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 Sixth Canadian Mathematical Olympiads
1974年第六届加拿大数学奥林匹克
  1. (1) given x = (1 + 1/n)n, y = (1 + 1/n)n+1, show that xy = yx. (2) Show that 12 - 22 + 32 - 42 + ... + (-1)n+1n2 = (-1)n+1(1 + 2 + ... + n).
  2. Given the points A (0, 1), B (0, 0), C (1, 0), D (2, 0), E (3, 0), F (3, 1). Show that angle FBE + angle FCE = angle FDE.
  3. All coefficients of the polynomial p(x) are non-negative and none exceed p(0). If p(x) has degree n, show that the coefficient of xn+1 in p(x)2 is at most p(1)2/2.
  4. What is the maximum possible value for the sum of the absolute values of the differences between each pair of n non-negative real numbers which do not exceed 1?
  5. AB is a diameter of a circle. X is a point on the circle other than the midpoint of the arc AB. BX meets the tangent at A at P, and AX meets the tangent at B at Q. Show that the line PQ, the tangent at X and the line AB are concurrent.
  6. What is the largest integer n which cannot be represented as 8a + 15b with a and b non-negative integers?
  7. Bus A leaves the terminus every 20 minutes, it travels a distance 1 mile to a circular road of length 10 miles and goes clockwise around the road, and then back along the same road to to the terminus (a total distance of 12 miles). The journey takes 20 minutes and the bus travels at constant speed. Having reached the terminus it immediately repeats the journey. Bus B does the same except that it leaves the terminus 10 minutes after Bus A and travels the opposite way round the circular road. The time taken to pick up or set down passengers is negligible. A man wants to catch a bus a distance 0 < x < 12 miles from the terminus (along the route of Bus A). Let f(x) the maximum time his journey can take (waiting time plus journey time to the terminus). Find f(2) and f(4). Find the value of x for which f(x) is a maximum. Sketch f(x).
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