Olympiad problems

2013 Bosnia and Herzegovina Junior BMO TST, 1

It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers

1988 Tournament Of Towns, (186) 3

Tags: number theory , divisible , sum , divide

Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.

2014 Regional Olympiad of Mexico Center Zone, 1

Tags: number theory , divisible , divide , product

Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.

2013 Flanders Math Olympiad, 1

Tags: number theory , digit , divisible , sum

A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.

2015 JBMO Shortlist, NT2

Tags: number theory , divisible , repunit

A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted as $\underbrace{{11\cdots1}}_{n}$ . Prove that: α) the repunit $\underbrace{{11\cdots1}}_{n}$is divisible by $37$ if and only if $n$ is divisible by $3$ b) there exists a positive integer $k$ such that the repunit $\underbrace{{11\cdots1}}_{n}$ is divisible by $41$ if $n$ is divisible by $k$

2006 Singapore Senior Math Olympiad, 1

Tags: mutliple , divide , divisible , number theory , prime

Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.

2019 Serbia JBMO TST, 1

Tags: divisible , factorial , number theory

Does there exist a positive integer $n$, such that the number of divisors of $n!$ is divisible by $2019$?

2010 All-Russian Olympiad Regional Round, 10.4

Tags: number theory , divisible

We call a natural number $b$ [i]lucky [/i] if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of [i]lucky [/i] natural numbers less than $2010$.

1932 Eotvos Mathematical Competition, 1

Tags: number theory , divisible

Let $a, b$ and $n$ be positive integers such that $ b$ is divisible by $a^n$. Prove that $(a+1)^b-1$ is divisible by $a^{n+1}$.

2003 Austrian-Polish Competition, 4

Tags: divide , divisible , product , number theory

A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.

1986 Polish MO Finals, 3

Tags: number theory , sum , divisible , prime , prime divisor

$p$ is a prime and $m$ is a non-negative integer $< p-1$. Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.

1965 Czech and Slovak Olympiad III A, 1

Tags: number theory , divisible , divisibility tests

Show that the number $5^{2n+1}2^{n+2}+3^{n+2}2^{2n+1}$ is divisible by $19$ for every non-negative integer $n$.

1947 Kurschak Competition, 1

Tags: number theory , divide , divisible

Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.

2008 Hanoi Open Mathematics Competitions, 1

Tags: number theory , divisible , sum of digits

How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?

2017 Switzerland - Final Round, 4

Tags: number theory , divide , divisible

Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold: $$pq | n^p + 2$$ $$n + 2 | n^p + q^p.$$ Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.

2010 QEDMO 7th, 9

Tags: number theory , divisible

Let $p$ be an odd prime number and $c$ an integer for which $2c -1$ is divisible by $p$. Prove that $$(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n$$ is divisible by $p$.

1978 All Soviet Union Mathematical Olympiad, 254

Tags: number theory , divisible

Prove that there is no $m$ such that ($1978^m - 1$) is divisible by ($1000^m - 1$).

2014 Chile National Olympiad, 4

Tags: number theory , divide , divisible

Prove that for every integer $n$ the expression $n^3-9n + 27$ is not divisible by $81$.

2010 Saudi Arabia BMO TST, 3

Tags: number theory , divisible , divide

How many integers in the set $\{1, 2 ,..., 2010\}$ divide $5^{2010!}- 3^{2010!}$?

2015 NZMOC Camp Selection Problems, 8

Tags: divide , divisible , divisor , number theory

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn + 1$ is a divisor of $d^2 + n^2$.

1996 Estonia National Olympiad, 4

Tags: number theory , divisible , divide , sum of powers , sum

Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.

2013 Saudi Arabia Pre-TST, 4.2

Tags: number theory , divide , divisible

Let $x, y$ be two integers. Prove that if $2013$ divides $x^{1433} + y^{1433}$ then $2013$ divides $x^7 + y^7$.

2000 Tuymaada Olympiad, 5

Tags: number theory , divisible , prime

Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?

2022 New Zealand MO, 5

Tags: number theory , multiple , divide , divisible , recurrence relation

The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.

2010 Thailand Mathematical Olympiad, 10

Tags: number theory , binomial , divisible , prime

Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.