Proving $\sqrt{x+y-x^2}+\sqrt{y+z-y^2}+\sqrt{z+w-z^2}+\sqrt{w+x-w^2}\le4\sqrt2-3$ I found this  inequality using unusual calculations in maths Olympics
  and I wonder if some clever teenager could prove it using their elementary knowledge of mathematics.
Let $x,y,z,w$ be non-negative numbers such that $$x+y+z+w=1$$ Prove that
$$\sqrt{x+y-x^2}+\sqrt{y+z-y^2}+\sqrt{z+w-z^2}+\sqrt{w+x-w^2}\le4\sqrt2-3$$

By Jensen $\sum\limits_{cyc}\sqrt{x+y-x^2}\geq2\sqrt{\sum\limits_{cyc}(2x-x^2)}=2\sqrt{2-\sum\limits_{cyc}x^2}\leq2\sqrt{2-\frac{1}{4}}=\sqrt7<4\sqrt2-3$
