2023 SSMO Relay Round 2 Problems/Problem 3

Problem

Let $T=$ TNYWR. In a committee of $2023$ people, $N$ are scientists and the rest are builders. In order to make a building, $\frac{N}{2}$ people must be choosen with at least one scientist and one builder. If $x$ is the number of ways to do this, find the largest integer $a$ such $2^a \mid x$.

Solution