ÖMO 2008

For every positive integer $n$ let $$a_n=\sum_{k=n}^{2n}\frac{(2k+1)^n}{k}$$ Show that there exists no $n$ , for which $a_n$ is a non-negative integer.

Which positive integers are missing in the sequence $\left\{a_n\right\}$ , with $a_n = n + \left[\sqrt n\right] +\left[\sqrt [3]n\right]$ for all $n \ge 1$ ? ( $\left[x\right]$ denotes the largest integer less than or equal to $x$ , i.e. $g$ with $g \le x < g + 1$ .)

We are given a square $ABCD$ . Let $P$ be a point not equal to a corner of the square or to its center $M$ . For any such $P$ , we let $E$ denote the common point of the lines $PD$ and $AC$ , if such a point exists. Furthermore, we let $F$ denote the common point of the lines $PC$ and $BD$ , if such a point exists. All such points $P$ , for which $E$ and $F$ exist are called acceptable points. Determine the set of all acceptable points, for which the line $EF$ is parallel to $AD$ .