PL-OM 1998

Prove that among the numbers of the form $50^n + (50n+1)^{50}$ , where $n$ is a natural number, there exist infinitely many composite numbers.

Show that for all real numbers $a,b,c,d,$ the following inequality holds:

$$(a+b+c+d)^2 \leq 3 (a^2 + b^2 + c^2 + d^2) + 6ab$$

In the isosceles triangle $ABC$ the angle $BAC$ is a right angle. Point $D$ lies on the side $BC$ and satisfies $BD = 2 \cdot CD$ . Point $E$ is the foot of the perpendicular of the point $B$ on the line $AD$ . Find the angle $CED$ .

Let $x,y$ be real numbers such that the numbers $x+y, x^2+y^2, x^3+y^3$ and $x^4+y^4$ are integers. Prove that for all positive integers $n$ , the number $x^n + y^n$ is an integer.