Found problems: 387
1987 Tournament Of Towns, (159) 3
Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .
2018 NZMOC Camp Selection Problems, 1
Suppose that $a, b, c$ and $d$ are four different integers. Explain why $$(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)$$ must be a multiple of $12$.
1986 Tournament Of Towns, (129) 4
We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even .
For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ .
Prove that $1986 !! + 1985 !!$ i s divisible by $1987$.
(V.V . Proizvolov , Moscow)
2016 Grand Duchy of Lithuania, 4
Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.
1945 Moscow Mathematical Olympiad, 094
Prove that it is impossible to divide a scalene triangle into two equal triangles.
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 $6$.
2013 India PRMO, 13
To each element of the set $S = \{1,2,... ,1000\}$ a colour is assigned. Suppose that for any two elements $a, b$ of $S$, if $15$ divides $a + b$ then they are both assigned the same colour. What is the maximum possible number of distinct colours used?
2023 Durer Math Competition Finals, 15
What is the biggest positive integer which divides $p^4 - q^4$ for all primes $p$ and $q$ greater than $10$?
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 $2020$?
2023 Peru MO (ONEM), 1
We define the set $M = \{1^2,2^2,3^2,..., 99^2, 100^2\}$.
a) What is the smallest positive integer that divides exactly two elements of $M$?
b) What is the largest positive integer that divides exactly two elements of $M$?
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)
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 $abc|(a+b+c)^{13}$
2015 Finnish National High School Mathematics Comp, 3
Determine the largest integer $k$ for which $12^k$ is a factor of $120! $
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 $.
2011 Indonesia TST, 4
Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.
2005 Greece JBMO TST, 4
Find all the positive integers $n , n\ge 3$ such that $n\mid (n-2)!$
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 $4!$.
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 $n | 4a^2 + 9b^2 -1$.
2019 Denmark MO - Mohr Contest, 3
Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?
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 $8$.