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 19th Indian National Mathematical Olympiad
2004年第19届印度奥林匹克数学竞赛
  1. ABCD is a convex quadrilateral. K, L, M, N are the midpoints of the sides AB, BC, CD, DA. BD bisects KM at Q. QA = QB = QC = QD, and LK/LM = CD/CB. Prove that ABCD is a square .
  2. p > 3 is a prime. Find all integers a, b, such that a2 + 3ab + 2p(a+b) + p2 = 0 .
  3. If α is a real root of x5 - x3 + x - 2 = 0, show that [α6] = 3 .
  4. ABC is a triangle, with sides a, b, c (as usual), circumradius R, and exradii ra, rb, rc. If 2R ≤ ra, show that a > b, a > c, 2R > rb, and 2R > rc .
  5. S is the set of all (a, b, c, d, e, f) where a, b, c, d, e, f are integers such that a2 + b2 + c2 + d2 + e2 = f2. Find the largest k which divides abcdef for all members of S.
  6. Show that the number of 5-tuples (a, b, c, d, e) such that abcde = 5(bcde + acde + abde + abce + abcd) is odd .
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