Olympiad problems

2022 Turkey MO (2nd round), 4

For which real numbers $a$, there exist pairwise different real numbers $x, y, z$ satisfying $$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$

2015 District Olympiad, 3

Tags: algebra , diophantine equation

Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $

2014 Belarus Team Selection Test, 4

Tags: number theory , diophantine , diophantine equation

Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$. (I. Gorodnin)

2008 Dutch Mathematical Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$.

1994 Abels Math Contest (Norwegian MO), 2a

Tags: number theory , diophantine , diophantine equation , prime

Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

2021 Taiwan TST Round 1, 3

Tags: number theory , diophantine equation

Find all triples $(x, y, z)$ of positive integers such that \[x^2 + 4^y = 5^z. \] [i]Proposed by Li4 and ltf0501[/i]

1979 IMO Longlists, 44

Tags: algebra , system of equations , cauchy-schwarz inequality , diophantine equation

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

1996 VJIMC, Problem 3

Tags: diophantine equation , number theory

Prove that the equation $$\frac x{1+x^2}+\frac y{1+y^2}+\frac z{1+z^2}=\frac1{1996}$$has finitely many solutions in positive integers.

2015 Saudi Arabia Pre-TST, 2.3

Tags: number theory , diophantine , diophantine equation

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

2009 Thailand Mathematical Olympiad, 4

Tags: number theory , diophantine , diophantine equation

Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to $(m - n)^2 = kmn + m + n$.

2022 IMO, 5

Tags: number theory , divisibility , lte lemma , diophantine equation

Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

2014 Stars Of Mathematics, 1

Tags: number theory , diophantine equation

Prove that for any integer $n>1$ there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^n+y \mid x+y^n$. ([i]Dan Schwarz[/i])

2011 Dutch Mathematical Olympiad, 1

Tags: diophantine , number theory , factorial , diophantine equation

Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$

1990 Austrian-Polish Competition, 4

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

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

2021 Mediterranean Mathematics Olympiad, 2

Tags: number theory , diophantine equation , diophantine

For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$: $$p_1\, p_2\, \cdots\, p_8 \left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right) ~~=~~ N $$ [i]Proposed by Gerhard Woeginger, Austria[/i]

2014 Contests, 2

Tags: diophantine equation , number theory

$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.

2022 Nigerian Senior MO Round 2, Problem 1

Tags: number theory , diophantine equation

Find all integer solutions of the equation $xy+5x-3y=27$.

2020 Francophone Mathematical Olympiad, 4

Tags: number theory , diophantine equation , diophantine

Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$

2024 Kyiv City MO Round 1, Problem 5

Tags: diophantine equation , number theory

Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$. [i]Proposed by Oleksii Masalitin[/i]

2013 USA Team Selection Test, 1

Tags: number theory , parallelogram , geometry , diophantine equation

Two incongruent triangles $ABC$ and $XYZ$ are called a pair of [i]pals[/i] if they satisfy the following conditions: (a) the two triangles have the same area; (b) let $M$ and $W$ be the respective midpoints of sides $BC$ and $YZ$. The two sets of lengths $\{AB, AM, AC\}$ and $\{XY, XW, XZ\}$ are identical $3$-element sets of pairwise relatively prime integers. Determine if there are infinitely many pairs of triangles that are pals of each other.

2002 Argentina National Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Determine the smallest positive integer $k$ so that the equation $$2002x+273y=200201+k$$ has integer solutions, and for that value of $k$, find the number of solutions $\left (x,y\right )$ with $x$, $y$ positive integers that have the equation.

2014 Swedish Mathematical Competition, 6

Tags: diophantine , number theory , diophantine equation

Determine all odd primes $p$ and $q$ such that the equation $x^p + y^q = pq$ at least one solution $(x, y)$ where $x$ and $y$ are positive integers.

PEN H Problems, 39

Tags: quadratic , diophantine equation

Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$. Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$, where $d(n)$ denotes the number of positive divisors of positive integer $n$.

PEN H Problems, 11

Tags: diophantine equation

Find all $(x,y,n) \in {\mathbb{N}}^3$ such that $\gcd(x, n+1)=1$ and $x^{n}+1=y^{n+1}$.

2024 Mozambican National MO Selection Test, P3

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

Find all triples of positive integers $(a,b,c)$ such that: $a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$