Olympiad problems

1987 Brazil National Olympiad, 3

Tags: combinatorics , game strategy , minimum , maximum , positive integers

Two players play alternately. The first player is given a pair of positive integers $(x_1, y_1)$. Each player must replace the pair $(x_n, y_n)$ that he is given by a pair of non-negative integers $(x_{n+1}, y_{n+1})$ such that $x_{n+1} = min(x_n, y_n)$ and $y_{n+1} = max(x_n, y_n)- k\cdot x_{n+1}$ for some positive integer $k$. The first player to pass on a pair with $y_{n+1} = 0$ wins. Find for which values of $x_1/y_1$ the first player has a winning strategy.

2024 Mozambican National MO Selection Test, P3

Tags: nt , number theory , Diophantine equation , factorization , positive integers , prime numbers

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

2006 MOP Homework, 7

Tags: function , positive integers , number theory

Let $n$ be a given integer greater than two, and let $S = \{1, 2,...,n\}$. Suppose the function $f : S^k \to S$ has the property that $f(a) \ne f(b)$ for every pair $a$ and $b$ of elements in $S^k$ with $a$ and $b$ differ in all components. Prove that $f$ is a function of one of its elements.

2006 Korea Junior Math Olympiad, 5

Tags: number theory , coprime , positive integers , Integers

Find all positive integers that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also, $m,n$ are relatively prime positive integers.

2010 Belarus Team Selection Test, 1.1

Tags: Sum , number theory , positive integers

Does there exist a subset $E$ of the set $N$ of all positive integers such that none of the elements in $E$ can be presented as a sum of at least two other (not necessarily distinct) elements from $E$ ? (E. Barabanov)

2024 CAPS Match, 6

Tags: combinatorics , number theory , international competitions , Combinatorial Number Theory , prime numbers , triplets , positive integers

Determine whether there exist infinitely many triples $(a, b, c)$ of positive integers such that every prime $p$ divides \[\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.\]

2014 Korea Junior Math Olympiad, 5

Tags: number theory , multiple , positive integers , divisible

For positive integers $x,y$, find all pairs $(x,y)$ such that $x^2y + x$ is a multiple of $xy^2 + 7$.

1983 Brazil National Olympiad, 1

Tags: number theory , Diophantine equation , positive integers

Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.

2024 Irish Math Olympiad, P1

Tags: irmo , recursion , positive integers

The [i]runcible[/i] positive integers are defined recursively as follows: [list] [*]$1$ and $2$ are runcible [*]If $a$ and $b$ are runcible (where $a$ and $b$ are not necessarily distinct) then $2a + 3b$ is runcible. [/list] Is $2024$ runcible?

2019 Tournament Of Towns, 5

Tags: Sequence , algebra , Sum , positive integers

Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence? (Alexandr Shapovalov)

2025 Alborz Mathematical Olympiad, P1

Tags: functional equation , number theory , positive integers , AlborzMO , algebra

Let $ \mathbb{Z^{+}} $ denote the set of all positive integers. Find all functions $ f: \mathbb{Z^{+}} \rightarrow \mathbb{Z^{+}} $ such that for every pair of positive integers $ a $ and $ b $, there exists a positive integer $ c $ satisfying: $$ f(a)f(b) - ab = 2^{c-1} - 1. $$ Proposed by Matin Yousefi

2002 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: Integer , number theory , positive integers

Determine all pairs $(a, b)$ of positive integers for which $\frac{a^2b+b}{ab^2+9}$ is an integer number.

1983 Brazil National Olympiad, 5

Tags: number theory , minimum , positive integers , Inequality

Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$. Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.