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

2014 Polish MO Finals, 2
Tags: diophantine equation , number theory
Find all pairs $(x,y)$ of positive integers that satisfy $$2^x+17=y^4$$.

1986 Poland - Second Round, 4
Tags: number theory , perfect square , diophantine , diophantine equation
Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation $$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$ Prove that $ x + y $ is the square of a natural number.

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

PEN H Problems, 68
Tags: diophantine equation , ratio , calculus
Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.

2005 Junior Tuymaada Olympiad, 3
Tags: diophantine equation , algebra , system of equations
Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?

PEN H Problems, 38
Tags: diophantine equation , abstract algebra
Suppose that $p$ is an odd prime such that $2p+1$ is also prime. Show that the equation $x^{p}+2y^{p}+5z^{p}=0$ has no solutions in integers other than $(0,0,0)$.

2010 District Olympiad, 4
Tags: diophantine equation , number theory , diophantine
Find all non negative integers $(a, b)$ such that $$a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|}.$$

2010 Saudi Arabia BMO TST, 4
Tags: diophantine equation , diophantine , number theory
Find all primes $p, q$ satisfying the equation $2p^q - q^p = 7.$

1995 Austrian-Polish Competition, 7
Tags: number theory , squarefree , perfect square , diophantine , diophantine equation
Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.

2025 Korea - Final Round, P6
Tags: number theory , diophantine equation
Positive integers $a, b$ satisfy both of the following conditions. For a positive integer $m$, if $m^2 \mid ab$, then $m = 1$. There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$. Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$, for each integer $n$.

2002 Croatia National Olympiad, Problem 3
Tags: diophantine equation , number theory
Find all triples $(x,y,z)$ of natural numbers that verify the equation $$2x^2y^2+2y^2z^2+2z^2x^2-x^4-y^4-z^4=576.$$

2016 Baltic Way, 3
Tags: number theory , diophantine equation
For which integers $n = 1, \ldots , 6$ does the equation $$a^n + b^n = c^n + n$$ have a solution in integers?

2017 Latvia Baltic Way TST, 14
Tags: diophantine equation , number theory , diophantine
Can you find three natural numbers $a, b, c$ whose greatest common divisor is $1$ and which satisfy the equality $$ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) ?$$

2018 Pan-African Shortlist, N5
Tags: number theory , diophantine equation
Find all quadruplets $(a, b, c, d)$ of positive integers such that \[ \left( 1 + \frac{1}{a} \right) \left( 1 + \frac{1}{b} \right) \left( 1 + \frac{1}{c} \right) \left( 1 + \frac{1}{d} \right) = 4. \]

2022 Bulgarian Autumn Math Competition, Problem 9.3
Tags: diophantine equation , factorial , number theory
Find all the pairs of natural numbers $(a, b),$ such that \[a!+1=(a+1)^{(2^b)}\]

2011 Baltic Way, 20
Tags: diophantine equation , number theory
An integer $n\ge 1$ is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers $n$ such that there are exactly two balanced numbers among $n,n+1,n+2$ and $n+3$.

1994 ITAMO, 2
Tags: number theory , diophantine equation
solve this diophantine equation y^2 = x^3 - 16

1983 Swedish Mathematical Competition, 3
Tags: even , number theory , system of equations , system , diophantine , diophantine equation
The systems of equations \[\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. \] has a solution in positive integers $x_i$. Show that $n$ must be even.

2022 Dutch IMO TST, 1
Tags: number theory , diophantine equation , diophantine
Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

2011 Croatia Team Selection Test, 4
Tags: number theory , relatively prime , diophantine equation , equation
Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

2022 Argentina National Olympiad, 5
Tags: number theory , diophantine equation , diophantine
Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$

1964 Dutch Mathematical Olympiad, 3
Tags: number theory , diophantine , diophantine equation , perfect square
Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.

2011 NZMOC Camp Selection Problems, 1
Tags: diophantine , diophantine equation , number theory
Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$ .

2013 Singapore Senior Math Olympiad, 2
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(m,n)$ such that $m^3-n^3=2mn +8$

2024 Euler Olympiad, Round 2, 1
Tags: euler , diophantine equation , number theory
Find all triples $(a, b,c) $ of positive integers, such that: \[ a! + b! = c!! \] where $(2k)!! = 2 \cdot 4 \cdot \ldots \cdot (2k)$ and $ (2k + 1)!! = 1 \cdot 3 \cdot \ldots \cdot (2k+1).$ [i]Proposed by Stijn Cambie, Belgium [/i]