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 25th International Mathematical Olympiad Shortlist Problems
1984年第25届国际数学奥林匹克备选题
  1. Given real A, find all solutions (x1, x2, ... , xn) to the n equations (i = 1, 2, ... , n):
    xi |xi|- (xi - A) |(xi - A)| = xi+1 |xi+1|, where we take xn+1 to mean x1. (France 1)
  2. Prove that there are infinitely many triples of positive integers (m, n, p) satisfying 4mn - m - n = p2 - 1, but none satisfying 4mn - m - n = p2. (Canada 2)
  3. Find all positive integers n such that n = d62 + d72 - 1, where 1 = d1 < d2 < ... < dk = n are all the positive divisors of n. (USSR 3)
  4. has been used in the Olympiad .
  5. has been used in the Olympiad .
  6. c is a positive integer. The sequence f1, f2, f3, ... is defined by f1 = 1, f2 = c, fn+1 = 2 fn - fn-1 + 2. Show that for each k there is an r such that fk fk+1 = fr. (Canada 3)
  7. Can we number the squares of an 8 x 8 board with the numbers 1, 2, ... , 64 so that any four squares with any of the following shapes

    have sum = 0 mod 4? Can we do it for the following shapes ?
    (German Federal Republic 5)
  8. has been used in the Olympiad .
  9. Let a, b, c be positive reals such that √a + √b + √c = √3/2. Show that the equations:
    √(y - a) + √(z - a) = 1
    √(z - b) + √(x - b) = 1
    √(x - c) + √(y - c) = 1
    have exactly one solution in reals x, y, z. (Poland 2)
  10. Prove that the product of five consecutive positive integers cannot be the square of an integer. (Great Britain 1)
  11. a1, a2, ... , a2n are distinct integers. Find all integers x which satisfy (x - a1)(x - a2) ... (x - a2n) = (-1)n(n!)2. (Canada 1)
  12. has been used in the Olympiad .
  13. A tetrahedron is inscribed in a straight circular cylinder of volume 1. Show that its volume cannot exceed 2/(3π). (Bulgaria 5)
  14. has been used in the Olympiad .
  15. The angles of the triangle ABC are all < 120o. Equilateral triangles are constructed on the outside of each side as shown. Show that the three lines AD, BE, CF are concurrent. Suppose they meet at S. Show that SD + SE + SF = 2(SA + SB + SC).
  16. has been used in the Olympiad .
  17. If (x1, x2, ... , xn) is a permutation of (1, 2, ... , n) we call the pair (xi, xj) discordant if i < j and xi > xj. Let d(n, k) be the number of permutations of (1, 2, ... , n) with just k discordant pairs. Find d(n, 2) and d(n, 3). (German Federal Republic 3)
  18. ABC is a triangle. A circles with the radii shown are drawn inside the triangle each touching two sides and the incircle. Find the radius of the incircle.(USA 5)
  19. The harmonic table is a triangular array:
    1
    1/2  1/2
    1/3  1/6  1/3
    1/4  1/12  1/12 1/4
    ...
    where an,1 = 1/n and an,k+1 = an-1,k - an,k. Find the harmonic mean of the 1985th row. (Canada 5)
  20. Find all pairs of positive reals (a, b) with a not 1 such that logab < loga+1(b+1). (USA 2)
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