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

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

1997 Singapore MO Open, 2
Tags: Binomial , divides , divisible , number theory
Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.

2019 Federal Competition For Advanced Students, P2, 6
Tags: number theory , divides , min
Find the smallest possible positive integer n with the following property: For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$. (Gerhard J. Woeginger)

2016 Saudi Arabia Pre-TST, 2.4
Tags: number theory , Product , divides , divisible
Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b > n$ such that $$\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)$$.

2003 Austrian-Polish Competition, 7
Tags: number theory , divides , 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 $(n^n)!$ .

2011 Saudi Arabia BMO TST, 4
Tags: fibonacci number , number theory , divides , divisible
Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$. Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .

2011 Tournament of Towns, 6
Tags: divides , divisor , number theory , power of 2
Prove that the integer $1^1 + 3^3 + 5^5 + .. + (2^n - 1)^{2^n-1}$ is a multiple of $2^n$ but not a multiple of $2^{n+1}$.

1999 Tournament Of Towns, 3
Tags: number theory , divides , divisible , Sum of Squares
Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

2011 QEDMO 10th, 3
Tags: multiple , number theory , divides
Let $a, b$ be positive integers such that $a^2 + ab + 1$ a multiple of $b^2 + ab + 1$. Prove that $a = b$.

1989 Poland - Second Round, 4
Tags: number theory , divides , 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.

2013 Thailand Mathematical Olympiad, 5
Tags: sum of digits , number theory , Digits , divides , 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 $n$.

2005 Chile National Olympiad, 2
Tags: number theory , divides
Let $p$ be a prime number greater than $2$ and let $m, n$ be integers such that: $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}.$$ Prove that $p$ divides $m$.

2008 Thailand Mathematical Olympiad, 4
Tags: number theory , Binomial , divisible , divides
Let $n$ be a positive integer. Show that $${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{n}{2n+1 \choose 2n+1}2008^n $$ is not divisible by $19$.

2009 Chile National Olympiad, 4
Tags: divides , divisible , number theory
Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$

2015 Thailand Mathematical Olympiad, 1
Tags: divides , recurrence relation , divisible , number theory
Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.

2018 Istmo Centroamericano MO, 1
Tags: number theory , recurrence relation , divides , divisible
A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.

1939 Eotvos Mathematical Competition, 2
Tags: number theory , factorial , power of 2 , divides
Determine the highest power of $2$ that divides $2^n!$.

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

1989 Tournament Of Towns, (205) 3
Tags: number theory , divisible , divides , Digits
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)

VMEO III 2006, 10.2
Tags: consecutive , dividisible , divides , number theory
Prove that among $39$ consecutive natural numbers, there is always a number that has sum of its digits divisible by $ 12$. Is it true if we replace $39$ with $38$?

2018 Saudi Arabia IMO TST, 1
Tags: number theory , divides , divisible , functional
Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

2006 Thailand Mathematical Olympiad, 5
Tags: divides , divisible , number theory
Show that there are coprime positive integers $m$ and $n$ such that $2549 | (25 \cdot 49)^m + 25^n - 2 \cdot 49^n$

2013 Balkan MO Shortlist, N8
Tags: number theory , divides , Integers
Suppose that $a$ and $b$ are integers. Prove that there are integers $c$ and $d$ such that $a+b+c+d=0$ and $ac+bd=0$, if and only if $a-b$ divides $2ab$.

2023 Durer Math Competition Finals, 15
Tags: number theory , divides
What is the biggest positive integer which divides $p^4 - q^4$ for all primes $p$ and $q$ greater than $10$?

2016 Costa Rica - Final Round, N2
Tags: divides , 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 $x^2 + y^2 + z^2$.