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Sequences of points

Source: 2004 China Second Round Olympiad

August 30, 2014
analytic geometrygeometry unsolvedgeometry

Problem Statement

In a planar rectangular coordinate system, a sequence of points An{A_n} on the positive half of the y-axis and a sequence of points Bn{B_n} on the curve y=2xy=\sqrt{2x} (x≥0)(x\ge0) satisfy the condition ∣OAn∣=∣OBn∣=1n|OA_n|=|OB_n|=\frac{1}{n}. The x-intercept of line AnBnA_nB_n is ana_n, and the x-coordinate of point BnB_n is bnb_n, n∈Nn\in\mathbb{N}. Prove that (1) an>an+1>4a_n>a_{n+1}>4, n∈Nn\in\mathbb{N}; (2) There is n0∈Nn_0\in\mathbb{N}, such that for any n>n0n>n_0, b2b1+b3b2+…+bnbn−1+bn+1bn<n−2004\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004.
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