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Three strictly increasing sequences

a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots

of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer

n

, the following conditions hold: (a)

c_{a_n}=b_n+1

; (b)

a_{n+1}>b_n

; (c) the number

c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n

is even. Find

a_{2010}

b_{2010}

and

c_{2010}

.
(4th Middle European Mathematical Olympiad, Team Competition, Problem 1)

Problem Statement