1997 USAMO Problems/Problem 4

Problem

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 $n\ge6$ .

Solution

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1997 USAMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5
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1997 USAMO Problems/Problem 4

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