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标题: 数学竞赛题(图) [打印本页]

作者: xyh    时间: 2013-8-14 18:50
标题: 数学竞赛题(图)





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作者: Xiangdang    时间: 2013-8-18 23:04
标题: 回复:数学竞赛题(图)

problems from 2013 International Math Olympiad, too hard to average people.



 

作者: sean9991    时间: 2013-9-25 23:17
标题: 回复:数学竞赛题(图)

answer to q1,
k = 2:
4/1 = 2/1 * 2/1, 6/3 = 3/2 * 4/3, 8/4 = 4/3 * 6/5
5/2 = 2/1 * 5/4, 7/4 = 3/2 * 7/6, 9/5 = 4/3 * 9/8, ...
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k = 3:
8/1 = 4/1 * 2/1, 10/3 = 5/2 * 4/3
9/2 = 4/1 * 9/8, 11/4 = 5/2 * 11/10, ...
 
prove by mathematical induction.  
Denote the LHS as F(n, k).
1. F(n,1) is trivial. 
2. assume true for F(n, K)
3. For F(n, K+1), there are two cases.
case 1. n even, let n = 2m,
F(2m, K+1) = F(m, K) * (1 + 1 / (2*(m+2^k-1)))
case 2. n odd, let n = 2m -1
F(2m-1, K+1) = F(m, k) * (1 + 1/(2m-1))
Either case, there is one more term and F(m, K) can be split as k terms by induction.
 
 
 
 
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