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 27th Vietnam Mathematical Olympiad
1989年第27届越南奥林匹克数学竞赛

A1. Show that the absolute value of sin(kx) /N + sin(kx + x) /(N+1) + sin(kx + 2x) /(N+2) + ... + sin(kx + nx) /(N+n) does not exceed the smaller of (n+1) |x| and 1/(N sin(x/2) ), where N is a positive integer and k is real and satisfies 0 ≤ k ≤ N.

A2. Let a1 = 1, a2 = 1, an+2 = an+1 + an be the Fibonacci sequence. Show that there are infinitely many terms of the sequence such that 1985an2 + 1956an + 1960 is divisible by 1989. Does there exist a term such that 1985an2 + 1956an + 1960 + 2 is divisible by 1989?

A3. ABCD is a square side 2. The segment AB is moved continuously until it coincides with CD (note that A is brought into coincidence with the opposite corner). Show that this can be done in such a way that the region passed over by AB during the motion has area < 5π/6.

B1. Do there exist integers m, n not both divisible by 5 such that m2 + 19n2 = 198·101989 ?

B2. Define the sequence of polynomials p0(x), p1(x), p2(x), ... by p0(x) = 0, pn+1(x) = pn(x) + (x - pn(x)2)/2. Show that for any 0 ≤ x ≤ 1, 0 < √x - pn(x) ≤ 2/(n+1).

B3. ABCDA'B'C'D' is a parallelepiped (with edges AB, BC, CD, DA, AA', BB', CC', DD', A'B', B'C', C'D', D'A'). Show that if a line intersects three of the lines AB', BC', CA', AD', then it also intersects the fourth.

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