作者darkseer (公假中)
看板IMO_Taiwan
标题Re: [问题] 一个印度朋友给我的组合题
时间Sun Aug 29 22:23:55 2004
※ 引述《chaogold (新扇不扇不擅讪新)》之铭言：
: ※ 引述《darkseer (公假中)》之铭言：
: : As title,
: : 给定一个n边形,一个操作 = 把该n边形对某一条边做镜射
: : 问是否对任意n边形,都存在一个由n种操作所构成的无穷数列
: : 使得在将该n边形依数列操作时该n边形会跑遍整个平面
: : ------------------------------------------------------------------------------
: : 刚看到以为不难
: : 结果三角形的case我都弄不出来XD
: : 不过已经证出若在三角形成立则原命题成立
: 每一次操作的正n边形是同一个?
: 镜射出来的正n边形是一直在的吗?
我的表达实在不好XD
直接po原文好了
Let P be an n-gon , lying on a plane. We name its edges 1,2,3,..........n.
If S = s_1, s_2,.............. be a finite or infinite sequence such that for each i s_i is in {1,2........n}.
We move P in accordance with the sequence S such that we first reflect P through s_1
( the number s_i corresponds to the s_i th edge of the polygon P ) and then through s_2
and so on. Show the following holds :
a) Show that there exists a infinite sequence S such that if we move P according to S then
then we can cover the whole plane.
b) Prove that the sequence S is'nt periodic.
c) Assume that P is a regular pentagon with the radius of its circumcircle as 1 and let D be
another circle with radius 1.00001 lying in the plane arbitrary. Does there exist a sequence
S such that we move P accordingly then P reside in D completely.
原题中的(a)就是我问的
(b)(c)简单的多(好奇怪的配置XD)
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