RU-SPB 2008

Point $O$ is the center of the circle into which quadrilateral $ABCD$ is inscribed. If angles $AOC$ and $BAD$ are both equal to $110$ degrees and angle $ABC$ is greater than angle $ADC$ , prove that $AB+AD>CD$ .

Fresh translation.

There are $2008$ trinomials $x^2-a_kx+b_k$ where $a_k$ and $b_k$ are all different numbers from set $(1,2,...,4016)$ . These trinomials has not common real roots. We mark all real roots on the $Ox$ -axis. Prove, that distance between some two marked points is $\leq \frac{1}{250}$

All faces of the tetrahedron $ABCD$ are acute-angled triangles. $AK$ and $AL$ -are altitudes in faces $ABC$ and $ABD$ . Points $C,D,K,L$ lies on circle. Prove, that $AB \perp CD$