Found problems: 536
2016 Taiwan TST Round 2, 2
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
2023 Turkey Junior National Olympiad, 3
Let $m,n$ be relatively prime positive integers. Prove that the numbers
$$\frac{n^4+m}{m^2+n^2} \qquad \frac{n^4-m}{m^2-n^2}$$
cannot be integer at the same time.
2015 Balkan MO Shortlist, N4
Find all pairs of positive integers $(x,y)$ with the following property:
If $a,b$ are relative prime and positive divisors of $ x^3 + y^3$, then $a+b - 1$ is divisor of $x^3+y^3$.
(Cyprus)
2024 Kyiv City MO Round 1, Problem 3
Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2023$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2023$ loses. Who wins if every player wants to win?
[i]Proposed by Mykhailo Shtandenko[/i]
2011 IMO Shortlist, 3
Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$
[i]Proposed by Mihai Baluna, Romania[/i]
2023 Grosman Mathematical Olympiad, 1
An arithmetic progression of natural numbers of length $10$ and with difference $11$ is given. Prove that the product of the numbers in this progression is divisible by $10!$.
2020-IMOC, N4
$\textbf{N4:} $ Let $a,b$ be two positive integers such that for all positive integer $n>2020^{2020}$, there exists a positive integer $m$ coprime to $n$ with
\begin{align*} \text{ $a^n+b^n \mid a^m+b^m$} \end{align*}
Show that $a=b$
[i]Proposed by ltf0501[/i]
Russian TST 2016, P1
Find all natural $n{}$ such that for every natural $a{}$ that is mutually prime with $n{}$, the number $a^n - 1$ is divisible by $2n^2$.
1988 IMO Longlists, 14
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.
2022 Lusophon Mathematical Olympiad, 3
The positive integers $x$ and $y$ are such that $x^{2022}+x+y^2$ is divisible by $xy$.
a) Give an example of such integers $x$ and $y$, with $x>y$.
b) Prove that $x$ is a perfect square.
1994 IMO Shortlist, 7
A wobbly number is a positive integer whose digits are alternately zero and non-zero with the last digit non-zero (for example, 201). Find all positive integers which do not divide any wobbly number.
2006 MOP Homework, 1
Find all functions $f : N \to N$ such that $f(m)+f(n)$ divides $m+n$ for all positive integers $m$ and $n$.
2016 Indonesia TST, 2
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
2003 Olympic Revenge, 2
Let $x_n$ the sequence defined by any nonnegatine integer $x_0$ and $x_{n+1}=1+\prod_{0 \leq i \leq n}{x_i}$
Show that there exists prime $p$ such that $p\not|x_n$ for any $n$.
2020/2021 Tournament of Towns, P3
A positive integer number $N{}$ is divisible by 2020. All its digits are different and if any two of them are swapped, the resulting number is not divisible by 2020. How many digits can such a number $N{}$ have?
[i]Sergey Tokarev[/i]
1992 IMO Longlists, 14
Integers $a_1, a_2, . . . , a_n$ satisfy $|a_k| = 1$ and
\[ \sum_{k=1}^{n} a_ka_{k+1}a_{k+2}a_{k+3} = 2,\]
where $a_{n+j} = a_j$. Prove that $n \neq 1992.$
2007 Germany Team Selection Test, 3
For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$.
[i]Proposed by Juhan Aru, Estonia[/i]
2012 Brazil Team Selection Test, 1
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$
[i]Proposed by Suhaimi Ramly, Malaysia[/i]
2015 Romania National Olympiad, 2
Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $
[b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $
[b]b)[/b] If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $
1998 Slovenia National Olympiad, Problem 1
Show that for any integter $a$, the number $\frac{a^5}5+\frac{a^3}3+\frac{7a}{15}$ is an integer.
2003 Olympic Revenge, 5
Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number.
Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$
2019 ELMO Shortlist, N4
A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$.
[i]Proposed by Carl Schildkraut and Holden Mui[/i]
2008 Germany Team Selection Test, 2
For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number
\[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}}
\]
but $ 2^{3k \plus{} 1}$ does not.
[i]Author: Waldemar Pompe, Poland[/i]
2003 IMO, 6
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.
2020 Indonesia MO, 3
The wording is just ever so slightly different, however the problem is identical.
Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$.