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A Sequence of +1&#039;s and -1&#039;s

Source: ISL 2022/C1

July 9, 2023

combinatoricsInteger sequenceISL 2022

Problem Statement

A

\pm 1

-sequence is a sequence of

2022

numbers

a_1, \ldots, a_{2022},

each equal to either

+1

or

-1

. Determine the largest

C

so that, for any

\pm 1

-sequence, there exists an integer

k

and indices

1 \le t_1 < \ldots < t_k \le 2022

so that

t_{i+1} - t_i \le 2

for all

i

, and

\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.

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