CZ-SK-MO 2012

Find all integers for which $n$ is $n^4 -3n^2 + 9$ prime

Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$ .

Prove that there are two numbers $u$ and $v$ , between any $101$ real numbers that apply $100 |u - v| \cdot |1 - uv| \le (1 + u^2)(1 + v^2)$

Inside the parallelogram $ABCD$ is a point $X$ . Make a line that passes through point $X$ and divides the parallelogram into two parts whose areas differ from each other the most.

In a group of $90$ children each has at least $30$ friends (friendship is mutual). Prove that they can be divided into three $30$ -member groups so that each child has its own a group of at least one friend.

In the set of real numbers solve the system of equations $x^4+y^2+4=5yz$ $y^4+z^2+4=5zx$ $z^4+x^2+4=5xy$