This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 367

2006 Switzerland - Final Round, 10
Tags: number theory , divisible , divides
Decide whether there is an integer $n > 1$ with the following properties: (a) $n$ is not a prime number. (b) For all integers $a$, $a^n - a$ is divisible by $n$

2011 Indonesia TST, 4
Tags: number theory , prime , sum of divisors , divides
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 $2^p - 1|N$.

2007 Austria Beginners' Competition, 1
Tags: number theory , divides , divisible
Prove that the number $9^n+8^n+7^n+6^n-4^n-3^n-2^n-1^n$ is divisible by $10$ for all non-negative $n$.

2017 Balkan MO Shortlist, N3
Tags: number theory , divides , exponential
Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.

1988 Tournament Of Towns, (186) 3
Tags: number theory , divisible , Sum , divides
Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.

1996 Estonia National Olympiad, 4
Tags: number theory , divisible , divides , Sum of powers , Sum
Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.

2013 Tournament of Towns, 6
Tags: prime , Fraction , divides , divisor , number theory , prime numbers
The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.

2017 Saudi Arabia IMO TST, 3
Tags: number theory , divides , divisible
Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.

1971 Spain Mathematical Olympiad, 8
Tags: number theory , divides
Among the $2n$ numbers $1, 2, 3, . . . , 2n$ are chosen in any way $n + 1$ different numbers. Prove that among the chosen numbers there are at least two, such that one divides the other.

2000 All-Russian Olympiad Regional Round, 9.2
Tags: number theory , divisible , divides
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 $a$?

1910 Eotvos Mathematical Competition, 2
Tags: number theory , multiple , divides
Let $a, b, c, d$ and $u$ be integers such that each of the numbers $$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$ is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.

2003 Austrian-Polish Competition, 4
Tags: divides , divisible , Product , number theory
A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.

2015 Balkan MO Shortlist, N2
Tags: recurrence relation , number theory with sequences , Sequence , divides , number theory , algebra
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)

1997 All-Russian Olympiad Regional Round, 10.3
Tags: number theory , divisible , divides
Natural numbers $m$ and $n$ are given. Prove that the number $2^n-1$ is divisible by the number $(2^m -1)^2$ if and only if the number $n$ is divisible by the number $m(2^m-1)$.

2023 Grand Duchy of Lithuania, 4
Tags: number theory , divides , factorial
Note that $k\ge 1$ for an odd natural number $$k! ! = k \cdot (k - 2) \cdot ... \cdot 1.$$ Prove that $2^n$ divides $(2^n -1)!! -1$ for all $n \ge 3$.

2003 Switzerland Team Selection Test, 9
Tags: number theory , combinatorics , divides
Given integers $0 < a_1 < a_2 <... < a_{101} < 5050$, prove that one can always choose for different numbers $a_k,a_l,a_m,a_n$ such that $5050 | a_k +a_l -a_m -a_n$

2005 Greece JBMO TST, 4
Tags: number theory , divides , factorial
Find all the positive integers $n , n\ge 3$ such that $n\mid (n-2)!$

1989 Tournament Of Towns, (215) 3
Tags: number theory , divides , divisible
Find six distinct positive integers such that the product of any two of them is divisible by their sum. (D. Fomin, Leningrad)

1975 Chisinau City MO, 102
Tags: combinatorics , number theory , divisible , divides , Digits
Two people write a $2k$-digit number, using only the numbers $1, 2, 3, 4$ and $5$. The first number on the left is written by the first of them, the second - the second, the third - the first, etc. Can the second one achieve this so that the resulting number is divisible by $9$, if the first seeks to interfere with it? Consider the cases $k = 10$ and $k = 15$.

2015 Saudi Arabia GMO TST, 4
Tags: number theory , divides , divisible
Let $p, q$ be two different odd prime numbers and $n$ an integer such that $pq$ divides $n^{pq} + 1$. Prove that if $p^3q^3$ divides $n^{pq} + 1$ then either $p^2$ divides $n + 1$ or $q^2$ divides $n + 1$. Malik Talbi

2018 Costa Rica - Final Round, 5
Tags: number theory , divisible , divides
Let $a$ and $ b$ be even numbers, such that $M = (a + b)^2-ab$ is a multiple of $5$. Consider the following statements: I) The unit digits of $a^3$ and $b^3$ are different. II) $M$ is divisible by $100$. Please indicate which of the above statements are true with certainty.

2021 Saudi Arabia Training Tests, 39
Tags: Sum , number theory , divides , divisible
Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.

VMEO III 2006 Shortlist, N3
Tags: number theory , divides
Given odd prime $p$. Sequence ${x_n}$ is defined by $x_{n+2}= 4x_{n+1}-x_n$. Choose $x_0,x_1$ such that for every random positive integer $k$, there exists $i\in \mathbb N$ such that $4p^2-8p+1|x_i - (2p)^k$.

2021 Saudi Arabia BMO TST, 3
Tags: number theory , divides , divisible
Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.

2022 Switzerland - Final Round, 2
Tags: number theory , remainder , divides
Let $n$ be a positive integer. Prove that the numbers $$1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}$$ all give different remainders when divided by $2^n$.