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

2022 Dutch IMO TST, 1
Tags: number theory , diophantine equation , diophantine
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 NZMOC Camp Selection Problems, 2
Tags: number theory , diophantine , diophantine equation
Find all pairs of integers $(a, b)$ such that $$a^2 + ab - b = 2018.$$

2015 Belarus Team Selection Test, 1
Tags: number theory , diophantine equation , diophantine
Solve the equation in nonnegative integers $a,b,c$: $3^a+2^b+2015=3c!$ I.Gorodnin

2012 Dutch BxMO/EGMO TST, 3
Tags: number theory , diophantine equation
Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

2010 Czech And Slovak Olympiad III A, 1
Tags: number theory , diophantine equation
Determine all pairs of integers $a, b$ for which they apply $4^a + 4a^2 + 4 = b^2$ .

1988 IMO Shortlist, 9
Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

2013 Czech And Slovak Olympiad IIIA, 1
Tags: number theory , diophantine equation , diophantine
Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$

1986 IMO Shortlist, 3
Tags: number theory , diophantine equation , geometry , algebra , pythagorean theorem
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

1983 Brazil National Olympiad, 1
Tags: number theory , diophantine equation , positive integer
Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.

2006 Hanoi Open Mathematics Competitions, 3
Tags: number theory , system of equations , diophantine equation
Find the number of different positive integer triples $(x, y,z)$ satisfying the equations $x^2 + y -z = 100$ and $x + y^2 - z = 124$:

2017 Junior Balkan Team Selection Tests - Moldova, Problem 1
Tags: number theory , diophantine equation
Find all natural numbers $x,y$ such that $$x^5=y^5+10y^2+20y+1.$$

1997 Brazil Team Selection Test, Problem 3
Tags: diophantine equation , number theory
Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.

2020 LIMIT Category 2, 15
Tags: number theory , limit , diophantine , diophantine equation
How many integer pairs $(x,y)$ satisfies $x^2+y^2=9999(x-y)$?

2020 Abels Math Contest (Norwegian MO) Final, 2a
Tags: diophantine equation , diophantine , number theory
Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider $0$ to be a natural number.

1979 Brazil National Olympiad, 4
Tags: number theory , diophantine equation
Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.

1975 Bulgaria National Olympiad, Problem 1
Tags: number theory , diophantine equation
Find all pairs of natural numbers $(m,n)$ bigger than $1$ for which $2^m+3^n$ is the square of whole number. [i]I. Tonov[/i]

2014 China Team Selection Test, 3
Tags: number theory , diophantine equation
Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

2011 China Northern MO, 3
Tags: number theory , diophantine equation , diophantine
Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.

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

PEN H Problems, 72
Tags: induction , number theory , relatively prime , absolute value , diophantine equation
Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.

2022 Austrian MO National Competition, 4
Tags: prime number , number theory , diophantine equation , diophantine
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

IV Soros Olympiad 1997 - 98 (Russia), 11.1
Tags: number theory , diophantine equation , diophantine
Solve the equation $xy =1997(x + y)$ in integers.

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 $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

2015 Thailand TSTST, 2
Tags: number theory , diophantine equation
Find all integer solutions to the equation $y^2=2x^4+17$.

2005 Estonia Team Selection Test, 3
Tags: diophantine equation , diophantine , number theory
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.