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 22nd United States of America Mathematics Olympiad
1993年第22届美国数学奥林匹克
  1. n > 1, and a and b are positive real numbers such that an - a - 1 = 0 and b2n - b - 3a = 0. Which is larger ?
  2. The diagonals of a convex quadrilateral meet at right angles at X. Show that the four points obtained by reflecting X in each of the sides are cyclic .
  3. Let S be the set of functions f defined on reals in the closed interval [0, 1] with non-negative real values such that f(1) = 1 and f(x) + f(y) ≤ f(x + y) for all x, y such that x + y ≤ 1. What is the smallest k such that f(x) ≤ kx for all f in S and all x ?
  4. The sequence an of odd positive integers is defined as follows: a1 = r, a2 = s, and an is the greatest odd divisor of an-1 + an-2. Show that, for sufficiently large n, an is constant and find this constant (in terms of r and s) .
  5. A sequence xn of positive reals satisfies xn-1xn+1 ≤ xn2. Let an be the average of the terms x0, x1, ... , xn and bn be the average of the terms x1, x2, ... , xn. Show that anbn-1 ≥ an-1bn .
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