Found problems: 387
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 $b(b+1)\cdots(b+2023)$?
2015 Balkan MO Shortlist, N2
Sequence $(a_n)_{n\geq 0}$ is defined as $a_{0}=0, a_1=1, a_2=2, a_3=6$,
and $ a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0$.
Prove that $n^2$ divides $a_n$ for infinite $n$.
(Romania)
1998 Rioplatense Mathematical Olympiad, Level 3, 3
Let $X$ be a finite set of positive integers.
Prove that for every subset $A$ of $X$, there is a subset $B$ of $X$, with the following property:
For each element $ e$ of $X$, $e$ divides an odd number of elements of $B$, if and only if $e$ is an element of $A$.
2014 Saudi Arabia GMO TST, 2
Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$
2009 Chile National Olympiad, 4
Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$
2008 Switzerland - Final Round, 6
Determine all odd natural numbers of the form $$\frac{p + q}{p - q},$$ where $p > q$ are prime numbers.
2021 Czech and Slovak Olympiad III A, 4
Find all natural numbers $n$ for which equality holds $n + d (n) + d (d (n)) +... = 2021$, where $d (0) = d (1) = 0$ and for $k> 1$, $ d (k)$ is the [i]superdivisor [/i] of the number $k$ (i.e. its largest divisor of $d$ with property $d <k$).
(Tomáš Bárta)
1995 Chile National Olympiad, 1
Let $a,b,c,d$ be integers. Prove that $ 12$ divides $ (a-b) (a-c) (a-d) (b- c) (b-d) (c-d)$.
2019 Durer Math Competition Finals, 11
What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?
2022 Chile Junior Math Olympiad, 4
Let $S$ be the sum of all products $ab$ where $a$ and $b$ are distinct elements of the set $\{1,2,...,46\}$. Prove that $47$ divides $S$.
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 $.
2005 Thailand Mathematical Olympiad, 2
Let $S $ be a set of three distinct integers. Show that there are $a, b \in S$ such that $a \ne b$ and $10 | a^3b - ab^3$.
2019 Junior Balkan Team Selection Tests - Romania, 2
Let $n$ be a positive integer and $A$ a set containing $8n + 1$ positive integers co-prime with $6$ and less than $30n$. Prove that there exist $a, b \in A$ two different numbers such that $a$ divides $b$.
1997 Chile National Olympiad, 2
Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$.
Prove that $a \equiv n \equiv 1 \mod 10$ .
2012 Brazil Team Selection Test, 2
Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.
2016 India Regional Mathematical Olympiad, 3
$a, b, c, d$ are integers such that $ad + bc$ divides each of $a, b, c$ and $d$. Prove that $ad + bc =\pm 1$
1995 Bulgaria National Olympiad, 1
Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.
2012 NZMOC Camp Selection Problems, 2
Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.
2000 All-Russian Olympiad Regional Round, 9.2
Are there different mutually prime natural numbers $a$, $b$ and $c$, greater than $1$, such that $2a + 1$ is divisible by $b$, $2b + 1$ is divisible by $c$ and $2c + 1$ is divisible by $a$?
2019 Ecuador NMO (OMEC), 3
For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$
2006 Singapore Senior Math Olympiad, 1
Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.
2010 Grand Duchy of Lithuania, 5
Find positive integers n that satisfy the following two conditions:
(a) the quotient obtained when $n$ is divided by $9$ is a positive three digit number, that has equal digits.
(b) the quotient obtained when $n + 36$ is divided by $4$ is a four digit number, the digits beeing $2, 0, 0, 9$ in some order.
2001 BAMO, 5
For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1, 2, ... , n\}$ such that $\tau (\tau (\tau (x))) = x$ for $x = 1, 2, ..., n$. The first few values are $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 9$.
Prove that $3^{334}$ divides $a_{2001}$.
(A permutation of $\{1, 2, ... , n\}$ is a rearrangement of the numbers $\{1, 2, ... , n\}$ or equivalently, a one-to-one and
onto function from $\{1, 2, ... , n\}$ to $\{1, 2, ... , n\}$. For example, one permutation of $\{1, 2, 3\}$ is the rearrangement $\{2, 1, 3\}$, which is equivalent to the function $\sigma : \{1, 2, 3\} \to \{1, 2, 3\}$ defined by $\sigma (1) = 2, \sigma (2) = 1, \sigma (3) = 3$.)
2015 Puerto Rico Team Selection Test, 8
Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.
1999 Tournament Of Towns, 3
Find all pairs $(x, y)$ of integers satisfying the following condition:
each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ .
(S Zlobin)