Question 5
(BUL 5) Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon $(n\ge3)$ inscribed in a circle of radius $r$. Prove the inequality
$$(n+1)h_{n+1}-nh_n>r.$$
Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n\ge3$.
$$(n+1)h_{n+1}-nh_n>r.$$
Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n\ge3$.