Olympiad problems

2012 NZMOC Camp Selection Problems, 2

Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

1970 Poland - Second Round, 3

Prove the theorem: There is no natural number $ n > 1 $ such that the number $ 2^n - 1 $ is divisible by $ n $.

divide divisible number theory inequalities

1977 Swedish Mathematical Competition, 1

$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$.

divide divisible number theory

1999 Switzerland Team Selection Test, 6

Prove that if $m$ and $n$ are positive integers such that $m^2 + n^2 - m$ is divisible by $2mn$, then $m$ is a perfect square.

divisible number theory perfect square perfect squares

2013 Saudi Arabia GMO TST, 4

Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$.

divide divisible number theory

1999 Estonia National Olympiad, 1

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.

odd prime divisible number theory prime numbers

2014 Czech-Polish-Slovak Junior Match, 4

The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

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1995 Singapore MO Open, 4

Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.

divide divisible number theory