parmenides51 wrote: 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. https://artofproblemsolving.com/community/c6h1202197p5915265: $$5x(1-x)+5y(1-y)\geq5xy+5yx=10xy>8xy.$$