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

2015 Singapore Junior Math Olympiad, 1
Tags: multiple , number theory , divide , digit
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 $37$.

2007 Cuba MO, 5
Tags: number theory , digit , divide , power of 2
Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.

2018 District Olympiad, 2
Tags: number theory , divide
Find the pairs of integers $(a, b)$ such that $a^2 + 2b^2 + 2a +1$ is a divisor of $2ab$.

1976 Spain Mathematical Olympiad, 4
Tags: number theory , divide
Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.

1960 Poland - Second Round, 4
Tags: number theory , divide , divisible
Prove that if $ n $ is a non-negative integer, then number $$ 2^{n+2} + 3^{2n+1}$$ is divisible by $7$.

2012 NZMOC Camp Selection Problems, 4
Tags: number theory , multiple , divide , divisible , prime
A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

2004 Estonia National Olympiad, 4
Tags: divide , divisible , odd , number theory
Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.

2010 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , divide
Let $p$ be a prime number, $p> 5$. Determine the non-zero natural numbers $x$ with the property that $5p + x$ divides $5p ^ n + x ^ n$, whatever $n \in N ^ {*} $.

1989 Poland - Second Round, 4
Tags: number theory , divide , divisible
The given integers are $ a_1, a_2, \ldots , a_{11} $ . Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $ such that the number $ x_1a_1 + \ldots x_{11}a_{ 11}$ is divisible by 1989.

2015 Costa Rica - Final Round, N3
Tags: number theory , divide
Find all the pairs $a,b \in N$ such that $ab-1 |a^2 + 1$.

2016 Czech-Polish-Slovak Junior Match, 2
Tags: number theory , divide , digit
Find the largest integer $d$ divides all three numbers $abc, bca$ and $cab$ with $a, b$ and $c$ being some nonzero and mutually different digits. Czech Republic

VMEO III 2006 Shortlist, N6
Tags: number theory , divide
Find all sets of natural numbers $(a, b, c)$ such that $$a+1|b^2+c^2\,\, , b+1|c^2+a^2\,\,, c+1|a^2+b^2.$$

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

2009 China Northern MO, 3
Tags: number theory , divide
Given $26$ different positive integers , in any six numbers of the $26$ integers , there are at least two numbers , one can be devided by another. Then prove : There exists six numbers , one of them can be devided by the other five numbers .

2016 Costa Rica - Final Round, N2
Tags: divide , divisible , number theory
Let $x, y, z$ be positive integers and $p$ a prime such that $x <y <z <p$. Also $x^3, y^3, z^3$ leave the same remainder when divided by $p$. Prove that $x + y + z$ divides $x^2 + y^2 + z^2$.

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 $3$.

1947 Moscow Mathematical Olympiad, 136
Tags: divide , 13-gon , parallelogram , convex , combinatorics
Prove that no convex $13$-gon can be cut into parallelograms.

2016 May Olympiad, 3
Tags: digit , number theory , divide
We say that a positive integer is [i]quad-divi[/i] if it is divisible by the sum of the squares of its digits, and also none of its digits is equal to zero. a) Find a quad-divi number such that the sum of its digits is $24$. b) Find a quad-divi number such that the sum of its digits is $1001$.

2020 Durer Math Competition Finals, 6
Tags: number theory , divide
Positive integers $a, b$ and $c$ are all less than $2020$. We know that $a$ divides $b + c$, $b$ divides $a + c$ and $c$ divides $a + b$. How many such ordered triples $(a, b, c)$ are there? Note: In an ordered triple, the order of the numbers matters, so the ordered triple $(0, 1, 2)$ is not the same as the ordered triple $(2, 0, 1)$.

1999 Singapore Team Selection Test, 3
Tags: divide , number theory , prime number
Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$

1910 Eotvos Mathematical Competition, 2
Tags: number theory , multiple , divide
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 $u$.

2014 Chile National Olympiad, 4
Tags: number theory , divide , divisible
Prove that for every integer $n$ the expression $n^3-9n + 27$ is not divisible by $81$.

2008 China Northern MO, 3
Tags: number theory , prime factor , divide
Prove that: (1) There are infinitely many positive integers $n$ such that the largest prime factor of $n^2+1$ is less than $n.$ (2) There are infinitely many positive integers $n$ such that $n^2+1$ divides $n!$.

2007 Austria Beginners' Competition, 1
Tags: number theory , divide , 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 $n$.

2015 Dutch BxMO/EGMO TST, 1
Tags: number theory , divisor , divide
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.