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.

AND:
OR:
NO:

Found problems: 436

2004 Singapore MO Open, 2
Tags: number theory , diophantine , diophantine equation
Find the number of ordered pairs $(a, b)$ of integers, where $1 \le a, b \le 2004$, such that $x^2 + ax + b = 167 y$ has integer solutions in $x$ and $y$. Justify your answer.

2002 Abels Math Contest (Norwegian MO), 1b
Tags: diophantine , diophantine equation , number theory
Find all integers $c$ such that the equation $(2a+b) (2b+a) =5^c$ has integer solutions.

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

2023 Puerto Rico Team Selection Test, 1
Tags: number theory , diophantine equation , diophantine
Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

2013 Czech-Polish-Slovak Junior Match, 1
Tags: diophantine equation , diophantine , number theory , radical
Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

2010 Saudi Arabia Pre-TST, 4.1
Tags: diophantine , number theory , system , system of equations
Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$

VI Soros Olympiad 1999 - 2000 (Russia), 9.2
Tags: number theory , diophantine equation , diophantine
Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds

1999 Singapore Team Selection Test, 1
Tags: diophantine equation , diophantine , number theory
Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.

2002 Denmark MO - Mohr Contest, 3
Tags: algebra , number theory , system of equations , diophantine
Two positive integers have the sum $2002$. Can $2002$ divide their product?

2018 Hanoi Open Mathematics Competitions, 6
Tags: algebra , diophantine , diophantine equation
Nam spent $20$ dollars for $20$ stationery items consisting of books, pens and pencils. Each book, pen, and pencil cost $3$ dollars, $1.5$ dollars and $0.5$ dollar respectively. How many dollars did Nam spend for books?

1990 All Soviet Union Mathematical Olympiad, 523
Tags: floor function , diophantine , number theory
Find all integers $n$ such that $\left[\frac{n}{1!}\right] + \left[\frac{n}{2!}\right] + ... + \left[\frac{n}{10!}\right] = 1001$.

2012 Dutch IMO TST, 3
Tags: diophantine equation , diophantine , number theory
Determine all pairs $(x, y)$ of positive integers satisfying $x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.

2002 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine , diophantine equation , divisor
Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$. a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$. b) Find the number of all non-negative solutions of the above equation.

1996 Estonia National Olympiad, 1
Tags: prime , diophantine equation , diophantine , number theory
Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

1994 Abels Math Contest (Norwegian MO), 2a
Tags: diophantine , number theory , diophantine equation , prime
Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

1991 All Soviet Union Mathematical Olympiad, 535
Tags: system of equations , diophantine , number theory
Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$

1988 All Soviet Union Mathematical Olympiad, 471
Tags: diophantine , diophantine equation , exponential , algebra , number theory
Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.

2022 Azerbaijan National Mathematical Olympiad, 3
Tags: diophantine equation , diophantine , number theory
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.

2020 JBMO Shortlist, 6
Tags: shortlist , number theory , diophantine
Are there any positive integers $m$ and $n$ satisfying the equation $m^3 = 9n^4 + 170n^2 + 289$ ?

1995 ITAMO, 6
Tags: diophantine equation , diophantine , number theory
Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$

2019 Hanoi Open Mathematics Competitions, 7
Tags: number theory , diophantine , diophantine equation
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 $n$

2014 Dutch Mathematical Olympiad, 4
Tags: number theory , prime , divisible , inequality , diophantine
A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if - $p$ is an odd prime number, - $a, b$, and $c$ are distinct and - $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$. a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ . b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3} $

2011 Cuba MO, 2
Tags: diophantine , diophantine equation , number theory
Determine all the integer solutions of the equation $3x^4-2024y+1= 0$.

2005 Austria Beginners' Competition, 2
Tags: number theory , diophantine , inequalities
Determine the number of integer pairs $(x, y)$ such that $(|x| - 2)^2 + (|y| - 2)^2 < 5$ .

2013 District Olympiad, 1
Tags: number theory , diophantine equation , diophantine
Find all triples of integers $(x, y, z)$ such that $$x^2 + y^2 + z^2 = 16(x + y + z).$$