Question 165
An interstellar hotel has 100 rooms, their capacities are $101$, $102$, $\ldots$, $200$ people. These rooms are occupied by $n$ people in total. Now a VIP guest is about to arrive, and the owner wants to provide him with a personal room. For that purpose, the owner wants to choose two rooms, $A$ and $B$, and move all guests from $A$ to $B$ without exceeding its capacity. Determine the largest $n$ for which the owner can be sure he is able to reach the goal whatever the initial distribution of the guests is. (D. Belov, A. Safiullina)