Question 95
Let $x$, $y$, $z$ be positive real numbers. Prove that
$$
\frac{x}{\sqrt{2(x^2 + y^2)}} + \frac{y}{\sqrt{2(y^2 + z^2)}} + \frac{z}{\sqrt{2(z^2 + x^2)}} \lt \frac{4x^2 + y^2}{x^2 + 4y^2} + \frac{4y^2 + z^2}{y^2 + 4z^2} + \frac{4z^2 + x^2}{z^2 + 4x^2} \lt 9.
$$
$$
\frac{x}{\sqrt{2(x^2 + y^2)}} + \frac{y}{\sqrt{2(y^2 + z^2)}} + \frac{z}{\sqrt{2(z^2 + x^2)}} \lt \frac{4x^2 + y^2}{x^2 + 4y^2} + \frac{4y^2 + z^2}{y^2 + 4z^2} + \frac{4z^2 + x^2}{z^2 + 4x^2} \lt 9.
$$