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: 387

1983 Austrian-Polish Competition, 8
Tags: sequence , divide , product , factorial , number theory , recurrence relation , divisible
(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n (b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.

1998 Estonia National Olympiad, 4
Tags: prime , divide , divisible , number theory
Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

2006 Peru MO (ONEM), 3
Tags: divide , number theory
A pair $(m, n)$ of positive integers is called “[i]linked[/i]” if $m$ divides $3n + 1$ and $n$ divides $3m + 1$. If $a, b, c$ are distinct positive integers such that $(a, b)$ and $( b, c)$ are linked pairs, prove that the number $1$ belongs to the set $\{a, b, c\}$

2013 Cuba MO, 8
Tags: divide , number theory
Prove that there are infinitely many pairs $(a, b)$ of positive integers with the following properties: $\bullet$ $a+b$ divides $ab+1$, $\bullet$ $a-b$ divides $ab -1$, $\bullet$ $b > 2$ and $a > b\sqrt3 - 1$.

2010 Cuba MO, 5
Tags: number theory , divide , divisible
Let $p\ge 2$ be a prime number and $a\ge 1$ be an integer different from $p$. Find all pairs $(a, p)$ such that $a + p | a^2 + p^2$.

2014 Saudi Arabia Pre-TST, 4.1
Tags: divide , number theory , inequalities , divisible
Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.

2019 Saudi Arabia Pre-TST + Training Tests, 1.2
Tags: divide , arithmetic sequence , number theory
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 Saudi Arabia IMO TST, 3
Tags: divide , number theory , divisible
Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.

2007 Switzerland - Final Round, 2
Tags: divisible , divide , number theory
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 $13$.

2003 Junior Balkan Team Selection Tests - Romania, 2
Tags: prime , consecutive , divide , sum of powers , number theory
Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.

2007 Bosnia and Herzegovina Junior BMO TST, 2
Tags: divisible , divide , number theory
Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$. .

2015 Singapore Junior Math Olympiad, 1
Tags: multiple , divide , digit , number theory
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

1991 Bundeswettbewerb Mathematik, 2
Tags: number theory , divide
Let $g$ be an even positive integer and $f(n) = g^n + 1$ , $(n \in N^* )$. Prove that for every positive integer $n$ we have: a) $f(n)$ divides each of the numbers $f(3n), f(5n), f(7n)$ b) $f(n)$ is relative prime to each of the numbers $f(2n), f(4n),f(6n),...$

1991 Chile National Olympiad, 4
Tags: divide , divisible , number theory
Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.

2019 Durer Math Competition Finals, 6
Tags: number theory , multiple , divide , divisible , digit
Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?

2004 Denmark MO - Mohr Contest, 2
Tags: divide , divisible , number theory
Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.

1989 Tournament Of Towns, (205) 3
Tags: number theory , divide , digit , divisible
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$? (M . I. Gusarov)

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

2010 Grand Duchy of Lithuania, 5
Tags: divide , number theory
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.

2022 Junior Balkan Team Selection Tests - Moldova, 11
Tags: divide , number theory
Find all ordered pairs of positive integers $(m, n)$ such that $2m$ divides the number $3n - 2$, and $2n$ divides the number $3m - 2$.

2019 Paraguay Mathematical Olympiad, 4
Tags: divide , number theory , divisible
Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.

2013 Korea Junior Math Olympiad, 4
Tags: prime number , divide , minimum , number theory
Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.

2014 Saudi Arabia Pre-TST, 2.1
Tags: divide , divisible , number theory
Prove that $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.

1996 Tournament Of Towns, (509) 2
Tags: number theory , divide , divisible
Do there exist three different prime numbers $p$, $q$ and $r$ such that $p^2 + d$ is divisible by $qr$, $q^2 + d$ is divisible by $rp$ and $r^2 + d$ is divisible by $pq$, if (a) $d = 10$; (b) $d = 11$? (V Senderov)

2019 Saudi Arabia Pre-TST + Training Tests, 5.1
Tags: prime , divide , divisible , number theory
Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$