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

2007 Argentina National Olympiad, 1
Tags: number theory , diophantine equation , diophantine
Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.

1984 Poland - Second Round, 1
Tags: number theory , diophantine , diophantine equation
For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.

2010 Grand Duchy of Lithuania, 2
Tags: number theory , diophantine , diophantine equation
Find all positive integers $n$ for which there are distinct integer numbers $a_1, a_2, ... , a_n$ such that $$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}$$

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$.

1982 Austrian-Polish Competition, 7
Tags: diophantine , diophantine equation , system of equations , number theory
Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties: (i) $x^y = a^b = c^d$ and $x > a > c$ (ii) $z = ab = cd$ (iii) $x + y = a + b$.

1948 Moscow Mathematical Olympiad, 154
Tags: diophantine , number theory , inequalities , combinatorics
How many different integer solutions to the inequality $|x| + |y| < 100$ are there?

2013 Junior Balkan Team Selection Tests - Romania, 1
Tags: diophantine , diophantine equation , number theory
Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$ Irish Mathematical Olympiad

2016 Abels Math Contest (Norwegian MO) Final, 2b
Tags: factorial , diophantine equation , diophantine , number theory
Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$

2016 Balkan MO Shortlist, N3
Tags: number theory , diophantine equation , diophantine
Find all the integer solutions $(x,y,z)$ of the equation $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$,

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

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

2019 Peru EGMO TST, 1
Tags: diophantine equation , diophantine , number theory , prime
Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.

2001 Croatia Team Selection Test, 3
Tags: diophantine equation , diophantine , number theory
Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

2019 Philippine TST, 2
Tags: number theory , diophantine
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?

2000 Kazakhstan National Olympiad, 4
Tags: diophantine equation , number theory , diophantine
Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $

2020 Austrian Junior Regional Competition, 4
Tags: diophantine equation , diophantine , number theory
Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$. (Richard Henner)

1995 Argentina National Olympiad, 2
Tags: number theory , diophantine equation , diophantine
For each positive integer $n$ let $p(n)$ be the number of ordered pairs $(x,y)$ of positive integers such that$$\dfrac{1}{x}+\dfrac{1}{y} =\dfrac{1}{n}.$$For example, for $n=2$ the pairs are $(3,6),(4,4),(6,3)$. Therefore $p(2)=3$. a) Determine $p(n)$ for all $n$ and calculate $p(1995)$. b) Determine all pairs $n$ such that $p(n)=3$.

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

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

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

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

1996 Singapore MO Open, 3
Tags: number theory , diophantine equation , diophantine
Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.

1996 Singapore MO Open, 4
Tags: diophantine equation , diophantine , number theory
Determine all the solutions of the equation $x^3 + y^3 + z^3 = wx^2y^2z^2$ in natural numbers $x, y, z, w$. Justify your answer

2020 Puerto Rico Team Selection Test, 2
Tags: diophantine equation , system of equations , number theory , digit , diophantine
The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

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 $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$