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 Fourth Canadian Mathematical Olympiads
1972年第四届加拿大数学奥林匹克
1. Three unit circles are arranged so that each touches the other two. Find the radii of the two circles which touch all three.
2. x1, x2, ... , xn are non-negative reals. Let s = ∑i<j xixj. Show that at least one of the xi has square not exceeding 2s/(n2 - n).
3. Show that 10201 is composite in base n > 2. Show that 10101 is composite in any base.
4. Show how to construct a convex quadrilateral ABCD given the lengths of each side and the fact that AB is parallel to CD.
5. Show that there are no positive integers m, n such that m3 + 113 = n3.
6. Given any distinct real numbers x, y, show that we can find integers m, n such that mx + ny > 0 and nx + my < 0.
7. Show that the roots of x2 - 198x + 1 lie between 1/198 and 197.9949494949... . Hence show that √2 < 1.41421356 (where the digits 421356 repeat). Is it true that √2 < 1.41421356?
8. X is a set with n elements. Show that we cannot find more than 2n-1 subsets of X such that every pair of subsets has non-empty intersection.
9. Given two pairs of parallel lines, find the locus of the point the sum of whose distances from the four lines is constant.
10. Find the longest possible geometric progression in {100, 101, 102, ... , 1000}.
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