Question 102
An olympic sequence is a sequence $s_{1}, s_{2}, \ldots, s_{2023}$ each of whose 2023 terms is equal to 1 or to -1. A slowly increasing sequence is a sequence of integers $t_{1}, t_{2}, \ldots, t_{n}$ such that $1 \leqslant t_{1}\lt t_{2}\lt \cdots\lt t_{n} \leqslant 2023$, and each of whose $n-1$ differences $t_{i+1}-t_{i}$ equals 1 or 2.
Find the largest integer $C$ such that, for every olympic sequence $s_{1}, s_{2}, \ldots, s_{2023}$, there exist an integer $n \geqslant 1$ and a slowly increasing sequence $t_{1}, t_{2}, \ldots, t_{n}$ of length $n$ such that
$$
\left|s_{t_{1}}+s_{t_{2}}+\cdots+s_{t_{n}}\right| \geqslant C
$$
Find the largest integer $C$ such that, for every olympic sequence $s_{1}, s_{2}, \ldots, s_{2023}$, there exist an integer $n \geqslant 1$ and a slowly increasing sequence $t_{1}, t_{2}, \ldots, t_{n}$ of length $n$ such that
$$
\left|s_{t_{1}}+s_{t_{2}}+\cdots+s_{t_{n}}\right| \geqslant C
$$