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: 408

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)

1999 Switzerland Team Selection Test, 6
Tags: divisible , perfect square , number theory
Prove that if $m$ and $n$ are positive integers such that $m^2 + n^2 - m$ is divisible by $2mn$, then $m$ is a perfect square.

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

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

2018 Stars of Mathematics, 2
Tags: perfect square , divisible , number theory
Show that, if $m$ and $n$ are non-zero integers of like parity, and $n^2 -1$ is divisible by $m^2 - n^2 + 1$, then $m^2 - n^2 + 1$ is the square of an integer. Amer. Math. Monthly

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 $S = 1^n + 2^n +...+ (p - 1)^n$

2003 Estonia Team Selection Test, 2
Tags: divisible , number theory
Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$. (H. Nestra)

2019 Saudi Arabia BMO TST, 1
Tags: divide , divisible , prime , number theory
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 Austrian Junior Regional Competition, 4
Tags: prime , divide , divisible , number theory
Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

2011 QEDMO 9th, 6
Tags: divide , divisible , number theory
Show that there are infinitely many pairs $(m, n)$ of natural numbers $m, n \ge 2$, for $m^m- 1$ is divisible by $n$ and $n^n- 1$ is divisible by $m$.

2008 Tournament Of Towns, 4
Tags: divisible , number theory , factorial
Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

2010 Hanoi Open Mathematics Competitions, 2
Tags: divisible , number theory
Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$ (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E) None of the above.

2017 Saudi Arabia BMO TST, 1
Tags: divide , factorial , number theory , divisible
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.

2003 Chile National Olympiad, 5
Tags: number theory , divide , divisible
Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)

2019 Saudi Arabia IMO TST, 2
Tags: polynomial , algebra , divisible
Let non-constant polynomial $f(x)$ with real coefficients is given with the following property: for any positive integer $n$ and $k$, the value of expression $$\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z$$ Prove that $f(x)$ is divisible by $x$

1999 Kazakhstan National Olympiad, 6
Tags: remainder , number theory with sequences , sequence , divisible , number theory
In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.

2023 Assara - South Russian Girl's MO, 2
Tags: divide , divisible , number theory
The natural numbers $a$ and $b$ are such that $a^a$ is divisible by $b^b$. Can we say that then $a$ is divisible by $b$?

2000 Singapore MO Open, 2
Tags: divisible , prime , divide , gcd , number theory
Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.

1981 Swedish Mathematical Competition, 3
Tags: algebra , polynomial , divisible
Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.

1940 Moscow Mathematical Olympiad, 065
Tags: number theory , divisible
How many pairs of integers $x, y$ are there between $1$ and $1000$ such that $x^2 + y^2$ is divisible by $7$?

1955 Moscow Mathematical Olympiad, 290
Tags: divide , number theory , divisible
Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?

2005 Thailand Mathematical Olympiad, 19
Tags: polynomial , divisible , algebra
Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.

1958 Kurschak Competition, 2
Tags: number theory , divide , divisible
Show that if $m$ and $n$ are integers such that $m^2 + mn + n^2$ is divisible by $9$, then they must both be divisible by $3$.

2018 Tuymaada Olympiad, 5
Tags: divisible , prime number , number theory
A prime $p$ and a positive integer $n$ are given. The product $$(1^3+1)(2^3+1)...((n-1)^3+1)(n^3+1)$$ is divisible by $p^3$. Prove that $p \leq n+1$. [i]Proposed by Z. Luria[/i]