Found problems: 408
2003 Bosnia and Herzegovina Junior BMO TST, 3
Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.
1941 Moscow Mathematical Olympiad, 072
Find the number $\overline {523abc}$ divisible by $7, 8$ and $9$.
2020 Swedish Mathematical Competition, 1
How many of the numbers $1\cdot 2\cdot 3$, $2\cdot 3\cdot 4$,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?
2009 Hanoi Open Mathematics Competitions, 11
Let $A = \{1,2,..., 100\}$ and $B$ is a subset of $A$ having $48$ elements.
Show that $B$ has two distint elements $x$ and $y$ whose sum is divisible by $11$.
2013 Flanders Math Olympiad, 1
A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.
2015 Saudi Arabia IMO TST, 1
Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$.
Trần Nam Dũng
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)
1940 Moscow Mathematical Olympiad, 070
How many positive integers $x$ less than $10 000$ are there such that $2^x - x^2$ is divisible by $7$ ?
2012 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$
1988 Poland - Second Round, 4
Prove that for every natural number $ n $, the number $ n^{2n} - n^{n+2} + n^n - 1 $ is divisible by $ (n - 1 )^3 $.
2013 Bosnia and Herzegovina Junior BMO TST, 1
It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers
2006 Austria Beginners' Competition, 1
Do integers $a, b$ exist such that $a^{2006} + b^{2006} + 1$ is divisible by $2006^2$?
1947 Moscow Mathematical Olympiad, 137
a) $101$ numbers are selected from the set $1, 2, . . . , 200$. Prove that among the numbers selected there is a pair in which one number is divisible by the other.
b) One number less than $16$, and $99$ other numbers are selected from the set $1, 2, . . . , 200$. Prove that among the selected numbers there are two such that one divides the other.
1992 Tournament Of Towns, (341) 3
Prove that for any positive integer $M$ there exists an integer divisible by $M$ such that the sum of its digits (in its decimal representation) is odd.
(D Fomin, St Petersburg)
2020 Kosovo National Mathematical Olympiad, 2
Let $a_1,a_2,...,a_n$ be integers such that $a_1^{20}+a_2^{20}+...+a_n^{20}$ is divisible by $2020$. Show that $a_1^{2020}+a_2^{2020}+...+a_n^{2020}$ is divisible by $2020$.
1993 Czech And Slovak Olympiad IIIA, 1
Find all natural numbers $n$ for which $7^n -1$ is divisible by $6^n -1$
2021 Durer Math Competition Finals, 8
Benedek wrote the following $300 $ statements on a piece of paper.
$2 | 1!$
$3 | 1! \,\,\, 3 | 2!$
$4 | 1! \,\,\, 4 | 2! \,\,\, 4 | 3!$
$5 | 1! \,\,\, 5 | 2! \,\,\, 5 | 3! \,\,\, 5 | 4!$
$...$
$24 | 1! \,\,\, 24 | 2! \,\,\, 24 | 3! \,\,\, 24 | 4! \,\,\, · · · \,\,\, 24 | 23!$
$25 | 1! \,\,\, 25 | 2! \,\,\, 25 | 3! \,\,\, 25 | 4! \,\,\, · · · \,\,\, 25 | 23! \,\,\, 25 | 24!$
How many true statements did Benedek write down?
The symbol | denotes divisibility, e.g. $6 | 4!$ means that $6$ is a divisor of number $4!$.
2016 Saudi Arabia IMO TST, 3
Let $n \ge 4$ be a positive integer and there exist $n$ positive integers that are arranged on a circle such that:
$\bullet$ The product of each pair of two non-adjacent numbers is divisible by $2015 \cdot 2016$.
$\bullet$ The product of each pair of two adjacent numbers is not divisible by $2015 \cdot 2016$.
Find the maximum value of $n$
2008 Postal Coaching, 3
Prove that for each natural number $m \ge 2$, there is a natural number $n$ such that $3^m$ divides $n^3 + 17$ but $3^{m+1}$ does not divide it.
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 $k \ge 3$?
2021 Dutch IMO TST, 3
Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.
1990 ITAMO, 5
Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.
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 $n$.
2018 Stanford Mathematics Tournament, 1
Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.
1972 Spain Mathematical Olympiad, 7
Prove that for every positive integer $n$, the number
$$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$
is a multiple of $8$.