HMMT 2012 U36

Maria is hopping up a flight of stairs with 100 steps. At every hop, she advances some integer number of steps. Each hop she makes has fewer steps. However, the positive difference between the length of consecutive hops decreases. Let $P$ be the number of distinct ways she can hop up the stairs. Find lower and upper bounds $L$ and $U$ for $P$. If $0<L \leq P \leq U$, your score will be $\left\lfloor\frac{23}{\sqrt{U / L}}\right\rfloor$. Otherwise, your score will be 0 .