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

2016 May Olympiad, 3
Tags: Digits , number theory , divides
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 $1001$.

2023 Francophone Mathematical Olympiad, 4
Tags: number theory , divides , divisible
Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$.

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

2008 Postal Coaching, 3
Tags: number theory , Product , divides , divisible
Prove that there exists an in nite sequence $<a_n>$ of positive integers such that for each $k \ge 1$ $(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.

2017 QEDMO 15th, 10
Tags: number theory , divisor , divides
Let $p> 3$ be a prime number and let $q = \frac{4^p-1}{3}$. Show that $q$ is a composite integer as well is a divisor of $2^{q-1}- 1$.

1958 Kurschak Competition, 2
Tags: number theory , divides , 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 $3$.

2004 Denmark MO - Mohr Contest, 2
Tags: number theory , divides , divisible
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 $11$.

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 $n$.

2017 Regional Olympiad of Mexico West, 6
Tags: number theory , divides
A [i]change [/i] in a natural number $n$ consists of adding a pair of zeros between two digits or at the end of the decimal representation of $n$. A [i]countryman [/i] of $n$ is a number that can be obtained from one or more changes in $n$. For example. $40041$, $4410000$ and $4004001$ are all countrymen from $441$. Determine all the natural numbers $n$ for which there is a natural number m with the property that $n$ divides $m$ and all the countrymen of $m$.

2016 JBMO Shortlist, 2
Tags: JBMO , number theory , divides
Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions: a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$.

2016 India Regional Mathematical Olympiad, 3
Tags: number theory , divides , divisor
$a, b, c, d$ are integers such that $ad + bc$ divides each of $a, b, c$ and $d$. Prove that $ad + bc =\pm 1$

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

2018 NZMOC Camp Selection Problems, 1
Tags: multiple , divides , number theory
Suppose that $a, b, c$ and $d$ are four different integers. Explain why $$(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)$$ must be a multiple of $12$.

1956 Polish MO Finals, 4
Tags: number theory , divides , divisible
Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation $$ a^2 + b^2 = c^2,$$ then: 1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $, 2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $, 3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.

2022 Chile Junior Math Olympiad, 4
Tags: number theory , divides
Let $S$ be the sum of all products $ab$ where $a$ and $b$ are distinct elements of the set $\{1,2,...,46\}$. Prove that $47$ divides $S$.

2015 Dutch IMO TST, 5
Tags: number theory , divides , min , max , positive integers
Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

2007 Cuba MO, 5
Tags: number theory , Digits , divides , 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 $2^{2007}$.

2020 Durer Math Competition Finals, 6
Tags: number theory , divides
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 $(2, 0, 1)$.

1974 Dutch Mathematical Olympiad, 2
Tags: number theory , divides , divisible
$n>2$ numbers, $ x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$ x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$

1997 Chile National Olympiad, 2
Tags: number theory , divisible , divides
Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$. Prove that $a \equiv n \equiv 1 \mod 10$ .

2014 Peru MO (ONEM), 3
Tags: number theory , divides
a) Let $a, b, c$ be positive integers such that $ab + b + 1$, $bc + c + 1$ and $ca + a + 1$ are divisors of the number $abc - 1$, prove that $a = b = c$. b) Find all triples $(a, b, c)$ of positive integers such that the product $$(ab - b + 1)(bc - c + 1)(ca - a + 1)$$ is a divisor of the number $(abc + 1)^2$.

2014 India PRMO, 13
Tags: number theory , Integer , divides
For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?

2014 Czech-Polish-Slovak Junior Match, 4
Tags: number theory , divides , divisible , Digits
The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

2022 Federal Competition For Advanced Students, P2, 1
Tags: function , number theory , divides
Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

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