HMMT 二月 2000 · 团队赛 · 第 15 题

英文解析

15. Consider a regular n -gon with radius r . Let x be the side length of the n -gon. So,
    sin e the en tral angle is (see diagram b elo w), use the La w of Cosines to nd that 2
    2 2 2 2 2 2 2 2 pqnx = r + r 2 r r os , so x = 2 r (1 os ). Th us, x = r 2 1 os .
    n n nq
    2 p
    So, the total p erimeter of the n -gon is nx = nr 2 1 os . No w, if we tak enlim of the p erimeter, the result will be 2 n , sin e the n -gon approa hes a irle,
    n !1
    q qp pso lim nr 2 1 os = 2 r , and so lim nr 1 os = r 2 .22
    n !1 n !1
    rxnnr
    2 πn