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x + 1 / x>= 2 if x>0 , sum x_i <= sum x_i/yi when sum x_i>=sum y_i and x_i,y_i>0

Source: Norwegian Mathematical Olympiad 2002 - Abel Competition p2ab

February 21, 2020
inequalitiesalgebra

Problem Statement

a) Let xx be a positive real number. Show that x+1/x≥2x + 1 / x\ge 2. b) Let n≥2n\ge 2 be a positive integer and let x1,y1,x2,y2,...,xn,ynx _1,y_1,x_2,y_2,...,x_n,y_n be positive real numbers such that x1+x2+...+xn≥x1y1+x2y2+...+xnynx _1+x _2+...+x _n \ge x _1y_1+x _2y_2+...+x _ny_n.
Show that x1+x2+...+xn≤x1y1+x2y2+...+xnynx _1+x _2+...+x _n \le \frac{x _1}{y_1}+\frac{x _2}{y_2}+...+\frac{x _n}{y_n}
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