Found problems: 408
2012 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$
2008 Estonia Team Selection Test, 4
Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.
2016 Saudi Arabia BMO TST, 3
Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.
2017 Saudi Arabia BMO TST, 1
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.
2012 Tournament of Towns, 5
Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.
2018 Istmo Centroamericano MO, 1
A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.
1940 Moscow Mathematical Olympiad, 065
How many pairs of integers $x, y$ are there between $1$ and $1000$ such that $x^2 + y^2$ is divisible by $7$?
1995 Singapore MO Open, 4
Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.
2000 Singapore MO Open, 2
Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.
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 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 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 $4!$.
1969 Swedish Mathematical Competition, 5
Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.
2016 Grand Duchy of Lithuania, 4
Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.
1970 Polish MO Finals, 3
Prove that an integer $n > 1$ is a prime number if and only if, for every integer $k$ with $1\le k \le n-1$, the binomial coefficient $n \choose k$ is divisible by $n$.
2003 Singapore Senior Math Olympiad, 1
It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.
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)
2015 Saudi Arabia BMO TST, 4
Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$.
Lê Anh Vinh
1980 Tournament Of Towns, (003) 3
If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.
1995 Poland - Second Round, 1
For a polynomial $P$ with integer coefficients, $P(5)$ is divisible by $2$ and $P(2)$ is divisible by $5$. Prove that $P(7)$ is divisible by $10$.
2019 Paraguay Mathematical Olympiad, 4
Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.
2018 Estonia Team Selection Test, 12
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
2016 IMAR Test, 1
Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the fi rst $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?
2018 Dutch Mathematical Olympiad, 1
We call a positive integer a [i]shuffle[/i] number if the following hold:
(1) All digits are nonzero.
(2) The number is divisible by $11$.
(3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$.
How many $10$-digit [i]shuffle[/i] numbers are there?
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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