OME 1984 Problem 3

If $p$ and $q$ are positive numbers with $p+q = 1$ , knowing that any real numbers $x,y$ satisfy $(x-y)^2 \ge 0$ , show that $\frac{x+y}{2} \ge \sqrt{xy}$ , $\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2$ ,

$\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}$