Let $x \geq y \geq z$ be real numbers such that $xy + yz + zx = 1$. Prove that $xz < \frac 12.$ Is it possible to improve the value of constant $\frac 12 \ ?$

The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$

In the triangle $ABC$, the angle $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$

Let $x > 1$ be a non-integer number. Prove that \[\biggl( \frac{x+\{x\}}{[x]} - \frac{[x]}{x+\{x\}} \biggr) + \biggl( \frac{x+[x]}{ \{x \} } - \frac{ \{ x \}}{x+[x]} \biggr) > \frac 92 \]