KURSCHAK 1971

A straight line cuts the side $AB$ of the triangle $ABC$ at $C_1$ , the side $AC$ at $B_1$ and the line $BC$ at $A_1$ . $C_2$ is the reflection of $C_1$ in the midpoint of $AB$ , and $B_2$ is the reflection of $B_1$ in the midpoint of $AC$ . The lines $B_2C_2$ and $BC$ intersect at $A_2$ . Prove that $$\frac{sen B_1A_1C}{sen C_2A_2B} = \frac{B_2C_2}{B_1C_1}$$

Given any $22$ points in the plane, no three collinear. Show that the points can be divided into $11$ pairs, so that the $11$ line segments defined by the pairs have at least five different intersections

There are $30$ boxes each with a unique key. The keys are randomly arranged in the boxes, so that each box contains just one key and the boxes are locked. Two boxes are broken open, thus releasing two keys. What is the probability that the remaining boxes can be opened without forcing them?