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 11th Indian National Mathematical Olympiad
1996年第11届印度奥林匹克数学竞赛
  1. Given any positive integer n, show that there are distinct positive integers a, b such that a + k divides b + k for k = 1, 2, ... , n. If a, b are positive integers such that a + k divides b + k for all positive integers k, show that a = b.
  2. C, C' are concentric circles with radii R, 3R respectively. Show that the orthocenter of any triangle inscribed in C must lie inside the circle C'. Conversely, show that any point inside C' is the orthocenter of some circle inscribed in C.
  3. Find reals a, b, c, d, e such that 3a = (b + c + d)3, 3b = (c + d + e)3, 3c = (d + e + a)3, 3d = (e + a + b)3, 3e = (a + b + c)3.
  4. X is a set with n elements. Find the number of triples (A, B, C) such that A, B, C are subsets of X, A is a subset of B, and B is a proper subset of C.
  5. The sequence a1, a2, a3, ... is defined by a1 = 1, a2 = 2, an+2 = 2an+1 - an + 2. Show that for any m, amam+1 is also a term of the sequence.
  6. A 2n x 2n array has each entry 0 or 1. There are just 3n 0s. Show that it is possible to remove all the 0s by deleting n rows and n columns.
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