数学竞赛题（图）
<br /="/"/><div></div><br /="/"/><br /="/"/><input /="/" src="http://static3.businessinsider.com/image/520b825b69bedd5e2a000006-745-543/screen%20shot%202013-08-14%20at%209.11.40%20am.png" style="CURSOR: default" type="image"/><br /="/"/><br /="/"/><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div>回复：数学竞赛题（图）
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/>problems from 2013 International Math Olympiad, too hard to average people.<div></div><div><br /="/"/></div><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：数学竞赛题（图）
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>answer to q1,</div><div>k = 2:</div><div>4/1 = 2/1 * 2/1, 6/3 = 3/2 * 4/3, 8/4 = 4/3 * 6/5</div><div>5/2 = 2/1 * 5/4, 7/4 = 3/2 * 7/6, 9/5 = 4/3 * 9/8, ...</div><div>----------</div><div>k = 3:</div><div>8/1 = 4/1 * 2/1, 10/3 = 5/2 * 4/3</div><div>9/2 = 4/1 * 9/8, 11/4 = 5/2 * 11/10, ...</div><div> </div><div>prove by mathematical induction. </div><div>Denote the LHS as F(n, k).</div><div>1. F(n,1) is trivial. </div><div>2. assume true for F(n, K)</div><div>3. For F(n, K+1), there are two cases.</div><div>case 1. n even, let n = 2m, </div><div>F(2m, K+1) = F(m, K) * (1 + 1 / (2*(m+2^k-1)))</div><div>case 2. n odd, let n = 2m -1</div><div>F(2m-1, K+1) = F(m, k) * (1 + 1/(2m-1))</div><div>Either case, there is one more term and F(m, K) can be split as k terms by induction.</div><div> </div><div> </div><div> </div><div> </div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>
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