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

2007 Estonia National Olympiad, 4
Tags: number theory , greatest common divisor , divide
Let $a, b,c$ be positive integers such that $gcd(a, b, c) = 1$ and each product of two is divided by the third. a) Prove that each of these numbers is equal to the least two remaining numbers the quotient of the coefficient and the highest coefficient. b) Give an example of one of these larger numbers $a, b$ and $c$

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

2021 Saudi Arabia Training Tests, 39
Tags: sum , number theory , divide , divisible
Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.

2013 Saudi Arabia Pre-TST, 3.2
Tags: divide , divisible , number theory
Let $a_1, a_2,..., a_9$ be integers. Prove that if $19$ divides $a_1^9+a_2^9+...+a_9^9$ then $19$ divides the product $a_1a_2...a_9$.

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

2008 Dutch Mathematical Olympiad, 3
Tags: combinatorics , number theory , divisible , divide
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 $8$.

2011 Saudi Arabia BMO TST, 4
Tags: fibonacci number , number theory , divide , divisible
Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$. Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .

1986 Tournament Of Towns, (108) 2
Tags: number theory , maximum , digit , divide
A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$? (S . Fomin, Leningrad)

1999 Singapore MO Open, 2
Tags: number theory , digit , divisible , divide
Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer

2024 Regional Competition For Advanced Students, 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]

2022 Regional Olympiad of Mexico West, 1
Tags: number theory , divide
Find a subset of $\{1,2, ...,2022\}$ with maximum number of elements such that it does not have two elements $a$ and $b$ such that $a = b + d$ for some divisor $d$ of $b$.

2005 Chile National Olympiad, 2
Tags: number theory , divide
Let $p$ be a prime number greater than $2$ and let $m, n$ be integers such that: $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}.$$ Prove that $p$ divides $m$.

2006 Thailand Mathematical Olympiad, 14
Tags: number theory , divide
Find the smallest positive integer $n$ such that $2549 | n^{2545} - 2$.

1947 Kurschak Competition, 1
Tags: number theory , divide , divisible
Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.

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

1999 Ukraine Team Selection Test, 8
Tags: divide , number theory
Find all pairs $(x,n)$ of positive integers for which $x^n + 2^n + 1$ divides $x^{n+1} +2^{n+1} +1$.

1997 German National Olympiad, 6a
Tags: polynomial , divide , algebra , prime
Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$. Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.

2007 Estonia National Olympiad, 1
Tags: number theory , digit , divide
The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers. a) Find all possibilities for the three numbers that are not included in this number. b) Give an example of such a number.

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

2010 Saudi Arabia BMO TST, 3
Tags: number theory , divisible , divide
How many integers in the set $\{1, 2 ,..., 2010\}$ divide $5^{2010!}- 3^{2010!}$?

2005 Thailand Mathematical Olympiad, 11
Tags: number theory , divide
Find the smallest positive integer $x$ such that $2^{254}$ divides $x^{2005} + 1$.

2016 Saudi Arabia Pre-TST, 2.4
Tags: number theory , product , divide , divisible
Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b > n$ such that $$\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)$$.

2023 Durer Math Competition Finals, 15
Tags: number theory , divide
What is the biggest positive integer which divides $p^4 - q^4$ for all primes $p$ and $q$ greater than $10$?

2016 Peru IMO TST, 4
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

2022 Regional Olympiad of Mexico West, 5
Tags: floor function , number theory , divide
Determine all positive integers $n$ such that $\lfloor \sqrt{n} \rfloor - 1$ divides $n + 1$ and $\lfloor \sqrt{n} \rfloor +2$ divides $ n + 4$.