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 United States of America Mathematics Olympiad
1981年第十届美国数学奥林匹克
  1. Prove that if n is not a multiple of 3, then the angle π/n can be trisected with ruler and compasses .
  2. What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a trian link. No three towns, A, B, C are such that the links between AB, AC and BC are all air, all bus or all train .
  3. Show that for any triangle, 3(√3)/2 ≥ sin 3A + sin 3B + sin 3C ≥ -2. When does equality hold ?
  4. A convex polygon has n sides. Each vertex is joined to a point P not in the same plane. If A, B, C are adjacent vertices of the polygon take the angle between the planes PBA and PBC. The sum of the n such angles equals the sum of the n angles subtended at P by the sides of the polygon (such as the angle APB). Show that n = 3 .
  5. Show that for any positive real x, [nx] ≥ ∑1n [kx]/k .
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