Found problems: 367
2012 Abels Math Contest (Norwegian MO) Final, 3b
Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?
2010 Saudi Arabia IMO TST, 1
Find all pairs $(m,n)$ of integers, $m ,n \ge 2$ such that $mn - 1$ divides $n^3 - 1$.
2007 Postal Coaching, 6
Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.
2015 Dutch IMO TST, 5
Let $N$ be the set of positive integers.
Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
Mathley 2014-15, 8
For every $n$ positive integers we denote $$\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}$$
where $x_n, y_n$ are coprime positive integers. Prove that $y_n$ is not divisible by $2^n$ for any positive integers $n$.
Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi
2013 Saudi Arabia Pre-TST, 3.2
Let $a_1, a_2,..., a_9$ be integers. Prove that if $19$ divides $a_1^9+a_2^9+...+a_9^9$ then $19$ divides the product $a_1a_2...a_9$.
2001 BAMO, 5
For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1, 2, ... , n\}$ such that $\tau (\tau (\tau (x))) = x$ for $x = 1, 2, ..., n$. The first few values are $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 9$.
Prove that $3^{334}$ divides $a_{2001}$.
(A permutation of $\{1, 2, ... , n\}$ is a rearrangement of the numbers $\{1, 2, ... , n\}$ or equivalently, a one-to-one and
onto function from $\{1, 2, ... , n\}$ to $\{1, 2, ... , n\}$. For example, one permutation of $\{1, 2, 3\}$ is the rearrangement $\{2, 1, 3\}$, which is equivalent to the function $\sigma : \{1, 2, 3\} \to \{1, 2, 3\}$ defined by $\sigma (1) = 2, \sigma (2) = 1, \sigma (3) = 3$.)
2006 Thailand Mathematical Olympiad, 14
Find the smallest positive integer $n$ such that $2549 | n^{2545} - 2$.
2019 Auckland Mathematical Olympiad, 2
Prove that among any $43$ positive integers there exist two $a$ and $b$ such that $a^2 - b^2$ is divisible by $100$.
1998 Tournament Of Towns, 3
Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.)
(G Galperin)
2021 Polish Junior MO Second Round, 3
Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.
2006 Thailand Mathematical Olympiad, 3
Let $P(x), Q(x)$ and $R(x)$ be polynomials satisfying the equation $2xP(x^3) + Q(-x -x^3) = (1 + x + x^2)R(x)$.
Show that $x - 1$ divides $P(x) - Q(x)$.
2012 China Northern MO, 3
Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that:
(1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$
(2) If $m,n \in S$ , then $mn\in S$.
1997 Tournament Of Towns, (537) 2
Let $a$ and $b$ be positive integers. If $a^2 + b^2$ is divisible by $ab$, prove that $a = b$.
(BR Frenkin)
2014 Regional Olympiad of Mexico Center Zone, 1
Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.
1977 Dutch Mathematical Olympiad, 3
From each set $ \{a_1,a_2,...,a_7\} \subset Z$ one can choose a number of elements whose sum is a multiple of $7$.
2019 Durer Math Competition Finals, 11
What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?
2016 Saudi Arabia Pre-TST, 1.3
Let $a, b$ be two positive integers such that $b + 1|a^2 + 1$,$ a + 1|b^2 + 1$. Prove that $a, b$ are odd numbers.
2001 Kazakhstan National Olympiad, 1
Prove that there are infinitely many natural numbers $ n $ such that $ 2 ^ n + 3 ^ n $ is divisible by $ n $.
2016 Saudi Arabia BMO TST, 3
For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.
2016 Peru IMO TST, 4
Let $N$ be the set of positive integers.
Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
2023 Chile National Olympiad, 1
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
2020 Argentina National Olympiad, 1
For every positive integer $n$, let $S (n)$ be the sum of the digits of $n$. Find, if any, a $171$-digit positive integer $n$ such that $7$ divides $S (n)$ and $7$ divides $S (n + 1)$.
2020 Swedish Mathematical Competition, 5
Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides
${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$+ ...+$ {p-1 \choose p-3} a^{p-5} .$
2012 QEDMO 11th, 12
Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.