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Part of 2016 Abels Math Contest (Norwegian MO) Final

Problems(1)

walking sequence

Source: Niels Henrik Abel mathematics competition 2015-16

8/16/2016
A "walking sequence" is a sequence of integers with ai+1=ai±1a_{i+1} = a_i \pm 1 for every ii .Show that there exists a sequence b1,b2,...,b2016b_1, b_2, . . . , b_{2016} such that for every walking sequence a1,a2,...,a2016a_1, a_2, . . . , a_{2016} where 1 \leq a_i \leq1010, there is for some jj for which aj=bja_j = b_j .
combinatorics