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 17th United States of America Mathematics Olympiad
1988年第17届美国数学奥林匹克
  1. The repeating decimal 0.ab ... k pq ... u = m/n, where m and n are relatively prime integers, and there is at least one decimal before the repeating part. Show that n is divisible by 2 or 5 (or both). [For example, 0.01136 = 0.01136363636 ... = 1/88 and 88 is divisible by 2.]
  2. The cubic x3 + ax2 + bx + c has real coefficients and three real roots r ≥ s ≥ t. Show that k = a2 - 3b ≥ 0 and that √k <= r - t.
  3. Let X be the set {1, 2, ... , 20} and let P be the set of all 9-element subsets of X. Show that for any map f: P → X we can find a 10-element subset Y of X, such that f(Y - {k}) ≠ k for any k in Y.
  4. ABC is a triangle with incenter I. Show that the circumcenters of IAB, IBC, ICA lie on a circle whose center is the circumcenter of ABC.
  5. Let p(x) be the polynomial (1 - x)a (1 - x2)b (1 - x3)c ... (1 - x32)k, where a, b, ..., k are integers. When expanded in powers of x, the coefficient of x1 is -2 and the coefficients of x2, x3, ... , x32 are all zero. Find k.
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