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4

Part of 1995 Abels Math Contest (Norwegian MO)

Problems(1)

\sum (x_i +y_i)^2 \sum 1/x_iy_i >= 4n^2

Source: Norwegian Mathematical Olympiad 1995 - Abel Competition p4

2/11/2020
Let xi,yix_i,y_i be positive real numbers, i=1,2,...,ni = 1,2,...,n. Prove that (∑i=1n(xi+yi)2)(∑i=1n1xiyi)≥4n2\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2
inequalitiesalgebra