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

2010 Saudi Arabia Pre-TST, 2.1
Tags: number theory , diophantine , system of equations , system
Find all triples $(x,y,z)$ of positive integers such that $$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$

2021 Malaysia IMONST 1, 12
Tags: diophantine , number theory , diophantine equation
Determine the number of positive integer solutions $(x,y, z)$ to the equation $xyz = 2(x + y + z)$.

2011 Grand Duchy of Lithuania, 3
Tags: diophantine , diophantine equation , number theory
Find all primes $p,q$ such that $p ^3-q^7=p-q$.

2019 Taiwan TST Round 2, 5
Tags: diophantine , number theory
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?

2016 Latvia Baltic Way TST, 19
Tags: diophantine , number theory , diophantine equation
Prove that for equation $$x^{2015} + y^{2015} = z^{2016}$$ there are infinitely many solutions where $x,y$ and $z$ are different natural numbers.

2009 IMAC Arhimede, 5
Tags: diophantine equation , diophantine , exponential equation , exponential , number theory
Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .

2010 Abels Math Contest (Norwegian MO) Final, 4a
Tags: diophantine equation , diophantine , number theory
Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.

2011 Junior Balkan Team Selection Tests - Moldova, 6
Tags: diophantine , diophantine equation , number theory
Find the sum of the numbers written with two digits $\overline{ab}$ for which the equation $3^{x + y} =3^x + 3^y + \overline{ab}$ has at least one solution $(x, y)$ in natural numbers.

2017 Thailand Mathematical Olympiad, 7
Tags: diophantine equation , diophantine , number theory
Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$. .

2014 Cuba MO, 7
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(a, b)$ that satisfy the equation $$(a + 1)(b- 1) = a^2b^2.$$

1998 Belarus Team Selection Test, 2
Tags: number theory , diophantine , perfect square , diophantine equation
a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer. b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?

2015 Dutch IMO TST, 2
Tags: diophantine equation , system of equations , diophantine , number theory
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

2018 Finnish National High School Mathematics Comp, 5
Tags: number theory , diophantine , diophantine equation
Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$ when $x$ and $y$ are non-negative integers.

2018 Hanoi Open Mathematics Competitions, 6
Tags: diophantine equation , diophantine , algebra
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?

2022 Dutch IMO TST, 1
Tags: diophantine equation , diophantine , number theory
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 $1$.

2018 Costa Rica - Final Round, N2
Tags: diophantine , diophantine equation , number theory
Determine all triples $(a, b, c)$ of nonnegative integers that satisfy: $$(c-1) (ab- b -a) = a + b-2$$

2011 Bundeswettbewerb Mathematik, 4
Tags: diophantine , diophantine equation , euclidean algorithm , number theory
Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$. Find all pairs $(a, b)$ for which $q^2 + r = 2011$.

2014 Dutch Mathematical Olympiad, 4
Tags: prime , diophantine , inequality , number theory , divisible
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} $

2015 Dutch Mathematical Olympiad, 4
Tags: number theory , prime , diophantine equation , diophantine
Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$

2014 Hanoi Open Mathematics Competitions, 10
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(x, y)$ satisfying the condition $12x^2 + 6xy + 3y^2 = 28(x + y)$.

2002 Junior Balkan Team Selection Tests - Romania, 2
Tags: diophantine , diophantine equation , number theory
Find all positive integers $a, b,c,d$ such that $a + b + c + d - 3 = ab + cd$.

1979 Bulgaria National Olympiad, Problem 1
Tags: diophantine equation , diophantine , number theory
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$. Daniel Harrer

2012 NZMOC Camp Selection Problems, 3
Tags: number theory , diophantine , diophantine equation
Find all triples of positive integers $(x, y, z)$ with $$\frac{xy}{z}+ \frac{yz}{x}+\frac{zx}{y}= 3$$

2000 Estonia National Olympiad, 3
Tags: diophantine , diophantine equation , number theory
Are there any (not necessarily positive) integers $m$ and $n$ such that a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ? b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$

2017 Hanoi Open Mathematics Competitions, 6
Tags: diophantine equation , prime , diophantine , number theory
Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime.