USAMO 1997 Problem 4

To clip a convex $n$ -gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$ , and $NC$ , where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$ . In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$ -gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$ . Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$ , and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$ , for all $n \geq 6$ .