SE-SKOLORNAS 1977

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

There is a point inside an equilateral triangle side $d$ whose distance from the vertices is $3, 4, 5$ . Find $d$ .

Show that the only integral solution to $$\left\{ \begin{array}{l} xy + yz + zx = 3n^2 - 1 x + y + z = 3n \end{array} \right.$$ with $x \geq y \geq z$ is $x=n+1$ , $y=n$ , $z=n-1$ .

Show that if $$\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1$$ then $$\frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1$$

The numbers $1, 2, 3, ... , 64$ are written in the cells of an $8 \times 8$ board (in some order, one per cell). Show that at least four $2 \times 2$ squares have sum greater than $100$ .

Show that there are positive reals $a$ , $b$ , $c$ such that $$\left\{ \begin{array}{l} a^2 + b^2 + c^2 > 2 a^3 + b^3 + c^3 <2 a^4 + b^4 + c^4 > 2 \end{array} \right.$$