BALKAN 1988

Let $ABC$ be a triangle and let $M,N,P$ be points on the line $BC$ such that $AM,AN,AP$ are the altitude, the angle bisector and the median of the triangle, respectively. It is known that

$\frac{[AMP]}{[ABC]}=\frac{1}{4}$ and $\frac{[ANP]}{[ABC]}=1-\frac{\sqrt{3}}{2}$ . Find the angles of triangle $ABC$ .

Find all polynomials of two variables $P(x,y)$ which satisfy $$P(a,b) P(c,d) = P (ac+bd, ad+bc), \forall a,b,c,d \in \mathbb{R}.$$

Let $ABCD$ be a tetrahedron and let $d$ be the sum of squares of its edges' lengths. Prove that the tetrahedron can be included in a region bounded by two parallel planes, the distances between the planes being at most $\frac{\sqrt{d}}{2\sqrt{3}}$

Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$ . Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$ , where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$ .