1997 USAMO 第 4 题

To clip a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by 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 $P_6$ of area $1$ is clipped to obtain a heptagon $P_7$. Then $P_7$ is clipped (in one of the seven possible ways) to obtain an octagon $P_8$, and so on. Prove that no matter how the clippings are done, the area of $P_n$ is greater than $\frac{1}{3}$, for all $n\ge6$.

裁剪凸 $n$-gon 意味着选择一对连续的边 $AB、BC$ 并用三个线段 $AM、MN、$ 和 $NC,$ 替换它们，其中 $M$ 是 $AB$ 的中点，$N$ 是 $BC$ 的中点。换句话说，将三角形 $MBN$ 切掉以获得凸 $(n+1)$-边形。面积 $1$ 的正六边形 $P_6$ 被裁剪以获得七边形 $P_7$。然后 $P_7$ 被剪裁(以七种可能的方式之一)以获得八边形 $P_8$，依此类推。证明无论如何剪裁，对于所有 $n\ge6$，$P_n$ 的面积都大于 $\frac{1}{3}$。