Question 189
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 $1 \leqslant t_{1}\lt \ldots\lt t_{k} \leqslant 2022$ so that $t_{i+1}-t_{i} \leqslant 2$ for all $i$, and
$$
\left|\sum_{i=1}^{k} a_{t_{i}}\right| \geqslant C.
$$
$$
\left|\sum_{i=1}^{k} a_{t_{i}}\right| \geqslant C.
$$