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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）

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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