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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.

question

A±1-sequenceA \pm 1 \text{-}sequence is a sequence of 2022 numbers a1,…,a2022a_{1}, \ldots, a_{2022}, each equal to either +1 or -1 . Determine the largest CC so that, for any ±1−sequence\pm 1 -sequence, there exists an integer kk and indices 1≤t1<…<tk≤20221 \leqslant t_{1}<\ldots<t_{k} \leqslant 2022 so that ti+1−ti≤2t_{i+1}-t_{i} \leqslant 2 for all ii, and |∑i=1kati|≥C \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

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

answer type

Numerical

is multiple answer

false

language

English

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
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import