Olympiad problems

1979 Chisinau City MO, 179

Prove that the equation $x^2 + y^2 = 1979$ has no integer solutions.

1999 Estonia National Olympiad, 1

Tags: diophantine , number theory , diophantine equation

Find all pairs of integers $(m, n)$ such that $(m - n)^2 =\frac{4mn}{m + n - 1}$

2012 Israel National Olympiad, 5

Tags: number theory , diophantine equation

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

2000 Singapore Senior Math Olympiad, 2

Tags: number theory , factorial , diophantine equation , diophantine

Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.

1995 Argentina National Olympiad, 2

Tags: diophantine equation , diophantine , number theory

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 $p(n)=3$.

1984 IMO Longlists, 12

Tags: polynomial , equation , diophantine equation , number theory

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

2016 Hanoi Open Mathematics Competitions, 8

Tags: diophantine equation , diophantine , number theory

Find all positive integers $x,y,z$ such that $x^3 - (x + y + z)^2 = (y + z)^3 + 34$

1996 Singapore MO Open, 4

Tags: diophantine , diophantine equation , 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

2023 Bulgaria JBMO TST, 3

Tags: prime number , diophantine equation , number theory

Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that: $\blacksquare$ $4\nmid c$ $\blacksquare$ $p\not\equiv 11\pmod{16}$ $\blacksquare$ $p^aq^b-1=(p+4)^c$

2023 Regional Competition For Advanced Students, 4

Tags: greatest common divisor , number theory , gcd , diophantine equation

Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$ holds. [i](Walther Janous)[/i]

2009 Vietnam Team Selection Test, 3

Tags: quadratic , diophantine equation , number theory

Let a, b be positive integers. a, b and a.b are not perfect squares. Prove that at most one of following equations $ ax^2 \minus{} by^2 \equal{} 1$ and $ ax^2 \minus{} by^2 \equal{} \minus{} 1$ has solutions in positive integers.

1994 ITAMO, 2

Tags: diophantine equation , number theory

solve this diophantine equation y^2 = x^3 - 16

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , diophantine , diophantine equation

Determine all positive integers $k$ for which there exist positive integers $n$ and $m, m\ge 2$, such that $3^k + 5^k = n^m$

PEN H Problems, 64

Tags: diophantine equation

Show that there is no positive integer $k$ for which the equation \[(n-1)!+1=n^{k}\] is true when $n$ is greater than $5$.

2024 Bundeswettbewerb Mathematik, 1

Tags: diophantine equation , number theory

Determine all pairs $(x,y)$ of integers satisfying \[(x+2)^4-x^4=y^3.\]

2020 Austrian Junior Regional Competition, 4

Tags: number theory , diophantine equation , diophantine

Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$. (Richard Henner)

1995 Tuymaada Olympiad, 3

Tags: algebra , number theory , diophantine equation

Prove that the equation $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.

2005 Thailand Mathematical Olympiad, 13

Tags: number theory , diophantine , diophantine equation , odd

Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation $k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.

2015 Hanoi Open Mathematics Competitions, 14

Tags: diophantine , number theory , diophantine equation

Determine all pairs of integers $(x, y)$ such that $2xy^2 + x + y + 1 = x^2 + 2y^2 + xy$.

2013 Irish Math Olympiad, 8

Tags: number theory , diophantine equation , divisor

Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satised by at least two pairs of integers $(x, y)$ with $1 < x \le y$.

2006 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , diophantine , diophantine equation

. Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$

PEN H Problems, 84

Tags: diophantine equation

For what positive numbers $a$ is \[\sqrt[3]{2+\sqrt{a}}+\sqrt[3]{2-\sqrt{a}}\] an integer?

2010 Benelux, 4

Tags: number theory , diophantine equation

Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and \[a^3 + b^3 = p^n\mbox{.}\] [i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]

1989 Tournament Of Towns, (226) 4

Tags: number theory , diophantine , diophantine equation

Find the positive integer solutions of the equation $$ x+ \frac{1}{y+ \frac{1}{z}}= \frac{10}{7}$$ (G. Galperin)

2017 Hanoi Open Mathematics Competitions, 6

Tags: number theory , diophantine , parametric equation , system of equations , diophantine equation

Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$