Question 51
Olya and Tolya have paints of two opposite colors – white and black. They play the following game on the segment $[0, 1]$. Each round of the game takes place in two stages: one of the players chooses a number $l \in [0, 1]$, and then the other player chooses some segment $J \subseteq [0, 1]$ of length $l$ and recolors all its points in the opposite color. In the next round they switch roles and so on.
After 2024 rounds, the total length $L$ of the white intervals is calculated. If $L \gt \frac{1}{2}$ then Olya wins, and if $L \le \frac{1}{2}$ then Tolya wins. Initially, the entire segment is painted white. Tolya chooses a number first. Who has a winning strategy?
(Andrey Naradzetski)
After 2024 rounds, the total length $L$ of the white intervals is calculated. If $L \gt \frac{1}{2}$ then Olya wins, and if $L \le \frac{1}{2}$ then Tolya wins. Initially, the entire segment is painted white. Tolya chooses a number first. Who has a winning strategy?
(Andrey Naradzetski)