NO-ABEL 2013

In a triangle $T$ , all the angles are less than $90^o$ , and the longest side has length $s$ . Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the distance from $p$ to $h$ is less than or equal to $s/\sqrt3$ .

A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$ . Show that $p$ divides $a$ .