Olympiad problems

2012 Junior Balkan Team Selection Tests - Romania, 3

Let $m$ and $n$ be two positive integers, $m, n \ge 2$. Solve in the set of the positive integers the equation $x^n + y^n = 3^m$.

IV Soros Olympiad 1997 - 98 (Russia), 11.1

Tags: number theory , diophantine equation , diophantine

Solve the equation $xy =1997(x + y)$ in integers.

1989 IMO Longlists, 83

Tags: number theory , diophantine equation , quadratic , equation

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

2013 Swedish Mathematical Competition, 1

Tags: number theory , diophantine , diophantine equation

For $r> 0$ denote by $B_r$ the set of points at distance at most $r$ length units from the origin. If $P_r$ is the set of the points in $B_r$ whit integer coordinates, show that the equation $$xy^3z + 2x^3z^3-3x^5y = 0$$ has an odd number of solutions $(x, y, z)$ in $P_r$.

2024 Mongolian Mathematical Olympiad, 1

Tags: number theory , diophantine equation , factorial

Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\] [i]Proposed by Otgonbayar Uuye.[/i]

2001 All-Russian Olympiad Regional Round, 11.1

Tags: number theory , diophantine , diophantine equation

Find all prime numbers $p$ and $q$ such that $p + q = (p -q)^3.$

2012 India Regional Mathematical Olympiad, 5

Tags: positive integer , number theory , diophantine equation , diophantine , prime

Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.

2011 China Northern MO, 3

Tags: number theory , diophantine equation , diophantine

Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.

2013 Balkan MO Shortlist, N5

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 $p^{q-1}- q^{p-1}=4n^2$

2012 Greece JBMO TST, 2

Tags: number theory , least common multiple , coprime , diophantine equation

Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .

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

1967 IMO Longlists, 44

Tags: algebra , polynomial , summation , equation , diophantine equation

Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

2012 Israel National Olympiad, 5

Tags: number theory , diophantine equation

Find all integer solutions of the equation $a^3+3ab^2+7b^3=2011$.

1979 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , diophantine equation , diophantine

Let $n$ be a given natural number. Determine the number of all orderer triples $(x, y, z)$ of non-negative integers $x, y, z$ that satisfy the equation $$x + 2y + 5z=10n.$$

2013 Greece National Olympiad, 2

Tags: quadratic , number theory , diophantine equation

Solve in integers the following equation: \[y=2x^2+5xy+3y^2\]

2015 USAMO, 5

Tags: diophantine equation , number theory

Let $a$, $b$, $c$, $d$, $e$ be distinct positive integers such that $a^4+b^4=c^4+d^4=e^5$. Show that $ac+bd$ is a composite number.

2019 Kosovo National Mathematical Olympiad, 5

Tags: diophantine equation , number theory

Find all positive integers $x,y$ such that $2^x+19^y$ is a perfect cube.

1989 IMO Shortlist, 25

Tags: number theory , diophantine equation , quadratic , equation

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

2016 Lusophon Mathematical Olympiad, 6

Tags: diophantine equation , power of 2 , number theory

Source: Lusophon MO 2016 Prove that any positive power of $2$ can be written as: $$5xy-x^2-2y^2$$ where $x$ and $y$ are odd numbers.

2020 South Africa National Olympiad, 4

Tags: number theory , diophantine equation

A positive integer $k$ is said to be [i]visionary[/i] if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020.$ How many visionary integers are there?

2015 India PRMO, 11

Tags: integer , number theory , algebra , diophantine equation

$11.$ Let $a,$ $b,$ and $c$ be real numbers such that $a-7b+8c=4.$ and $8a+4b-c=7.$ What is the value of $a^2-b^2+c^2 ?$

2017 JBMO Shortlist, NT4

Tags: number theory , diophantine equation

Solve in nonnegative integers the equation $5^t + 3^x4^y = z^2$.

2019 South Africa National Olympiad, 6

Tags: number theory , diophantine equation

Determine all pairs $(m, n)$ of non-negative integers that satisfy the equation $$ 20^m - 10m^2 + 1 = 19^n. $$

2013 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , diophantine equation , diophantine

Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$

2003 Moldova Team Selection Test, 1

Tags: quadratic , number theory , relatively prime , diophantine equation

Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called [i]quadratic [/i] iff at least one of the numbers $ a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.