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

2019 Austrian Junior Regional Competition, 1
Tags: number theory , diophantine equation , diophantine
Let $x$ and $y$ be integers with $x + y \ne 0$. Find all pairs $(x, y)$ such that $$\frac{x^2 + y^2}{x + y}= 10.$$ (Walther Janous)

2022 Indonesia TST, N
Tags: integer , quadratic , greatest common divisor , number theory , diophantine
For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation: \[ x^2 - xy + y^2 = n. \] a) Determine $f(2022)$. b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.

2008 Hanoi Open Mathematics Competitions, 3
Tags: number theory , diophantine , diophantine equation
Show that the equation $x^2 + 8z = 3 + 2y^2$ has no solutions of positive integers $x, y$ and $z$.

1966 Polish MO Finals, 1
Tags: diophantine , diophantine equation , number theory
Solve in integers the equation $$x^4 +4y^4 = 2(z^4 +4u^4)$$

2008 Dutch Mathematical Olympiad, 2
Tags: number theory , diophantine equation , diophantine
Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$.

1965 Kurschak Competition, 1
Tags: number theory , diophantine , inequalities
What integers $a, b, c$ satisfy $a^2 + b^2 + c^2 + 3 < ab + 3b + 2c$ ?

1926 Eotvos Mathematical Competition, 1
Tags: diophantine , diophantine equation , system of equations , system , number theory
Prove that, if $a$ and $b$ are given integers, the system of equatìons $$x + y + 2z + 2t = a$$ $$2x - 2y + z- t = b$$ has a solution in integers $x, y,z,t$.

2002 Singapore Senior Math Olympiad, 3
Tags: diophantine , radical , number theory , diophantine equation
Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that $$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$ (As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)

I Soros Olympiad 1994-95 (Rus + Ukr), 9.5
Tags: number theory , diophantine equation , diophantine
Find the triplets of natural numbers $(p,q,r)$ that satisfy the equality $$\frac{1}{p}+\frac{q}{q^r -1}=1.$$

2016 Latvia Baltic Way TST, 17
Tags: number theory , diophantine equation , diophantine , prime
Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?

2013 Balkan MO Shortlist, N6
Tags: number theory , diophantine equation , diophantine , prime
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$

2014 Contests, 1
Tags: diophantine equation , diophantine , number theory
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

2009 Postal Coaching, 2
Tags: diophantine , diophantine equation , number theory
Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.

2014 Belarus Team Selection Test, 4
Tags: number theory , diophantine , diophantine equation
Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$. (I. Gorodnin)

2008 Regional Olympiad of Mexico Center Zone, 1
Tags: diophantine equation , diophantine , number theory
Find all pairs of integers $ a, b $ that satisfy $a ^2-3a = b ^3-2$.

2010 Federal Competition For Advanced Students, P2, 2
Tags: number theory , diophantine equation , diophantine
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

1999 Swedish Mathematical Competition, 3
Tags: number theory , diophantine equation , diophantine
Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.

1904 Eotvos Mathematical Competition, 2
Tags: number theory , diophantine
If a is a natural number, show that the number of positive integral solutions of the indeterminate equation $$x_1 + 2x_2 + 3x_3 + ... + nx_n = a \ \ (1) $$ is equal to the number of non-negative integral solutions of $$y_1 + 2y_2 + 3y_3 + ... + ny_n = a - \frac{n(n + 1)}{2} \ \ (2)$$ [By a solution of equation (1), we mean a set of numbers $\{x_1, x_2,..., x_n\}$ which satisfies equation (1)].

2006 Abels Math Contest (Norwegian MO), 3
Tags: integer , number theory , diophantine , rational
(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.

1997 Denmark MO - Mohr Contest, 4
Tags: number theory , diophantine , diophantine equation
Find all pairs $x,y$ of natural numbers that satisfy the equation $$x^2-xy+2x-3y=1997$$

2013 Danube Mathematical Competition, 3
Tags: diophantine equation , diophantine , exponential equation , exponential , number theory
Determine the natural numbers $m,n$ such as $85^m-n^4=4$

2010 Saudi Arabia BMO TST, 1
Tags: diophantine , diophantine equation , number theory
Find all triples $(x,y,z)$ of positive integers such that $3^x + 4^y = 5^z$.

1997 Estonia National Olympiad, 1
Tags: diophantine , diophantine equation , number theory
Find: a) Any quadruple of positive integers $(a, k, l, m)$ such that $a^k = a^l + a^m,$ b) Any quintuple of positive integers $(a, k, l, m, n)$ for which $a^k = a^l + a^m+a^n$

1975 Chisinau City MO, 92
Tags: number theory , diophantine equation , diophantine
Solve in natural numbers the equation $x^2-y^2=105$.

2015 Saudi Arabia Pre-TST, 1.3
Tags: diophantine equation , diophantine , number theory
Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$. (Malik Talbi)