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

2018 Saudi Arabia IMO TST, 1
Tags: number theory , divide , 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 $m, n \in Z^+$.

OMMC POTM, 2022 1
Tags: divisible , number theory
The digits $2,3,4,5,6,7,8,9$ are written down in some order. When read in that order, the digits form an $8$-digit, base $10$ positive integer. if this integer is divisible by $44$, how many ways could the digits have been initially ordered? [i]Proposed by Evan Chang (squareman), USA[/i]

2007 BAMO, 4
Tags: divisible , diophantine , number theory
Let $N$ be the number of ordered pairs $(x,y)$ of integers such that $x^2+xy+y^2 \le 2007$. Remember, integers may be positive, negative, or zero! (a) Prove that $N$ is odd. (b) Prove that $N$ is not divisible by $3$.

2015 JBMO Shortlist, NT1
Tags: number theory , consecutive , divisible , difference , maximum , integer
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?

2015 Saudi Arabia GMO TST, 4
Tags: divide , number theory , 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

2003 Singapore Senior Math Olympiad, 1
Tags: number theory , perfect square , divide , divisible
It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.

1980 Tournament Of Towns, (003) 3
Tags: permutation , divisible , combinatorics , number theory
If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.

2011 Belarus Team Selection Test, 1
Tags: digit , sum , divisible , number theory
Let $g(n)$ be the number of all $n$-digit natural numbers each consisting only of digits $0,1,2,3$ (but not nessesarily all of them) such that the sum of no two neighbouring digits equals $2$. Determine whether $g(2010)$ and $g(2011)$ are divisible by $11$. I.Kozlov

2019 Ecuador NMO (OMEC), 3
Tags: power of 2 , divide , divisible , number theory
For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$

1983 Austrian-Polish Competition, 8
Tags: sequence , divide , product , factorial , number theory , recurrence relation , divisible
(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n (b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.

1998 Estonia National Olympiad, 4
Tags: prime , divide , divisible , number theory
Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

2002 Estonia National Olympiad, 2
Tags: digit , divisible , number theory
Do there exist distinct non-zero digits $a, b$ and $c$ such that the two-digit number $\overline{ab}$ is divisible by $c$, the number $\overline{bc}$ is divisible by $a$ and $\overline{ca}$, is divisible by $b$?

2020 Czech and Slovak Olympiad III A, 6
Tags: inequalities , divisible , number theory , binomial
For each positive integer $k$, denote by $P (k)$ the number of all positive integers $4k$-digit numbers which can be composed of the digits $2, 0$ and which are divisible by the number $2 020$. Prove the inequality $$P (k) \ge \binom{2k - 1}{k}^2$$ and determine all $k$ for which equality occurs. (Note: A positive integer cannot begin with a digit of $0$.) (Jaromir Simsa)

2007 Switzerland - Final Round, 8
Tags: number theory , combinatorics , divisible
Let $M\subset \{1, 2, 3, . . . , 2007\}$ a set with the following property: Among every three numbers one can always choose two from $M$ such that one is divisible by the other. How many numbers can $M$ contain at most?

2010 Cuba MO, 5
Tags: number theory , divide , divisible
Let $p\ge 2$ be a prime number and $a\ge 1$ be an integer different from $p$. Find all pairs $(a, p)$ such that $a + p | a^2 + p^2$.

2014 Saudi Arabia Pre-TST, 4.1
Tags: divide , number theory , inequalities , divisible
Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.

1957 Moscow Mathematical Olympiad, 347
Tags: cubic , divisible , polynomial , algebra
a) Let $ax^3 + bx^2 + cx + d$ be divisible by $5$ for given positive integers $a, b, c, d$ and any integer $x$. Prove that $a, b, c$ and $d$ are all divisible by $5$. b) Let $ax^4 + bx^3 + cx^2 + dx + e$ be divisible by $7$ for given positive integers $a, b, c, d, e$ and all integers $x$. Prove that $a, b, c, d$ and $e$ are all divisible by $7$.

2017 Saudi Arabia IMO TST, 3
Tags: divide , number theory , divisible
Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.

2007 Switzerland - Final Round, 2
Tags: divisible , divide , number theory
Let $a, b, c$ be three integers such that $a + b + c$ is divisible by $13$. Prove that $$a^{2007}+b^{2007}+c^{2007}+2 \cdot 2007abc$$ is divisible by $13$.

2007 Bosnia and Herzegovina Junior BMO TST, 2
Tags: divisible , divide , number theory
Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$. .

2011 Hanoi Open Mathematics Competitions, 7
Tags: number theory , divisible
How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$.

2000 Tuymaada Olympiad, 5
Tags: number theory , prime , divisible
Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?

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 $y$.

2019 Durer Math Competition Finals, 6
Tags: number theory , multiple , divide , divisible , digit
Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?

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