NO-ABEL 2008 Problem 3

a) Let $x$ and $y$ be positive numbers such that $x + y = 2$ . Show that $\frac{1}{x}+\frac{1}{y} \le \frac{1}{x^2}+\frac{1}{y^2}$ b) Let $x,y$ and $z$ be positive numbers such that $x + y +z= 2$ . Show that $\frac{1}{x}+\frac{1}{y} +\frac{1}{z} +\frac{9}{4} \le \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$ .