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if p,q>0 and p+q = 1 show (p+1/p)^2+ (q+q)^2 >=25/2, also 2 inequalities

Source: Spanish Mathematical Olympiad 1984 P3

August 3, 2018
inequalitiesalgebra

Problem Statement

If pp and qq are positive numbers with p+q=1p+q = 1, knowing that any real numbers x,yx,y satisfy (x−y)2≥0(x-y)^2 \ge 0, show that
x+y2≥xy\frac{x+y}{2} \ge \sqrt{xy},
x2+y22≥(x+y2)2\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2,
(p+1p)2+(q+1q)2≥252\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}
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