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.

AND:
OR:
NO:

Found problems: 408

2012 Tournament of Towns, 5
Tags: prime , sum , divisible , number theory
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 Czech-Polish-Slovak Junior Match, 2
Tags: number theory , divisible , divide
Let the numbers $x_i \in \{-1, 1\}$ be given for $i = 1, 2,..., n$, satisfying $$x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0.$$ Prove that $n$ is divisible by $4$.

2016 Dutch IMO TST, 2
Tags: number theory , divisible , divisor , exponential
Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

2007 BAMO, 4
Tags: number theory , divisible , diophantine
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 $3$.

2005 Estonia National Olympiad, 2
Tags: number theory , sum of powers , divisible , divide
Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.

1956 Polish MO Finals, 4
Tags: number theory , divide , 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 $.

2015 Caucasus Mathematical Olympiad, 2
Tags: number theory , divisible , maximum
There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?

1964 Czech and Slovak Olympiad III A, 1
Tags: number theory , divisible , last digit
Show that the number $11^{100}-1$ is both divisible by $6000$ and its last four decimal digits are $6000$.

1908 Eotvos Mathematical Competition, 1
Tags: number theory , divide , divisible
Given two odd integers $a$ and $b$; prove that $a^3 -b^3$ is divisible by $2^n$ if and only if $a-b$ is divisible by $2^n$.

1987 All Soviet Union Mathematical Olympiad, 444
Tags: divisible , number theory
Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$.

1996 Tournament Of Towns, (509) 2
Tags: number theory , divide , divisible
Do there exist three different prime numbers $p$, $q$ and $r$ such that $p^2 + d$ is divisible by $qr$, $q^2 + d$ is divisible by $rp$ and $r^2 + d$ is divisible by $pq$, if (a) $d = 10$; (b) $d = 11$? (V Senderov)

2019 Lusophon Mathematical Olympiad, 1
Tags: number theory , digit , divisible
Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by $7$ or $13$.

2020 Swedish Mathematical Competition, 1
Tags: number theory , divide , divisible
How many of the numbers $1\cdot 2\cdot 3$, $2\cdot 3\cdot 4$,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?

2019 New Zealand MO, 4
Tags: number theory , divide , divisible , multiple
Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.

2002 Singapore Team Selection Test, 2
Tags: floor function , number theory , integer , divide , divisible
For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.

1949-56 Chisinau City MO, 9
Tags: number theory , divisible
Prove that for any integer $n$ the number $n (n^2 + 5)$ is divisible by $6$.

2024 Austrian MO Regional Competition, 4
Tags: number theory , divisible , divide
Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

2001 Tuymaada Olympiad, 2
Tags: infinite chessboard , number theory , sum , divisible
Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any $10$ numbers in a row vertically or horizontal, would be divisible by $101$?

2002 Estonia National Olympiad, 1
Tags: gcd , lcm , divisible , divide , number theory
The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.

2007 Switzerland - Final Round, 9
Tags: number theory , integer , divide , divisible
Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.

2017 Bundeswettbewerb Mathematik, 1
Tags: blackboard , divisible , divisibility , combinatorics , number theory
The numbers $1,2,3,\dots,2017$ are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by $8$, then Amelie wins. Else Boris wins. Who can force a victory?

2013 Costa Rica - Final Round, 5
Tags: algebra , polynomial , divisible
Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.

2001 Kazakhstan National Olympiad, 1
Tags: divide , divisible , number theory
Prove that there are infinitely many natural numbers $ n $ such that $ 2 ^ n + 3 ^ n $ is divisible by $ n $.

1925 Eotvos Mathematical Competition, 1
Tags: number theory , divide , divisible
Let $a,b, c,d$ be four integers. Prove that the product of the six differences $$b - a,c - a,d - a,d - c,d - b, c - b$$ is divisible by $12$.

2008 Thailand Mathematical Olympiad, 4
Tags: number theory , binomial , divisible , divide
Let $n$ be a positive integer. Show that $${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{n}{2n+1 \choose 2n+1}2008^n $$ is not divisible by $19$.