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 34th Vietnam Mathematical Olympiad
1996年第34届越南数学奥林匹克

A1. Find all real x, y such that √(3x) (1 + 1/(x + y) ) = 2 and √(7y) (1 - 1/(x + y) ) = 4√2.

A2. SABC is a tetrahedron. DAB, EBC, FCA are triangles in the plane of ABC congruent to SAB, SBC, SCA respectively. O is the circumcenter of DEF. Let K be the exsphere of SABC opposite O (which touches the planes SAB, SBC, SCA, ABC, lies on the opposite side of ABC to S, but on the same side of SAB as C, the same side of SBC as A, and the same side of SCA as B). Show that K touches the plane ABC at O.

A3. Let n be a positive integer and k a positive integer not greater than n. Find the number of ordered k-tuples (a1, a2, ... , an) such that: (1) all ai are different, but all belong to {1, 2, ... , n}; (2) ar > as for some r < s; (3) ar has the opposite parity to r for some r.

B1. Find all functions f(n) on the positive integers with positive integer values, such that f(n) + f(n+1) = f(n+2) f(n+3) - 1996 for all n.

B2. The triangle ABC has BC = 1 and angle A = x. Let f(x) be the shortest possible distance between its incenter and its centroid. Find f(x). What is the largest value of f(x) for 60o < x < 180o?

B3. Let w, x, y, z be non-negative reals such that 2(wx + wy + wz + xy + xz + yz) + wxy + xyz + yzw + zwx = 16. Show that 3(w + x + y + z) ≥ 2(wx + wy + wz + xy + xz + yz).

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