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OlympiadBench / 2004 / A±1-sequence is a sequence of 2022 numbers …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

answer type

Numerical

is multiple answer

false

language

English

question

A \pm 1 \text{-}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 \leqslant t_{1}<\ldots<t_{k} \leqslant 2022

so that

t_{i+1}-t_{i} \leqslant 2

for all

i

, and

\left|\sum_{i=1}^{k} a_{t_{i}}\right| \geqslant C

Plain-text mathematical notation (without MathML)

A±1-sequence is a sequence of 2022 numbers a₁,…,a₂₀₂₂, each equal to either +1 or -1 . Determine the largest C so that, for any ±1−sequence, there exists an integer k and indices 1≤t₁<…<t_(k)≤2022 so that t_(i+1)−t_(i)≤2 for all i, and

|∑_(i=1)^(k)a_(t_(i))|≥C

Original LaTeX notation

$A \pm 1 \text{-}sequence$ 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 $1 \leqslant t_{1}<\ldots<t_{k} \leqslant 2022$ so that $t_{i+1}-t_{i} \leqslant 2$ for all $i$, and

$$
\left|\sum_{i=1}^{k} a_{t_{i}}\right| \geqslant C
$$

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2004
task_id: 54a339f0-0d88-55ca-9b1b-5076ddf6ffac
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2004
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

initial import