<s>[INST] 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
$$\sqrt{x+y-x^2}+\sqrt{y+z-y^2}+\sqrt{z+w-z^2}+\sqrt{w+x-w^2}\le4\sqrt2-3$$
 [/INST] By applying the generalized mean inequality twice,
$$\left(\frac{\sqrt{x + y - x^2} + \sqrt{y + z - y^2} + \sqrt{z + w - z^2} + \sqrt{w + x - w^2}}{4}\right)^2\\
\le \frac{1}{4}(2x + 2y + 2z + 2w - x^2 - y^2 - z^2 - w^2)\\
= \frac{1}{2} - \frac{x^2 + y^2 + z^2 + w^2}{4}\\
\le \frac{1}{2} - \left(\frac{x + y + z + w}{4}\right)^2\\
=\frac{1}{2} - \frac{1}{16}\\
= \frac{7}{16}$$
so that 
$$\sqrt{x + y - x^2} + \sqrt{y + z - y^2} + \sqrt{z + w - z^2} + \sqrt{w + x - w^2}\le\sqrt7 $$
with equality at $$x = y = z = w = \frac{1}{4}$$
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