Problem

Source: 2019 Spain Mathematic Olympiad P5

Tags: minimum value, inequalities



We consider all pairs (x, y) of real numbers such that $0\leq x \leq y \leq 1$.Let $M (x,y)$ the maximum value of the set $$A=\{xy, 1-x-y+xy, x+y-2xy\}.$$Find the minimum value that $M(x,y)$ can take for all these pairs $(x,y)$.