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Some average sequences

Source: Austrian-Polish 2004, Problem 9

July 5, 2015
Sequencesalgebra

Problem Statement

Given are the sequences (...,a−2,a−1,a0,a1,a2,...);(...,b−2,b−1,b0,b1,b2,...);(...,c−2,c−1,c0,c1,c2,...) (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...) of positive real numbers. For each integer nn the following inequalities hold: an≥12(bn+1+cn−1)a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1}) bn≥12(cn+1+an−1)b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1}) cn≥12(an+1+bn−1)c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1}) Determine a2005a_{2005}, b2005b_{2005}, c2005c_{2005}, if a0=26,b0=6,c0=2004a_0 = 26, b_0 = 6, c_0 = 2004.
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