KURSCHAK 1974

A library has one exit and one entrance and a blackboard at each. Only one person enters or leaves at a time. As he does so he records the number of people found/remaining in the library on the blackboard. Prove that at the end of the day exactly the same numbers will be found on the two blackboards (possibly in a different order).

$S_n$ is a square side $\frac{1}{n}$ . Find the smallest $k$ such that the squares $S_1, S_2,S_3, ...$ can be put into a square side $k$ without overlapping.

Let $$p_k(x) = 1 -x + \frac{x^2}{2! } - \frac{x^3}{3!}+ ... + \frac{(-x)^{2k}}{(2k)!}$$ Show that it is non-negative for all real $x$ and all positive integers $k$ .