Found problems: 408
2016 Peru IMO TST, 16
Find all pairs $ (m, n)$ of positive integers that have the following property:
For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.
2013 Thailand Mathematical Olympiad, 5
Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.
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 Korea Junior Math Olympiad, 5
For positive integers $x,y$, find all pairs $(x,y)$ such that $x^2y + x$ is a multiple of $xy^2 + 7$.
2005 Estonia National Olympiad, 3
How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?
2018 Czech-Polish-Slovak Junior Match, 1
For natural numbers $a, b c$ it holds that $(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc$.
Prove that $(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2$
2017 May Olympiad, 5
We will say that two positive integers $a$ and $b$ form a [i]suitable pair[/i] if $a+b$ divides $ab$ (its sum divides its multiplication). Find $24$ positive integers that can be distribute into $12$ suitable pairs, and so that each integer number appears in only one pair and the largest of the $24$ numbers is as small as possible.
1978 All Soviet Union Mathematical Olympiad, 254
Prove that there is no $m$ such that ($1978^m - 1$) is divisible by ($1000^m - 1$).
2013 Saudi Arabia BMO TST, 4
Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.
2006 MOP Homework, 2
Determine all pairs of positive integers $(m,n)$ such that m is but divisible by every integer from $1$ to $n$ (inclusive), but not divisible by $n + 1, n + 2$, and $n + 3$.
2016 Dutch IMO TST, 2
Determine all pairs $(a, b)$ of integers having the following property:
there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.
2021 Saudi Arabia JBMO TST, 3
We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.
1955 Moscow Mathematical Olympiad, 299
Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.
2017 Auckland Mathematical Olympiad, 3
The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.
2021 Saudi Arabia Training Tests, 40
Given $m, n$ such that $m > n^{n-1}$ and the number $m+1$, $m+2$,$ ...$, $m+n$ are composite. Prove that there exist distinct primes $p_1, p_2, ..., p_n$ such that $m + k$ is divisible by $p_k$ for each $k = 1, 2, ...$
2011 Saudi Arabia BMO TST, 4
Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$.
Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .
2013 Saudi Arabia GMO TST, 4
Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$.
1982 All Soviet Union Mathematical Olympiad, 329
a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$.
b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?
2011 Belarus Team Selection Test, 1
Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with
$a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$).
Prove that
a) $(a-1)\vdots p_i$ for some $i=1,..,n$
b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$?
I. Bliznets
2012 Switzerland - Final Round, 7
Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.
2009 Hanoi Open Mathematics Competitions, 5
Prove that $m^7- m$ is divisible by $42$ for any positive integer $m$.
2021 Bosnia and Herzegovina Team Selection Test, 2
Let $p > 2$ be a prime number. Prove that there is a permutation $k_1, k_2, ..., k_{p-1}$ of numbers $1,2,...,p-1$ such that the number $1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}$ is divisible by $p$.
Note: The numbers $k_1, k_2, ..., k_{p-1}$ are a permutation of the numbers $1,2,...,p-1$ if each of of numbers $1,2,...,p-1$ appears exactly once among the numbers $k_1, k_2, ..., k_{p-1}$.
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 $100$.
2003 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$.
(H. Nestra)
1998 Singapore Senior Math Olympiad, 1
Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.