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

2020 Swedish Mathematical Competition, 5
Tags: number theory , combinatorics , divide , divisible
Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides ${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$+ ...+$ {p-1 \choose p-3} a^{p-5} .$

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

2016 Saudi Arabia BMO TST, 3
Tags: number theory , divide , divisible
For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.

1952 Moscow Mathematical Olympiad, 232
Tags: polynomial , algebra , divide
Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.

2009 China Northern MO, 8
Tags: number theory , sum of digits , divide
Find the smallest positive integer $N$ satisfies : 1 . $209$│$N$ 2 . $ S (N) = 209 $ ( # Here $S(m)$ means the sum of digits of number $m$ )

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

2007 Thailand Mathematical Olympiad, 11
Tags: number theory , combinatorics , divisible , divide
Compute the number of functions $f : \{1, 2,... , 2550\} \to \{61, 80, 84\}$ such that $\sum_{k=1}^{2550} f(k)$ is divisible by $3$.

2015 NZMOC Camp Selection Problems, 8
Tags: divide , divisible , divisor , number theory
Determine all positive integers $n$ which have a divisor $d$ with the property that $dn + 1$ is a divisor of $d^2 + n^2$.

2003 Austrian-Polish Competition, 7
Tags: number theory , divide , factorial , divisible
Put $f(n) = \frac{n^n - 1}{n - 1}$. Show that $n!^{f(n)}$ divides $(n^n)! $. Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .

2010 Saudi Arabia Pre-TST, 1.3
Tags: number theory , divide , digit , divisible
1) Let $a$ and $b$ be relatively prime positive integers. Prove that there is a positive integer $n$ such that $1 \le n \le b$ and $b$ divides $a^n - 1$. 2) Prove that there is a multiple of $7^{2010}$ of the form $99... 9$ ($n$ nines), for some positive integer $n$ not exceeding $7^{2010}$.

1969 Swedish Mathematical Competition, 5
Tags: number theory , divide , divisible
Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.

2012 Switzerland - Final Round, 7
Tags: number theory , divide , divisible
Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.

1998 Tournament Of Towns, 3
Tags: combinatorics , divisible , divide
Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.) (G Galperin)

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

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 $2^p - 1|N$.

2017 Gulf Math Olympiad, 4
Tags: ceiling function , power of 2 , divide , maximum , number theory
1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ . 2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$

2016 Dutch BxMO TST, 5
Tags: number theory , divide
Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$

2018 Czech-Polish-Slovak Junior Match, 1
Tags: number theory , divide , divisible
For natural numbers $a, b c$ it holds that $(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc$. Prove that $(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2$

2020 Argentina National Olympiad, 1
Tags: number theory , divide , sum of digits
For every positive integer $n$, let $S (n)$ be the sum of the digits of $n$. Find, if any, a $171$-digit positive integer $n$ such that $7$ divides $S (n)$ and $7$ divides $S (n + 1)$.

1979 Chisinau City MO, 174
Tags: number theory , odd , divide , divisible
Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.

2017 Saudi Arabia Pre-TST + Training Tests, 1
Tags: number theory , divide
Let $m, n, k$ and $l$ be positive integers with $n \ne 1$ such that $n^k + mn^l + 1$ divides $n^{k+l }- 1$. Prove that either $m = 1$ and $l = 2k$, or $l | k$ and $m =\frac{n^{k-l} - 1}{n^l - 1}$.

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

1951 Kurschak Competition, 2
Tags: number theory , divide , divisible
For which $m > 1$ is $(m -1)!$ divisible by $m$?

2012 India Regional Mathematical Olympiad, 2
Tags: power , number theory , divide , divisible
Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$

2013 Thailand Mathematical Olympiad, 5
Tags: sum of digits , number theory , digit , divide , divisible
Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.