Olympiad problems

1981 All Soviet Union Mathematical Olympiad, 316

Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.

1992 IMO Longlists, 36

Tags: diophantine equation , algebra , system of equations , polynomial

Find all rational solutions of \[a^2 + c^2 + 17(b^2 + d^2) = 21,\]\[ab + cd = 2.\]

PEN H Problems, 34

Tags: diophantine equation , quadratic

Are there integers $m$ and $n$ such that $5m^2 -6mn+7n^2 =1985$?

1988 Bundeswettbewerb Mathematik, 4

Tags: number theory , diophantine equation , equation , algebra

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

2022 Nigerian Senior MO Round 2, Problem 1

Tags: diophantine equation , number theory

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

PEN H Problems, 37

Tags: diophantine equation , modular arithmetic

Prove that for each positive integer $n$ there exist odd positive integers $x_n$ and $y_n$ such that ${x_{n}}^2 +7{y_{n}}^2 =2^n$.

2016 Regional Olympiad of Mexico Northeast, 1

Tags: number theory , diophantine equation , diophantine

Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that: $$a^3 + b^3 + c^3 = 2016$$

2018 Pan-African Shortlist, N1

Tags: number theory , diophantine equation

Does there exist positive integers $a, b, c$ such that $4(ab - a - c^2) = b$?

2005 Austria Beginners' Competition, 1

Tags: number theory , diophantine , diophantine equation

Show that there are no positive integers $a$ und $b$ such that $4a(a + 1) = b(b + 3)$

1991 Romania Team Selection Test, 8

Tags: function , number theory , greatest common divisor , diophantine equation , algebra

Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$. If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.

PEN H Problems, 91

Tags: diophantine equation , geometry , rectangle , perimeter

If $R$ and $S$ are two rectangles with integer sides such that the perimeter of $R$ equals the area of $S$ and the perimeter of $S$ equals the area of $R$, then we call $R$ and $S$ a friendly pair of rectangles. Find all friendly pairs of rectangles.

2014 Turkey EGMO TST, 2

Tags: number theory , diophantine equation

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

2022 China Team Selection Test, 4

Tags: diophantine equation , number theory

Find all positive integers $a,b,c$ and prime $p$ satisfying that \[ 2^a p^b=(p+2)^c+1.\]

JOM 2015 Shortlist, N4

Tags: number theory , diophantine equation

Determine all triplet of non-negative integers $ (x,y,z) $ satisfy $$ 2^x3^y+1=7^z $$

1996 Israel National Olympiad, 7

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

Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$

1993 Swedish Mathematical Competition, 3

Tags: even , number theory , diophantine , diophantine equation

Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.

1973 Swedish Mathematical Competition, 4

Tags: diophantine , number theory , diophantine equation

$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that \[ \frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p} \]

PEN H Problems, 77

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

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

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 $2^x + 9 \cdot 7^y = z^3$

1998 Poland - Second Round, 4

Tags: diophantine equation , diophantine , number theory

Find all pairs of integers $(x,y)$ satisfying $x^2 +3y^2 = 1998x$.

2015 Israel National Olympiad, 1

Tags: diophantine equation , number theory , 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]

2018 Hong Kong TST, 3

Tags: number theory , diophantine equation

Find all primes $p$ and all positive integers $a$ and $m$ such that $a\leq 5p^2$ and $(p-1)!+a=p^m$

PEN A Problems, 2

Tags: arithmetic sequence , diophantine equation , divisibility theory , pell s equation

Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.

PEN H Problems, 55

Tags: diophantine equation , floor function

Given that \[34! = 95232799cd96041408476186096435ab000000_{(10)},\] determine the digits $a, b, c$, and $d$.

2012 IMAC Arhimede, 4

Tags: diophantine equation , number theory

Solve the following equations in the set of natural numbers: a) $(5+11\sqrt2)^p=(11+5\sqrt2)^q$ b) $1005^x+2011^y=1006^z$