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: 367

2012 Tournament of Towns, 3
Tags: number theory , divides , polynomial , divisible
Let $n$ be a positive integer. Prove that there exist integers $a_1, a_2,..., a_n$ such that for any integer $x$, the number $(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n$ is divisible by $2n - 1$.

1953 Kurschak Competition, 2
Tags: number theory , Perfect Square , divides
$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a square.

2008 Dutch Mathematical Olympiad, 3
Tags: combinatorics , number theory , divisible , divides
Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ($1$ and $2008$ included). Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$.

2013 Abels Math Contest (Norwegian MO) Final, 3
Tags: number theory , prime , divides
A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.

2020 Malaysia IMONST 2, 3
Tags: number theory , divisible , divides
Given integers $a$ and $b$ such that $a^2+b^2$ is divisible by $11$. Prove that $a$ and $b$ are both divisible by $11$.

2005 iTest, 25
Tags: number theory , divides , divisible
Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?

1982 Poland - Second Round, 5
Tags: number theory , divides , divisible
Let $ q $ be an even positive number. Prove that for every natural number $ n $ number $q^{(q+1)^n}+1$ is divisible by $ (q + 1)^{n+1} $ but not divisible by $ (q + 1)^{n+2} $.

2009 China Northern MO, 3
Tags: number theory , divides
Given $26$ different positive integers , in any six numbers of the $26$ integers , there are at least two numbers , one can be devided by another. Then prove : There exists six numbers , one of them can be devided by the other five numbers .

2001 Austria Beginners' Competition, 1
Tags: number theory , divides , divisible
Prove that for every odd positive integer $n$ the number $n^n-n$ is divisible by $24$.

2015 Saudi Arabia IMO TST, 1
Tags: number theory , divides , divisible , GCD
Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$. Trần Nam Dũng

1984 Bundeswettbewerb Mathematik, 1
Tags: number theory , divisible , divides
The natural numbers $n$ and $z$ are relatively prime and greater than $1$. For $k = 0, 1, 2,..., n - 1$ let $s(k) = 1 + z + z^2 + ...+ z^k.$ Prove that: a) At least one of the numbers $s(k)$ is divisible by $n$. b) If $n$ and $z - 1$ are also coprime, then already one of the numbers $s(k)$ with $k = 0,1, 2,..., n- 2$ is divisible by $n$.

2012 NZMOC Camp Selection Problems, 2
Tags: number theory , consecutive , divides , divisible , Sum
Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

2017 India PRMO, 1
Tags: number theory , sum of digits , divides , divisible
How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

2016 Saudi Arabia BMO TST, 3
Tags: number theory , divides , divisible
Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.

2021 Saudi Arabia JBMO TST, 3
Tags: number theory , divisible , divides
We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.

2002 Singapore Team Selection Test, 2
Tags: floor function , number theory , Integer , divides , 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.

2000 Chile National Olympiad, 3
Tags: number theory , divides , divisible , periodic
A number $N_k$ is defined as [i]periodic[/i] if it is composed in number base $N$ of a repeated $k$ times . Prove that $7$ divides to infinite periodic numbers of the set $N_1, N_2, N_3,...$

1959 Poland - Second Round, 4
Tags: number theory , perfect cube , divides
Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula $$a_n =3(n^2 + n) + 7$$ Prove that this sequence has the following properties: 1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $, 2( No term of the sequence is the cube of an integer.

2002 Kazakhstan National Olympiad, 7
Tags: prime divisor , divisor , number theory , divides
Prove that for any integers $ n> m> 0 $ the number $ 2 ^n-1 $ has a prime divisor not dividing $ 2 ^m-1 $.

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

2017 Gulf Math Olympiad, 4
Tags: ceiling function , power of 2 , divides , maximum , number theory
1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ . 2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$

2008 Switzerland - Final Round, 3
Tags: number theory , divides , divisible
Show that each number is of the form $$2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}$$ is divisible by $2008$, where the exponential towers can be any independent ones have height $\ge 3$.

2015 Saudi Arabia BMO TST, 4
Tags: divides , divisible , number theory
Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$. Lê Anh Vinh

2019 Durer Math Competition Finals, 5
Tags: divisible , divides , number theory
We want to write down as many distinct positive integers as possible, so that no two numbers on our list have a sum or a difference divisible by $2019$. At most how many integers can appear on such a list?

2022 IFYM, Sozopol, 5
Tags: algebra , divides
Find all functions $f : N \to N$ such that $f(p)$ divides $f(n)^p -n$ by any natural number $n$ and prime number $p$.