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

2014 Junior Balkan Team Selection Tests - Moldova, 4
Tags: number theory , divide , divisible
A set $A$ contains $956$ natural numbers between $1$ and $2014$, inclusive. Prove that in the set $A$ there are two numbers $a$ and $b$ such that $a + b$ is divided by $19$.

2017 Czech-Polish-Slovak Junior Match, 3
Tags: number theory , digit , divisible
How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

1911 Eotvos Mathematical Competition, 3
Tags: number theory , divide , divisible
Prove that $3^n + 1$ is not divisible by $2^n$ for any integer $n > 1$.

2022 Brazil EGMO TST, 3
Tags: number theory , divisible , sum of digits
A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?

1999 Abels Math Contest (Norwegian MO), 2b
Tags: number theory , divide , divisible
If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$

2015 Puerto Rico Team Selection Test, 8
Tags: divisible , number theory , multiple , divide
Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values ​​of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

1984 Tournament Of Towns, (071) T5
Tags: number theory , sum of digits , divisible , consecutive
Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.

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

1983 All Soviet Union Mathematical Olympiad, 360
Tags: divisible , exponential , number theory
Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$.

2011 May Olympiad, 5
Tags: number theory , digit , divisible
We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?

2020 Tournament Of Towns, 1
Tags: number theory , digit , divisible
Does there exist a positive integer that is divisible by $2020$ and has equal numbers of digits $0, 1, 2, . . . , 9$ ? Mikhail Evdokimov

2015 Saudi Arabia GMO TST, 4
Tags: number theory , divide , divisible
Let $p$ be an odd prime number. Prove that there exists a unique integer $k$ such that $0 \le k \le p^2$ and $p^2$ divides $k(k + 1)(k + 2) ... (k + p - 3) - 1$. Malik Talbi

1975 Chisinau City MO, 101
Tags: number theory , divisible , difference
Prove that among any $k + 1$ natural numbers there are two numbers whose difference is divisible by $k$.

2017 Hanoi Open Mathematics Competitions, 14
Tags: divisible , number theory
Put $P = m^{2003}n^{2017} - m^{2017}n^{2003}$ , where $m, n \in N$. a) Is $P$ divisible by $24$? b) Do there exist $m, n \in N$ such that $P$ is not divisible by $7$?

2018 Dutch Mathematical Olympiad, 1
Tags: number theory , digit , divisible
We call a positive integer a [i]shuffle[/i] number if the following hold: (1) All digits are nonzero. (2) The number is divisible by $11$. (3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$. How many $10$-digit [i]shuffle[/i] numbers are there?

1996 Austrian-Polish Competition, 1
Tags: number theory , divide , divisible , digit
Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$, (iv) the number $m = n/5$ has $k$ odd digits

2000 Tournament Of Towns, 5
Tags: number theory , divisible , consecutive , divide
What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le k \le N$ ? (S Tokarev)

2015 Thailand Mathematical Olympiad, 1
Tags: number theory , divide , recurrence relation , divisible
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 $n$.

2022 Czech-Polish-Slovak Junior Match, 5
Tags: number theory , divisible
An integer $n\ge1$ is [i]good [/i] if the following property is satisfied: If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$. How many good integers are $n\ge 1$?

1965 Dutch Mathematical Olympiad, 2
Tags: number theory , perfect square , divide , divisible
Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.

2012 Tournament of Towns, 5
Tags: number theory , prime , sum , divisible
Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.

2021 Girls in Mathematics Tournament, 3
Tags: number theory , divisible , sum of digits
A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?

2012 Brazil Team Selection Test, 2
Tags: number theory , divide , divisible
Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.

2011 Argentina National Olympiad, 5
Tags: number theory , divide , divisible
Find all integers $n$ such that $1<n<10^6$ and $n^3-1$ is divisible by $10^6 n-1$.

2011 Danube Mathematical Competition, 3
Tags: prime number , divisible , number theory
Determine all positive integer numbers $n$ satisfying the following condition: the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.