一道外文题

葡萄糖

In a circle we enscribe a regular 2001-gon and inside it a regular 667-gon with shared vertices.
Prove that the surface in the 2001-gon but not in the 667-gon is of the form  \( k\*\sin^3\left(\frac{\pi}{2001}\right)*\cos^3\left(\frac{\pi}{2001}\right) \)  with \( k \) a positive integer. Find \( k \).

wayne

这个很好计算了:
\(s= 667*(1/2R^2)*\Big(3\sin(a)-\sin(3a)\Big)= 2*667*R^2*\sin^3(a) .\)
而 \(2001a=2\pi\)
所以, \(k=8\*667 =5336\)