Olympiad problems

1926 Eotvos Mathematical Competition, 1

Prove that, if $a$ and $b$ are given integers, the system of equatìons $$x + y + 2z + 2t = a$$ $$2x - 2y + z- t = b$$ has a solution in integers $x, y,z,t$.

2022 Korea Junior Math Olympiad, 8

Tags: number theory , rational number , diophantine equation , pythagorean triple , discriminant

Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$

1993 Tournament Of Towns, (389) 1

Tags: diophantine , diophantine equation , number theory

Consider the set of solutions of the equation $$x^2+y^3=z^2.$$ in positive integers. Is it finite or infinite? (Folklore)

2019 India PRMO, 6

Tags: diophantine equation , number theory

Let $\overline{abc}$ be a three digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$

2013 Switzerland - Final Round, 9

Tags: diophantine , diophantine equation , number theory

Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$

2021 Austrian MO National Competition, 3

Tags: number theory , diophantine equation , diophantine

Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)

2015 Turkey Junior National Olympiad, 3

Tags: number theory , diophantine equation , prime number

Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds.

2021 Cyprus JBMO TST, 2

Tags: number theory , diophantine equation , gcd , lcm

Find all pairs of natural numbers $(\alpha,\beta)$ for which, if $\delta$ is the greatest common divisor of $\alpha,\beta$, and $\varDelta$ is the least common multiple of $\alpha,\beta$, then \[ \delta + \Delta = 4(\alpha + \beta) + 2021\]

1980 IMO Longlists, 12

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

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

1999 French Mathematical Olympiad, Problem 2

Tags: number theory , diophantine equation

Find all natural numbers $n$ such that $$(n+3)^n=\sum_{k=3}^{n+2}k^n.$$

2015 Israel National Olympiad, 1

Tags: number theory , diophantine equation , algebra

[list=a] [*] Find an example of three positive integers $a,b,c$ satisfying $31a+30b+28c=365$. [*] Prove that any triplet $a,b,c$ satisfying the above condition, also satisfies $a+b+c=12$. [/list]

2010 Saudi Arabia BMO TST, 1

Tags: diophantine , number theory , diophantine equation

Find all triples $(x,y,z)$ of positive integers such that $3^x + 4^y = 5^z$.

2017 QEDMO 15th, 1

Tags: number theory , diophantine , diophantine equation

Find all integers $x, y, z$ satisfy the $x^4-10y^4 + 3z^6 = 21$.

Mathley 2014-15, 2

Tags: diophantine , number theory , diophantine equation

Let $n$ be a positive integer and $p$ a prime number $p > n + 1$. Prove that the following equation does not have integer solution $$1 + \frac{x}{n + 1} + \frac{x^2}{2n + 1} + ...+ \frac{x^p}{pn + 1} = 0$$ Luu Ba Thang, Department of Mathematics, College of Education

2015 Thailand TSTST, 2

Tags: number theory , diophantine equation

Find all integer solutions to the equation $y^2=2x^4+17$.

2016 Costa Rica - Final Round, N3

Tags: number theory , diophantine equation , diophantine

Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$

1995 Israel Mathematical Olympiad, 4

Tags: diophantine , diophantine equation , number theory

Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$.

2005 Czech-Polish-Slovak Match, 6

Tags: modular arithmetic , number theory , diophantine equation , prime factorization

Determine all pairs of integers $(x, y)$ satisfying the equation \[y(x + y) = x^3- 7x^2 + 11x - 3.\]

2020 Azerbaijan Senior NMO, 2

Tags: number theory , perfect cube , diophantine equation

$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$

2021 Final Mathematical Cup, 1

Tags: diophantine , number theory , diophantine equation

Find all integer $n$ such that the equation $2x^2 + 5xy + 2y^2 = n$ has integer solution for $x$ and $y$.

2005 China National Olympiad, 6

Tags: number theory , diophantine equation

Find all nonnegative integer solutions $(x,y,z,w)$ of the equation\[2^x\cdot3^y-5^z\cdot7^w=1.\]

2022 Dutch IMO TST, 1

Tags: number theory , diophantine equation , diophantine

Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.

1989 IMO Shortlist, 15

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

Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\] \[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$

2015 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory , diophantine equation

Solve in nonnegative integers the following equation : $$21^x+4^y=z^2$$

2023 Philippine MO, 1

Tags: algebra , diophantine equation , number theory

Find all ordered pairs $(a, b)$ of positive integers such that $a^2 + b^2 + 25 = 15ab$ and $a^2 + ab + b^2$ is prime.