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 Fifth British Mathematical Olympiad
1969年第五届英国奥林匹克数学竞赛
  1. Find the condition on the distinct real numbers a, b, c such that (x - a)(x - b)/(x - c) takes all real values. Sketch a graph where the condition is satisfied and another where it is not .
  2. Find all real solutions to cos x + cos5x + cos 7x = 3 .
  3. For which positive integers n can we find distinct integers a, b, c, d, a', b', c', d' greater than 1 such that n2 - 1 = aa' + bb' + cc' + dd'? Give the solution for the smallest n .
  4. Find all integral solutions to a2 - 3ab - a + b = 0 .
  5. A long corridor has unit width and a right-angle corner. You wish to move a pipe along the corridor and round the corner. The pipe may have any shape, but every point must remain in contact with the floor. What is the longest possible distance between the two ends of the pipe ?
  6. If a, b, c, d, e are positive integers, show that any divisor of both ae + b and ce + d also divides ad - bc .
  7. (1) f is a real-valued function on the reals, not identically zero, and differentiable at x = 0. It satisfies f(x) f(y) = f(x+y) for all x, y. Show that f(x) is differentiable arbitrarily many times for all x and that if f(1) < 1, then f(0) + f(1) + f(2) + ... = 1/( 1 - f(1) ) .
    (2) Find the real-valued function f on the reals, not identically zero, and differentiable at x = 0 which satisfies f(x) f(y) = f(x-y) for all x , y .
  8. A square side x has its vertices on the sides of a triangle with inradius r. Show that 2r > x > r√2 .
  9. Let An be an n x n array of lattice points (n > 3). Is there a polygon with n2 sides whose vertices are the points of An such that no two sides intersect except adjacent sides at a vertex? You should prove the result for n = 4 and 5, but merely state why it is plausible for n > 5 .
  10. Given a triangle, construct an equilateral triangle with the same area using ruler and compasses .
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