TUYMAADA 2001

On the side $AB$ of an isosceles triangle $AB$ ( $AC=BC$ ) lie points $P$ and $Q$ such that $\angle PCQ \le \frac{1}{2} \angle ACB$ . Prove that $PQ \le \frac{1}{2} AB$ .

Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$ , that $x-\{x\}$ is an integer.)

Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h.