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 18th Vietnam Mathematical Olympiad
1980年第18届越南奥林匹克数学竞赛

A1. Let x1, x2, ... , xn be real numbers in the interval [0, π] such that (1 + cos x1) + (1 + cos x2) + ... + (1 + cos xn) is an odd integer. Show that sin x1 + sin x2 + ... + sin xn ≥ 1.

A2. Let x1, x2, ... , xn be positive reals with sum s. Show that (x1 + 1/x1)2 + (x2 + 1/x2)2 + ... + (xn + 1/xn)2 ≥ n(n/s + s/n)2.

A3. P is a point inside the triangle A1A2A3. The ray AiP meets the opposite side at Bi. Ci is the midpoint of AiBi and Di is the midpoint of PBi. Show that area C1C2C3 = area D1D2D3.

B1. Show that for any tetrahedron it is possible to find two perpendicular planes such that if the projection of the tetrahedron onto the two planes has areas A and A', then A'/A > √2.

B2 Does there exist real m such that the equation x3 - 2x2 - 2x + m has three different rational roots ?

B3. Given n > 1 and real s > 0, find the maximum of x1x2 + x2x3 + x3x4 + ... + xn-1xn for non-negative reals xi such that x1 + x2 + ... + xn = s.

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