In a convex quadrilateral it is known $ABCD$ that $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^{\circ}$ and $AD = BC$. Prove that from the lengths $DB$, $CA$ and $DC$, you can make a right triangle.

Let $N = 2010!+1$. Prove that a) $N$ is not divisible by $4021$; b) $N$ is not divisible by $2027,2029,2039$; c)$ N$ has a prime divisor greater than $2050$.

For positive real numbers $a, b, c, d,$ satisfying the following conditions: $a(c^2 - 1)=b(b^2+c^2)$ and $d \leq 1$, prove that : $d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}$

In country there are two capitals ($A$ and $B$) and finite number of towns. Some towns (or town with one of capital) connected with roads (one-way). (between every two towns or capital and town there are arbitrary number of roads) such that exist at least one way from $A$ to $B$. Given, that any two ways from $A$ to $B$ have at least one common road. Prove, that exist one road, such that all ways from $A$ to $B$ pass through this road.