Found problems: 408
2007 Abels Math Contest (Norwegian MO) Final, 4
Let $a, b$ and $c$ be integers such that $a + b + c = 0$.
(a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$.
(b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$.
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1957 Moscow Mathematical Olympiad, 357
For which integer $n$ is $N = 20^n + 16^n - 3^n - 1$ divisible by $323$?
2011 IMAR Test, 4
Given an integer number $n \ge 3$, show that the number of lists of jointly coprime positive integer numbers that sum to $n$ is divisible by $3$.
(For instance, if $n = 4$, there are six such lists: $(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)$ and $(1, 1, 1, 1)$.)
2018 Saudi Arabia GMO TST, 2
Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.
1949 Moscow Mathematical Olympiad, 168
Prove that some (or one) of any $100$ integers can always be chosen so that the sum of the chosen integers is divisible by $100$.
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 $m$?
OMMC POTM, 2022 1
The digits $2,3,4,5,6,7,8,9$ are written down in some order. When read in that order, the digits form an $8$-digit, base $10$ positive integer. if this integer is divisible by $44$, how many ways could the digits have been initially ordered?
[i]Proposed by Evan Chang (squareman), USA[/i]
2021 Regional Competition For Advanced Students, 4
Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$.
(Walther Janous)
2015 JBMO Shortlist, NT1
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
1986 Polish MO Finals, 3
$p$ is a prime and $m$ is a non-negative integer $< p-1$.
Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.
2015 JBMO Shortlist, NT3
a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$
b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$
PS. a) original from Albania
b) modified by problem selecting committee
1968 Swedish Mathematical Competition, 4
For $n\ne 0$, let f(n) be the largest $k$ such that $3^k$ divides $n$. If $M$ is a set of $n > 1$ integers, show that the number of possible values for $f(m-n)$, where $m, n$ belong to $M$ cannot exceed $n-1$.
1991 Chile National Olympiad, 4
Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.
2011 QEDMO 9th, 6
Show that there are infinitely many pairs $(m, n)$ of natural numbers $m, n \ge 2$, for $m^m- 1$ is divisible by $n$ and $n^n- 1$ is divisible by $m$.
2015 Saudi Arabia GMO TST, 4
Let $p, q$ be two different odd prime numbers and $n$ an integer such that $pq$ divides $n^{pq} + 1$. Prove that if $p^3q^3$ divides $n^{pq} + 1$ then either $p^2$ divides $n + 1$ or $q^2$ divides $n + 1$.
Malik Talbi
1988 Tournament Of Towns, (186) 3
Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.
1994 Austrian-Polish Competition, 7
Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.
2021 Girls in Mathematics Tournament, 3
A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?
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)
2021 Saudi Arabia Training Tests, 39
Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.
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 $n$.
2016 Chile National Olympiad, 1
The natural number $a_n$ is obtained by writing together and ordered, in decimal notation , all natural numbers between $1$ and $n$. So we have for example that $a_1 = 1$,$a_2 = 12$, $a_3 = 123$, $. . .$ , $a_{11} = 1234567891011$, $...$ . Find all values of $n$ for which $a_n$ is not divisible by $3$.
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 $a_1a_2...a_9$.
2008 Dutch Mathematical Olympiad, 3
Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ($1$ and $2008$ included).
Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$.
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 $F_{2p} - F_p$ .