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x,y>0 and x + y <= 1 prove 8xy <= 5x(1 - x) + 5y(1 - y)

Source: Irmo 2015 p2 q6

September 16, 2018

algebrainequalitiesalgebraic inequality

Problem Statement

Suppose

x,y

are nonnegative real numbers such that

x + y \le 1

. Prove that

8xy \le 5x(1 - x) + 5y(1 - y)

and determine the cases of equality.