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

2012 Abels Math Contest (Norwegian MO) Final, 3b
Tags: number theory , divisible , divide
Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?

2015 Dutch IMO TST, 5
Tags: positive integer , number theory , divide , min , max
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

2011 Tournament of Towns, 6
Tags: divide , 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 $2^{n+1}$.

2022 New Zealand MO, 5
Tags: number theory , multiple , divide , divisible , recurrence relation
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.

2021 Polish Junior MO Second Round, 3
Tags: divisible , divide , number theory
Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.

2003 Chile National Olympiad, 5
Tags: number theory , divide , divisible
Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)

2005 All-Russian Olympiad Regional Round, 10.5
Tags: number theory , arithmetic progression , divisible , divide
Arithmetic progression $a_1, a_2, . . . , $ consisting of natural numbers is such that for any $n$ the product $a_n \cdot a_{n+31}$ is divisible by $2005$. Is it possible to say that all terms of the progression are divisible by $2005$?

2023 Greece Junior Math Olympiad, 4
Tags: number theory , divide , divisor
Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.

2018 Saudi Arabia GMO TST, 2
Tags: number theory , divide , divisible , prime
Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.

2022 Saudi Arabia IMO TST, 1
Tags: number theory , divide , number theory with sequences , recurrence relation
Let $(a_n)$ be the integer sequence which is defined by $a_1= 1$ and $$ a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1.$$ Let $S$ be the set of all primes $p$ such that there exists an index $i$ such that $p|a_i$. Prove that the set $S$ is an infinite set and it is not equal to the set of all primes.

2007 IMAC Arhimede, 4
Tags: divide , sum , sums and products , combinatorics , pigeonhole principle , number theory
Prove that for any given number $a_k, 1 \le k \le 5$, there are $\lambda_k \in \{-1, 0, 1\}, 1 \le k \le 5$, which are not all equal zero, such that $11 | \lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2+\lambda_4a_4^2+\lambda_5a_5^2$

1986 Austrian-Polish Competition, 7
Tags: prime divisor , factorial , divide
Let $k$ and $n$ be integers with $0 < k < n^2/4$ such that k has no prime divisor greater than $n$. Prove that $k$ divides $n!$.

2021 Regional Competition For Advanced Students, 4
Tags: number theory , divide , divisible
Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

2020 Denmark MO - Mohr Contest, 3
Tags: digit , number theory , divide
Which positive integers satisfy the following three conditions? a) The number consists of at least two digits. b) The last digit is not zero. c) Inserting a zero between the last two digits yields a number divisible by the original number.

2022 Austrian MO National Competition, 1
Tags: function , number theory , divide
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]

2017 Puerto Rico Team Selection Test, 3
Tags: number theory , multiple , divide
Given are $n$ integers. Prove that at least one of the following conditions applies: 1) One of the numbers is a multiple of $n$. 2) You can choose $k\le n$ numbers whose sum is a multiple of $ n$.

1991 Chile National Olympiad, 4
Tags: divide , divisible , number theory
Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.

2011 QEDMO 9th, 6
Tags: number theory , divide , divisible
Show that there are infinitely many pairs $(m, n)$ of natural numbers $m, n \ge 2$, for $m^m- 1$ is divisible by $n$ and $n^n- 1$ is divisible by $m$.

2022 Switzerland - Final Round, 2
Tags: number theory , remainder , divide
Let $n$ be a positive integer. Prove that the numbers $$1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}$$ all give different remainders when divided by $2^n$.

2014 Czech-Polish-Slovak Junior Match, 5
Tags: game , number theory , divide , divisor
There is the number $1$ on the board at the beginning. If the number $a$ is written on the board, then we can also write a natural number $b$ such that $a + b + 1$ is a divisor of $a^2 + b^2 + 1$. Can any positive integer appear on the board after a certain time? Justify your answer.

2018 Grand Duchy of Lithuania, 4
Tags: number theory , divide , divisor
Find all positive integers $n$ for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

2018 Singapore Junior Math Olympiad, 1
Tags: multiple , number theory , divide , digit
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

2015 Saudi Arabia GMO TST, 4
Tags: number theory , divide , divisible
Let $p, q$ be two different odd prime numbers and $n$ an integer such that $pq$ divides $n^{pq} + 1$. Prove that if $p^3q^3$ divides $n^{pq} + 1$ then either $p^2$ divides $n + 1$ or $q^2$ divides $n + 1$. Malik Talbi

1988 Tournament Of Towns, (186) 3
Tags: number theory , divisible , sum , divide
Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.

1994 Austrian-Polish Competition, 7
Tags: number theory , divide , divisible
Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.