奥数之家
奥数论坛
简短留言
| 首页 | 竞赛大纲 | 优秀前辈 | 视频提示 | 专题讲座 | 论文锦集 | 综合训练 | 修身养性 | 家教平台 | 奥数论坛 |
 
The 10th Vietnam Mathematical Olympiad
1971年第十届越南数学奥林匹克
A1. m, n, r, s are positive integers such that: (1) m < n and r < s; (2) m and n are relatively prime, and r and s are relatively prime; and (3) tan-1m/n + tan-1r/s = π/4. Given m and n, find r and s. Given n and s, find m and r. Given m and s, find n and r.
B1. ABCDA'B'C'D' is a cube (with ABCD and A'B'C'D' faces, and AA', BB', CC', DD' edges). L is a line which intersects or is parallel to the lines AA', BC and DB'. L meets the line BC at M (which may be the point at infinity). Let m = |BM|. The plane MAA' meets the line B'C' at E. Show that |B'E| = m. The plane MDB' meets the line A'D' at F. Show that |D'F| = m. Hence or otherwise show how to construct the point P at the intersection of L and the plane A'B'C'D'. Find the distance between P and the line A'B' and the distance between P and the line A'D' in terms of m. Find a relation between these two distances that does not depend on m. Find the locus of M. Let S be the envelope of the line L as M varies. Find the intersection of S with the faces of the cube.
点击此处查看相关视频讲解
在方框内输入单词或词组
建议使用:IE 6.0及以上版本浏览器。不支持 Netscape浏览器。 本站空间由北京师范大学提供
Copyright © 2005-2007 aoshoo.com All Rights Reserved 滇ICP备05000048号
MSN:shuxvecheng@hotmail.com QQ:316180036 E-mail:aoshoo@sina.com 电话:15810289082