Question 84
Quentin and Timothé play a game. First, Quentin chooses a prime number $p \gt 2$, then Timothé chooses a positive integer $n_0$. Quentin then chooses an integer $n_1 \gt n_0$ and computes $s_1 = n_0^{n_1} + n_1^{n_0}$; then Timothé chooses an integer $n_2 \gt n_1$ and computes $s_2 = n_1^{n_2} + n_2^{n_1}$. The players keep playing in turn, choosing at turn $k$ an integer $n_k \gt n_{k-1}$ and computing $s_k = n_k^{n_{k-1}} + n_{k-1}^{n_k}$. The first player to choose an integer $n_k$ such that $p$ divides $s_k (s_1 + 2 s_2 + 3 s_3 + \ldots + k s_k)$ wins the game. Determine which of Quentin and Timothé has a winning strategy.