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 12th Indian National Mathematical Olympiad
1997年第12届印度奥林匹克数学竞赛
  1. ABCD is a parallelogram. A line through C does not pass through the interior of ABCD and meets the lines AB, AD at E, F respectively. Show that AC2 + CE·CF = AB·AE + AD·AF.
  2. Show that there do not exist positive integers m, n such that m/n + (n+1)/m = 4.
  3. a, b, c are distinct reals such that a + 1/b = b + 1/c = c + 1/a = t for some real t. Show that t = -abc.
  4. In a unit square, 100 segments are drawn from the center to the perimeter, dividing the square into 100 parts. If all parts have equal perimeter p, show that 1.4 < p < 1.5 .
  5. Find the number of 4 x 4 arrays with entries from {0, 1, 2, 3} such that the sum of each row is divisible by 4, and the sum of each column is divisible by 4.
  6. a, b are positive reals such that the cubic x3 - ax + b = 0 has all its roots real. α is the root with smallest absolute value. Show that b/a < α ≤ 3b/2a.
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