Found problems: 367
1925 Eotvos Mathematical Competition, 1
Let $a,b, c,d$ be four integers. Prove that the product of the six differences
$$b - a,c - a,d - a,d - c,d - b, c - b$$
is divisible by $12$.
2015 NZMOC Camp Selection Problems, 8
Determine all positive integers $n$ which have a divisor $d$ with the property that $dn + 1$ is a divisor of $d^2 + n^2$.
1999 Estonia National Olympiad, 5
On the squares $a1, a2,... , a8$ of a chessboard there are respectively $2^0, 2^1, ..., 2^7$ grains of oat, on the squares $b8, b7,..., b1$ respectively $2^8, 2^9, ..., 2^{15}$ grains of oat, on the squares $c1, c2,..., c8$ respectively $2^{16}, 2^{17}, ..., 2^{23}$ grains of oat etc. (so there are $2^{63}$ grains of oat on the square $h1$). A knight starts moving from some square and eats after each move all the grains of oat on the square to which it had jumped, but immediately after the knight leaves the square the same number of grains of oat reappear. With the last move the knight arrives to the same square from which it started moving. Prove that the number of grains of oat eaten by the knight is divisible by $3$.
2014 Saudi Arabia Pre-TST, 2.1
Prove that $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.
1994 Poland - Second Round, 6
Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.
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 Saudi Arabia BMO TST, 1
Let $p$ be an odd prime number.
a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n.
b) Find all $n$ satisfy condition above when $p = 3$
2019 Saudi Arabia Pre-TST + Training Tests, 1.2
Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .
2017 Abels Math Contest (Norwegian MO) Final, 2
Let the sequence an be defined by $a_0 = 2, a_1 = 15$, and $a_{n+2 }= 15a_{n+1} + 16a_n$ for $n \ge 0$.
Show that there are infinitely many integers $k$ such that $269 | a_k$.
1977 Swedish Mathematical Competition, 1
$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$.
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.
1986 Austrian-Polish Competition, 7
Let $k$ and $n$ be integers with $0 < k < n^2/4$ such that k has no prime divisor greater than $n$. Prove that $k$ divides $n!$.
2007 Estonia National Olympiad, 4
Let $a, b,c$ be positive integers such that $gcd(a, b, c) = 1$ and each product of two is divided by the third.
a) Prove that each of these numbers is equal to the least two remaining numbers the quotient of the coefficient and the highest coefficient.
b) Give an example of one of these larger numbers $a, b$ and $c$
2023 Francophone Mathematical Olympiad, 4
Do there exist integers $a$ and $b$ such that none of the numbers $a,a+1,\ldots,a+2023,b,b+1,\ldots,b+2023$ divides any of the $4047$ other numbers, but $a(a+1)(a+2)\cdots(a+2023)$ divides $b(b+1)\cdots(b+2023)$?
2011 Argentina National Olympiad, 5
Find all integers $n$ such that $1<n<10^6$ and $n^3-1$ is divisible by $10^6 n-1$.
2007 Switzerland - Final Round, 2
Let $a, b, c$ be three integers such that $a + b + c$ is divisible by $13$. Prove that $$a^{2007}+b^{2007}+c^{2007}+2 \cdot 2007abc$$ is divisible by $13$.
2021 Durer Math Competition Finals, 14
How many functions $f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 16\}$ have the property that $f(f(x))-4x$ is divisible by $17$ for all integers $1 \le x \le 16$?
2003 May Olympiad, 3
Find all pairs of positive integers $(a,b)$ such that $8b+1$ is a multiple of $a$ and $8a+1$ is a multiple of $b$.
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 $y$.
2010 Saudi Arabia Pre-TST, 1.3
1) Let $a$ and $b$ be relatively prime positive integers. Prove that there is a positive integer $n$ such that $1 \le n \le b$ and $b$ divides $a^n - 1$.
2) Prove that there is a multiple of $7^{2010}$ of the form $99... 9$ ($n$ nines), for some positive integer $n$ not exceeding $7^{2010}$.
2008 Regional Olympiad of Mexico Center Zone, 5
Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_ {32} = 6$, $p_ {203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$. Find the largest prime that divides $S $.
2021 Auckland Mathematical Olympiad, 4
Prove that there exist two powers of $7$ whose difference is divisible by $2021$.
2015 Balkan MO Shortlist, N3
Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $
Let $s,t$ be two different positive integers with the following property:
If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$.
Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer.
(FYROM)
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?
2003 Cuba MO, 2
Prove that if $$\frac{p}{q}=1-\frac{1}{2} + \frac{1}{3}- \frac{1}{4} + ... -\frac{1}{1334} + \frac{1}{1335}$$ where $p, q \in Z_+$ then $p$ is divisible by $2003$.