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

1967 IMO Shortlist, 3
Tags: equation , algebra , summation , diophantine equation , polynomial
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

2019 Chile National Olympiad, 3
Tags: number theory , diophantine equation , diophantine
Find all solutions $x,y,z$ in the positive integers of the equation $$3^x -5^y = z^2$$

2001 Moldova National Olympiad, Problem 8
Tags: diophantine equation , number theory
Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$. (a) Find all of the elements of $S$. (b) Find all $x$ such that both $x$ and $x+1$ are in $S$.

2014 VTRMC, Problem 5
Tags: diophantine equation , number theory
Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$.

1995 Canada National Olympiad, 4
Tags: diophantine equation , number theory
Let $n$ be a constant positive integer. Show that for only non-negative integers $k$, the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$.

2015 Kazakhstan National Olympiad, 2
Tags: diophantine equation , number theory
Solve in positive integers $x^yy^x=(x+y)^z$

PEN H Problems, 83
Tags: diophantine equation
Find all pairs $(a, b)$ of positive integers such that \[(\sqrt[3]{a}+\sqrt[3]{b}-1 )^{2}= 49+20 \sqrt[3]{6}.\]

2017 Saudi Arabia Pre-TST + Training Tests, 7
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(x, y)$ such that $y^3 = 8x^6 + 2x^3 y -y^2$.

1997 India National Olympiad, 2
Tags: special factorizations , number theory , quadratic , diophantine equation , modular arithmetic
Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]

2018 Thailand Mathematical Olympiad, 6
Tags: diophantine , number theory , diophantine equation
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

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

1997 Slovenia National Olympiad, Problem 1
Tags: diophantine equation , number theory
Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.

1985 IMO Longlists, 82
Tags: algebra , diophantine equation , polynomial
Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

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 $x \leq y$.

1967 IMO Shortlist, 2
Tags: algebra , diophantine equation , parametric equation , root , polynomial
The equation \[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\] is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$

1986 Greece Junior Math Olympiad, 1
Tags: diophantine equation , diophantine , number theory
Find all pairs of integers $(x,y)$ such that $$(x+1)(y+1)(x+y)(x^2+y^2)=16x^2y^2$$

2016 Costa Rica - Final Round, N3
Tags: number theory , diophantine equation , diophantine
Find all natural values of $n$ and $m$, such that $(n -1)2^{n - 1} + 5 = m^2 + 4m$.

2022 VTRMC, 3
Tags: diophantine equation , number theory
Find all positive integers $a, b, c, d,$ and $n$ satisfying $n^a + n^b + n^c = n^d$ and prove that these are the only such solutions.

2017 OMMock - Mexico National Olympiad Mock Exam, 4
Tags: diophantine equation , number theory
Show that the equation $$a^2b=2017(a+b)$$ has no solutions for positive integers $a$ and $b$. [i]Proposed by Oriol Solé[/i]

2018 Philippine MO, 4
Tags: diophantine equation , number theory
Determine all ordered pairs $(x, y)$ of nonnegative integers that satisfy the equation $$3x^2 + 2 \cdot 9^y = x(4^{y+1}-1).$$

2004 Regional Competition For Advanced Students, 1
Tags: number theory , diophantine equation
Determine all integers $ a$ and $ b$, so that $ (a^3\plus{}b)(a\plus{}b^3)\equal{}(a\plus{}b)^4$

1998 Poland - Second Round, 4
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(x,y)$ satisfying $x^2 +3y^2 = 1998x$.

2022 JBMO Shortlist, N3
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 $$p^a = 1 + 5q^b.$$

2014 NIMO Problems, 15
Tags: diophantine equation
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. [i]Proposed by Lewis Chen[/i]

2012 India Regional Mathematical Olympiad, 5
Tags: prime , number theory , diophantine equation , positive integer , diophantine
Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.