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 14th Asian Pacific Mathematical Olympiad
2002年第14届亚太地区数学奥林匹克
  1. Let a1, a2, a3, ... , an be a sequence of non-negative integers, where n is a positive integer.
    Let An = (a1 + a2 + a3 + ... + an)/n
    Prove that a1!a2!a3! ... an! ≥ ([An]!)n
    where [An] is the greatest integer less than or equal to Anand a! = 1 × 2 × ... × a for a ≥ 1 (and 0! = 1)
    When does equality hold ?
  2. Find all positive integers a and b such that (a2 + b)/(b2-a) and (b2 + a)/(a2-b) are both integers.
  3. Let ABC be an equilateral triangle. Let P be a point on the side AC and Q be a point on the side AB so that both triangles ABP and ACQ are acute. Let R be the orthocentre of triangle ABP and S be the orthocentre of triangle ACQ. Let T be the point common to the segments BP and CQ. Find all possible values of ∠CBP and ∠BCQ such that triangle TRS is equilateral.
  4. Let x, y, z be positive numbers such that 1/x + 1/y +1/z =1
    Show that √(x + yz) + √(y + zx) + √(z + xy) ≥ √(xyz) + √x + √y + √z .
  5. Let R denote the set of all real numbers. Find all functions f from R to R satisfying:
    (i) there are only finitely many s in R such that f(s) = 0, and
    (ii) f(x4 + y) = x3f(x) + f(f(y)) for all x, y in R.
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