b) Find all positive rational solutions of the equation $x^y = y^x$ ($x \ne y$).

2019 Azerbaijan IMO TST, 3
Tags: number theory , diophantine
Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2006 Alexandru Myller, 1
Tags: diophantine , number theory
Find a countable family of natural solutions to $ \frac{1}{a} +\frac{1}{b} +\frac{1}{ab}=\frac{1}{c} . $

2024 India Iran Friendly Math Competition, 4
Tags: number theory , diophantine
Prove that there are no integers $x, y, z$ satisfying the equation $$x^2+y^2-z^2=xyz-2.$$ [i]Proposed by Navid Safaei[/i]

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

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

2008 Thailand Mathematical Olympiad, 7
Tags: diophantine , diophantine equation , lcm , gcd
Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.

2021 Austrian MO Beginners' Competition, 4
Tags: diophantine , number theory
Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$. Prove that $m> p$. (Karl Czakler)