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 15th United States of America Mathematics Olympiad
1986年第15届美国数学奥林匹克
  1. Do there exist 14 consecutive positive integers each divisible by a prime less than 13 ? What about 21 consecutive positive integers each divisible by a prime less than 17 ?
  2. Five professors attended a lecture. Each fell asleep just twice. For each pair there was a moment when both were asleep. Show that there was a moment when three of them were asleep.
  3. What is the smallest n > 1 for which the average of the first n (non-zero) squares is a square ?
  4. A T-square allows you to construct a straight line through two points and a line perpendicular to a given line through a given point. Circles C and C' intersect at X and Y. XY is a diameter of C. P is a point on C' inside C. Using only a T-square, find points Q,R on C such that QR is perpendicular to XY and PQ is perpendicular to PR.
  5. A partition of n is an increasing sequence of integers with sum n. For example, the partitions of 5 are: 1, 1, 1, 1, 1; 1, 1, 1, 2; 1, 1, 3; 1, 4; 5; 1, 2, 2; and 2, 3. If p is a partition, f(p) = the number of 1s in p, and g(p) = the number of distinct integers in the partition. Show that ∑ f(p) = ∑ g(p), where the sum is taken over all partitions of n.
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