Olympiad problems

2010 Junior Balkan Team Selection Tests - Romania, 1

Let $p$ be a prime number, $p> 5$. Determine the non-zero natural numbers $x$ with the property that $5p + x$ divides $5p ^ n + x ^ n$, whatever $n \in N ^ {*} $.

2010 Grand Duchy of Lithuania, 5

Tags: number theory , divided

Find positive integers n that satisfy the following two conditions: (a) the quotient obtained when $n$ is divided by $9$ is a positive three digit number, that has equal digits. (b) the quotient obtained when $n + 36$ is divided by $4$ is a four digit number, the digits beeing $2, 0, 0, 9$ in some order.

2007 Estonia National Olympiad, 1

Tags: number theory , divided , Digits

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.

1998 Estonia National Olympiad, 4

Tags: primes , divided , number theory , divisible

Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.