ISL 1976

In $ABC$ , the inscribed circle is tangent to side $BC$ at $X$ . Segment $AX$ is drawn. Prove that the line joining the midpoint of $AX$ to the midpoint of side $BC$ passes through center $I$ of the inscribed circle.

Let $a,b,c,d$ be nonnegative real numbers. Prove that $$a^4+b^4+c^4+d^4+2abcd \ge a^2b^2+a^2c^2+a^2d^2+b^2c^2+b^2d^2+c^2d^2.$$