Mathematical Olympiad in China (2009–2010): Problems and Solutions by Xiong Bin Lee Peng Yee

Author:Xiong Bin, Lee Peng Yee
Language: eng
Format: epub, pdf
Publisher: World Scientific
Published: 2013-06-15T16:00:00+00:00

Fig. 2

where .

Since OO1Q = BAP = CAM, and OO2Q = CAP = BAM, it follows that

and thus

Note that M is the midpoint of BC, and therefore O1Q = QO2, i.e. line AO bisects segment O1O2.

Let A = {a1, a2, …, a2010} and B = {b1, b2,…, b2010} be two sets of complex numbers, such that the equality

holds for every n = 1, 2,…, 2010. Prove that A = B. (Posed by Leng Gangsong)

Proof Let Sk = and . We first show by induction that Sk = k for k = 1, 2,…, 2010.

Setting n = 1 in the given equality, we have 2009S1 = 20091, and hence S1 = 1. Assume that Sj = j for j = 1, 2,…, k − 1, where 2 k 2010; we are going to show that Sk = k.

By the binomial expansion theorem, we have

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