Olympiad problems

2011 Swedish Mathematical Competition, 1

Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$

2017 USAJMO, 2

Tags: diophantine equation

Consider the equation \[(3x^3+xy^2)(x^2y+3y^3)=(x-y)^7\] (a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation. (b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.

1998 Swedish Mathematical Competition, 1

Tags: number theory , diophantine , diophantine equation

Find all positive integers $a, b, c$, such that $(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2$.

2014 Swedish Mathematical Competition, 6

Tags: number theory , diophantine , 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, 56

Tags: diophantine equation

Prove that the equation $\prod_{cyc} (x_1-x_2)= \prod_{cyc} (x_1-x_3)$ has a solution in natural numbers where all $x_i$ are different.

2007 Moldova National Olympiad, 9.4

Tags: diophantine equation

Find all rational terms of sequence defined by formula $ a_n=\sqrt{\frac{9n-2}{n+1}}, n \in N $

2009 Korea Junior Math Olympiad, 8

Tags: diophantine equation , number theory , prime number

Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?

PEN H Problems, 74

Tags: diophantine equation

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{a^{a}}= b^{b}.\]

2015 Hanoi Open Mathematics Competitions, 14

Tags: diophantine equation , number theory , diophantine

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

2016 Greece National Olympiad, 1

Tags: number theory , diophantine equation

Find all triplets of nonnegative integers $(x,y,z)$ and $x\leq y$ such that $x^2+y^2=3 \cdot 2016^z+77$

2014 JBMO Shortlist, 2

Tags: diophantine equation , modular arithmetic

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

2022 District Olympiad, P3

Tags: number theory , diophantine equation , induction

$a)$ Solve over the positive integers $3^x=x+2.$ $b)$ Find pairs $(x,y)\in\mathbb{N}\times\mathbb{N}$ such that $(x+3^y)$ and $(y+3^x)$ are consecutive.

PEN H Problems, 3

Tags: diophantine equation , quadratic

Does there exist a solution to the equation \[x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65\] in integers with $x, y, z, u, v$ greater than $1998$?

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.

2016 Korea Winter Program Practice Test, 1

Tags: diophantine equation , number theory

Solve: $a, b, m, n\in \mathbb{N}$ $a^2+b^2=m^2-n^2, ab=2mn$

PEN H Problems, 32

Tags: diophantine equation

Let $n$ be a natural number. Solve in whole numbers the equation \[x^{n}+y^{n}=(x-y)^{n+1}.\]

1997 Estonia National Olympiad, 1

Tags: number theory , diophantine , diophantine equation

Prove that for every integer $n\ge 3$ there are such positives integers $x$ and $y$ such that $2^n = 7x^2 + y^2$

1991 AIME Problems, 6

Tags: function , floor function , modular arithmetic , inequalities , number theory , diophantine equation , algebra

Suppose $r$ is a real number for which \[ \left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546. \] Find $\lfloor 100r \rfloor$. (For real $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

2016 Tuymaada Olympiad, 4

Tags: diophantine equation , number theory

For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation $$8^k=x^3+y^3+z^3-3xyz$$

2022 USAJMO, 5

Tags: number theory , diophantine equation , prime number

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

PEN H Problems, 79

Tags: diophantine equation , modular arithmetic

Find all positive integers $m$ and $n$ for which \[1!+2!+3!+\cdots+n!=m^{2}\]

1997 Tournament Of Towns, (558) 3

Tags: number theory , diophantine , diophantine equation

Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$. (N Vassiliev)

1967 IMO Shortlist, 3

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 $S(x,n) = 0.$

2022 Dutch IMO TST, 1

Tags: number theory , diophantine equation , diophantine

Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

1996 Greece Junior Math Olympiad, 4b

Tags: diophantine equation , diophantine , prime , number theory

Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.