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: 387

1958 Polish MO Finals, 1
Tags: perfect cube , divide , number theory
Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by $ 504 $ .

2009 District Olympiad, 1
Tags: divide , number theory
Let $m$ and $n$ be positive integers such that $5$ divides $2^n + 3^m$. Prove that $5$ divides $2^m + 3^n$.

2019 Saudi Arabia BMO TST, 1
Tags: divide , divisible , prime , number theory
Let $p$ be an odd prime number. a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n. b) Find all $n$ satisfy condition above when $p = 3$

2019 Austrian Junior Regional Competition, 4
Tags: prime , divide , divisible , number theory
Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

2019 Costa Rica - Final Round, 5
Tags: divide , number theory
We have an a sequence such that $a_n = 2 \cdot 10^{n + 1} + 19$. Determine all the primes $p$, with $p \le 19$, for which there exists some $n \ge 1$ such that $p$ divides $a_n$.

2011 QEDMO 9th, 6
Tags: divide , divisible , number theory
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 $m$.

2019 Federal Competition For Advanced Students, P2, 6
Tags: divide , min , number theory
Find the smallest possible positive integer n with the following property: For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$. (Gerhard J. Woeginger)

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

2017 Saudi Arabia BMO TST, 1
Tags: divide , factorial , number theory , divisible
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.

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

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: $1,2,3,...$)

2003 Switzerland Team Selection Test, 9
Tags: number theory , combinatorics , divide
Given integers $0 < a_1 < a_2 <... < a_{101} < 5050$, prove that one can always choose for different numbers $a_k,a_l,a_m,a_n$ such that $5050 | a_k +a_l -a_m -a_n$

2013 Balkan MO Shortlist, N8
Tags: divide , integer , number theory
Suppose that $a$ and $b$ are integers. Prove that there are integers $c$ and $d$ such that $a+b+c+d=0$ and $ac+bd=0$, if and only if $a-b$ divides $2ab$.

1952 Moscow Mathematical Olympiad, 232
Tags: polynomial , divide , algebra
Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.

2023 Assara - South Russian Girl's MO, 2
Tags: divide , divisible , number theory
The natural numbers $a$ and $b$ are such that $a^a$ is divisible by $b^b$. Can we say that then $a$ is divisible by $b$?

2000 Singapore MO Open, 2
Tags: divisible , prime , divide , gcd , number theory
Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.

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

2018 Saudi Arabia BMO TST, 3
Tags: number theory , divide , divisor
Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

1955 Moscow Mathematical Olympiad, 290
Tags: divide , number theory , divisible
Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?

1958 Kurschak Competition, 2
Tags: number theory , divide , divisible
Show that if $m$ and $n$ are integers such that $m^2 + mn + n^2$ is divisible by $9$, then they must both be divisible by $3$.

2002 Estonia National Olympiad, 1
Tags: divisible , gcd , lcm , 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 $n$.

2008 Switzerland - Final Round, 3
Tags: divide , divisible , number theory
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 $\ge 3$.

2023 Francophone Mathematical Olympiad, 4
Tags: divide , divisible , number theory
Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$.

2008 Regional Olympiad of Mexico Center Zone, 5
Tags: product , digit , divide , number theory
Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_ {32} = 6$, $p_ {203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$. Find the largest prime that divides $S $.

2010 Saudi Arabia Pre-TST, 1.3
Tags: divide , digit , divisible , number theory
1) Let $a$ and $b$ be relatively prime positive integers. Prove that there is a positive integer $n$ such that $1 \le n \le b$ and $b$ divides $a^n - 1$. 2) Prove that there is a multiple of $7^{2010}$ of the form $99... 9$ ($n$ nines), for some positive integer $n$ not exceeding $7^{2010}$.