Question 4
To clip a convex $n$-gon, choose consecutive sides $AB,BC$, let $M,N$ be their respective midpoints, and replace $AB,BC$ by $AM,MN,NC$. Thus triangle $MBN$ is cut off, giving a convex $(n+1)$-gon. Start with a regular hexagon $\mathcal P_6$ of area $1$ and repeatedly clip, obtaining $\mathcal P_7,\mathcal P_8,\ldots$. Prove that every possible sequence of clippings satisfies $\operatorname{area}(\mathcal P_n)\gt 1/3$ for all $n\ge6$.