Olympiad problems

1984 IMO Longlists, 13

Tags: diophantine equation , polynomial , equation , quadratic , number theory

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

2019 Danube Mathematical Competition, 1

Tags: equation , diophantine equation , algebra

Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $ [i]Lucian Petrescu[/i]

2012 IFYM, Sozopol, 3

Tags: number theory , diophantine equation

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

2015 Swedish Mathematical Competition, 2

Tags: diophantine , diophantine equation , number theory

Determine all integer solutions to the equation $x^3 + y^3 + 2015 = 0$.

1974 Czech and Slovak Olympiad III A, 3

Tags: number theory , digit , sum of digits , permutation , equation , diophantine equation

Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]

2002 Dutch Mathematical Olympiad, 2

Tags: diophantine equation , diophantine , number theory

Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$

2014 Hanoi Open Mathematics Competitions, 11

Tags: number theory , diophantine equation , diophantine

Find all pairs of integers $(x,y)$ satisfying the following equality $8x^2y^2 + x^2 + y^2 = 10xy$

2015 Saudi Arabia Pre-TST, 2.3

Tags: diophantine , diophantine equation , number theory

Find all integer solutions of the equation $14^x - 3^y = 2015$. (Malik Talbi)

1999 Austrian-Polish Competition, 7

Tags: diophantine , diophantine equation , number theory

Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.

2020 Korea National Olympiad, 4

Tags: number theory , diophantine equation

Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.

1989 IMO Longlists, 94

Tags: algebra , diophantine equation , linear equation , counting

Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation \[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\] where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]

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.

2017 Hanoi Open Mathematics Competitions, 6

Tags: number theory , diophantine equation , prime , diophantine

Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime.

2017 Singapore Junior Math Olympiad, 2

Tags: diophantine , diophantine equation , number theory , factorial

Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$

1990 Austrian-Polish Competition, 4

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

Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$

2015 India PRMO, 14

Tags: diophantine equation , algebra , number theory

$14.$ If $3^x+2^y=985.$ and $3^x-2^y=473.$ What is the value of $xy ?$

1987 Dutch Mathematical Olympiad, 1

Tags: number theory , diophantine , diophantine equation

Solve into $N$: $$a^2 = 2^b +c^4$$

2013 Bulgaria National Olympiad, 1

Tags: modular arithmetic , diophantine equation , number theory

Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

2008 Hanoi Open Mathematics Competitions, 2

Tags: diophantine equation , diophantine , number theory

Find all pairs $(m, n)$ of positive integers such that $m^2 + 2n^2 = 3(m + 2n)$

2021 Malaysia IMONST 1, 12

Tags: number theory , diophantine equation , diophantine

Determine the number of positive integer solutions $(x,y, z)$ to the equation $xyz = 2(x + y + z)$.

2023 Bangladesh Mathematical Olympiad, P3

Tags: number theory , factorization , diophantine equation

Solve the equation for the positive integers: $$(x+2y)^2+2x+5y+9=(y+z)^2$$

2020 Jozsef Wildt International Math Competition, W55

Tags: number theory , diophantine equation , fibonacci , fibonacci number

Prove that the equation $$1320x^3=(y_1+y_2+y_3+y_4)(z_1+z_2+z_3+z_4)(t_1+t_2+t_3+t_4+t_5)$$ has infinitely many solutions in the set of Fibonacci numbers. [i]Proposed by Mihály Bencze[/i]

2007 Ukraine Team Selection Test, 12

Tags: polynomial , diophantine equation , number theory

Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{2}\plus{}n\plus{}1$ are not more then $ \sqrt{n}$. [hide] Stronger one. Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{3}\minus{}1$ are not more then $ \sqrt{n}$.[/hide]

VMEO III 2006 Shortlist, N5

Tags: number theory , diophantine equation , diophantine

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

2016 Croatia Team Selection Test, Problem 4

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

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$