Olympiad problems

2017 QEDMO 15th, 6

Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.

2003 Federal Math Competition of S&M, Problem 1

Tags: diophantine equation , number theory

Find the number of solutions to the equation$$x_1^4+x_2^4+\ldots+x_{10}^4=2011$$in the set of positive integers.

2013 India IMO Training Camp, 1

Tags: diophantine equation , modular arithmetic , number theory

A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

2020 South Africa National Olympiad, 4

Tags: number theory , diophantine equation

A positive integer $k$ is said to be [i]visionary[/i] if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020.$ How many visionary integers are there?

1984 IMO Longlists, 13

Tags: polynomial , diophantine equation , 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 $4mn - m- n = p^2.$

2001 JBMO ShortLists, 1

Tags: 3d geometry , diophantine equation , number theory , geometry

Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube. [hide="Note"] [color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$.[/color][/hide]

2022 Kosovo & Albania Mathematical Olympiad, 4

Tags: algebra , diophantine equation

Let $A$ be the set of natural numbers $n$ such that the distance of the real number $n\sqrt{2022} - \frac13$ from the nearest integer is at most $\frac1{2022}$. Show that the equation $$20x + 21y = 22z$$ has no solutions over the set $A$.

PEN H Problems, 88

Tags: diophantine equation , induction , number theory

(Leo Moser) Show that the Diophantine equation \[\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}+\frac{1}{x_{1}x_{2}\cdots x_{n}}= 1\] has at least one solution for every positive integers $n$.

2022 Dutch IMO TST, 1

Tags: diophantine equation , diophantine , number theory

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 $1$.

2009 Irish Math Olympiad, 3

Tags: number theory , diophantine equation , modular arithmetic

Find all pairs $(a,b)$ of positive integers such that $(ab)^2 - 4(a+b)$ is the square of an integer.

2019 Peru EGMO TST, 1

Tags: prime , number theory , diophantine equation , diophantine

Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.

PEN H Problems, 10

Tags: modular arithmetic , diophantine equation

Prove that there are unique positive integers $a$ and $n$ such that \[a^{n+1}-(a+1)^{n}= 2001.\]

2003 Singapore MO Open, 3

Tags: diophantine , number theory , diophantine equation

For any given prime $p$, determine whether the equation $x^2 + y^2 + p^z = 2003$ always has integer solutions in $x, y, z$. Justify your answer

2003 Turkey MO (2nd round), 1

Tags: modular arithmetic , greatest common divisor , diophantine equation , number theory

Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$.

1979 IMO Longlists, 56

Tags: function , floor function , algebra , diophantine equation

Show that for every $n\in\mathbb{N}$, $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$, there exists an $n\in\mathbb{N}$ such that $ n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$.

1980 IMO Longlists, 3

Tags: algebra , number theory , equation , diophantine equation

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

2021 Malaysia IMONST 1, 12

Tags: diophantine , number theory , diophantine equation

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

1990 French Mathematical Olympiad, Problem 3