HMIC 2014 Problem 1

Consider a regular $n$-gon with $n>3$, and call a line acceptable if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons. (a) Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having 3 or 4 sides. (b) Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have 3 or 4 sides.