HMMT November 2024 H5

Alf, the alien from the 1980s TV show, has a big appetite for the mineral apatite. However, he's currently on a diet, so for each integer $k \geq 1$, he can eat exactly $k$ pieces of apatite on day $k$. Additionally, if he eats apatite on day $k$, he cannot eat on any of days $k+1, k+2, \ldots, 2 k-1$. Compute the maximum total number of pieces of apatite Alf could eat over days $1,2, \ldots, 99,100$.