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: 536

2014-2015 SDML (High School), 2
Tags: SDML High School , 2014-15 SDML Round 2 , Divisibility , SDML Middle School
Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number?

2025 VJIMC, 1
Tags: number theory , primes , Divisibility
Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$, there is a prime number $p$ (possibly depending on $a,b,n$) such that $a^n + b$ is divisible by $p$, but not divisible by $p^2$.

1999 IMO, 4
Tags: number theory , Divisibility , prime , IMO Shortlist , IMO , IMO 1999 , Hi
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

2022 CIIM, 4
Tags: permutations , counting , number theory , Divisibility
Given a positive integer $n$, determine how many permutations $\sigma$ of the set $\{1, 2, \ldots , 2022n\}$ have the following property: for each $i \in \{1, 2, \ldots , 2021n + 1\}$, the number $$\sigma(i) + \sigma(i + 1) + \cdots + \sigma(i + n - 1)$$ is a multiple of $n$.

2019 Tournament Of Towns, 3
Tags: number theory , Tournament of Towns , ToT , Divisibility
The product of two positive integers $m$ and $n$ is divisible by their sum. Prove that $m + n \le n^2$. (Boris Frenkin)

2016 Rioplatense Mathematical Olympiad, Level 3, 6
Tags: number theory , sum of digits , Sequence , Divisibility
When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions: (i) $n$ divides $A_m$, (ii) $n$ divides $m$, (iii) $n$ divides the sum of the digits of $A_m$.

2021 Balkan MO Shortlist, C2
Tags: Balkan , shortlist , 2021 , combinatorics , Divisibility , maximum
Let $K$ and $N > K$ be fixed positive integers. Let $n$ be a positive integer and let $a_1, a_2, ..., a_n$ be distinct integers. Suppose that whenever $m_1, m_2, ..., m_n$ are integers, not all equal to $0$, such that $\mid{m_i}\mid \le K$ for each $i$, then the sum $$\sum_{i = 1}^{n} m_ia_i$$ is not divisible by $N$. What is the largest possible value of $n$? [i]Proposed by Ilija Jovcevski, North Macedonia[/i]

2018 OMMock - Mexico National Olympiad Mock Exam, 4
Tags: number theory , sum of digits , Divisibility
For each positive integer $n$ let $s(n)$ denote the sum of the decimal digits of $n$. Find all pairs of positive integers $(a, b)$ with $a > b$ which simultaneously satisfy the following two conditions $$a \mid b + s(a)$$ $$b \mid a + s(b)$$ [i]Proposed by Victor Domínguez[/i]

1969 IMO Longlists, 54
Tags: algebra , polynomial , modular arithmetic , number theory , Divisibility , IMO Shortlist , IMO Longlist
$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

2008 India Regional Mathematical Olympiad, 5
Tags: number theory , Divisibility , positive integer
Let $N$ be a ten digit positive integer divisible by $7$. Suppose the first and the last digit of $N$ are interchanged and the resulting number (not necessarily ten digit) is also divisible by $7$ then we say that $N$ is a good integer. How many ten digit good integers are there?

2016 Greece Team Selection Test, 1
Tags: number theory , Sequence , Divisibility
Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

2024 Middle European Mathematical Olympiad, 8
Tags: number theory , number theory proposed , Sequence , Divisibility , constant
Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that \[a_ia_{i+1} \mid k-a_i^2\] for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all integers $n \ge M$.

1967 IMO Shortlist, 1
Tags: number theory , Divisibility , binomial coefficients , IMO , IMO 1967
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

2008 Germany Team Selection Test, 3
Tags: modular arithmetic , number theory , Divisibility , Extremal combinatorics , Additive combinatorics , IMO Shortlist
Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

2016 Greece Team Selection Test, 1
Tags: number theory , Sequence , Divisibility
Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

2022 China Team Selection Test, 3
Tags: inequalities , number theory , Divisibility
Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers that are not divisible by each other, i.e. for any $i \neq j$, $a_i$ is not divisible by $a_j$. Show that \[ a_1+a_2+\cdots+a_n \ge 1.1n^2-2n. \] [i]Note:[/i] A proof of the inequality when $n$ is sufficient large will be awarded points depending on your results.

2022 Poland - Second Round, 3
Tags: number theory , prime numbers , modular congruences , Divisibility
Positive integers $a,b,c$ satisfying the equation $$a^3+4b+c = abc,$$ where $a \geq c$ and the number $p = a^2+2a+2$ is a prime. Prove that $p$ divides $a+2b+2$.

1998 IMO, 4
Tags: number theory , Divisibility , polynomial , IMO , IMO 1998 , david monk , Hi
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

2010 Belarus Team Selection Test, 2.2
Tags: number theory , prime , Divisibility
Let $p$ be a positive prime integer, $S(p)$ be the number of triples $(x,y,z)$ such that $x,y,z\in\{0,1,..., p-1\}$ and $x^2+y^2+z^2$ is divided by $p$. Prove that $S(p) \ge 2p- 1$. (I. Bliznets)

2010 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: number theory , Divisibility
If $a$ and $b$ are positive integers such that $ab \mid a^2+b^2$ prove that $a=b$

1969 IMO Shortlist, 49
Tags: number theory , Divisibility , combinatorics , invariant , IMO Shortlist , IMO Longlist
$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$

1989 Bundeswettbewerb Mathematik, 1
Tags: algebra , polynomial , number theory , Divisibility , degree
Determine the polynomial $$f(x) = x^k + a_{k-1} x^{k-1}+\cdots +a_1 x +a_0 $$ of smallest degree such that $a_i \in \{-1,0,1\}$ for $0\leq i \leq k-1$ and $f(n)$ is divisible by $30$ for all positive integers $n$.

1988 IMO Shortlist, 1
Tags: modular arithmetic , linear algebra , algebra , Sequence , Divisibility , Linear Recurrences , IMO Shortlist
An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.

2018 Dutch BxMO TST, 3
Tags: number theory , prime numbers , Divisibility
Let $p$ be a prime number. Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.

2014 Contests, 3
Tags: Divisibility , Quadratic Residues , Hensel , Surjective
Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$. [i]Warut Suksompong, Thailand[/i]