a) Let $x$ be a positive real number. Show that $x + 1 / x\ge 2$. b) Let $n\ge 2$ be a positive integer and let $x _1,y_1,x_2,y_2,...,x_n,y_n$ be positive real numbers such that $x _1+x _2+...+x _n \ge x _1y_1+x _2y_2+...+x _ny_n$. Show that $x _1+x _2+...+x _n \le \frac{x _1}{y_1}+\frac{x _2}{y_2}+...+\frac{x _n}{y_n}$
Problem
Source: Norwegian Mathematical Olympiad 2002 - Abel Competition p2ab
Tags: inequalities, algebra