数学竞赛题（图）

xyh

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Xiangdang

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

sean9991

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, ... ---------- 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.         www.ddhw.com