10.3. In an intergalactic hotel, there are 100 rooms with capacities of 
$101, 102, \ldots, 200$ people. In these rooms, a total of $n$ people live. A VIP guest has arrived at the hotel, and a whole room needs to be vacated for them. To do this, the hotel manager selects one room and relocates all its occupants to another single room. For what largest $n$ can the hotel manager always free up a room regardless of the current occupancy? (D. Belov, A. Safiullina)
Answer: 8824.

Solution: Suppose that with 8824 guests, the director cannot carry out the relocation. Let's divide the rooms into pairs by capacity: $101-200, 102-199, \ldots, 150-151$. Note that for each pair of rooms, the total number of people living in the two rooms is greater than the capacity of the larger room in the pair; otherwise, all the people from this pair could be gathered in the room with the larger capacity. Thus, the total number of people is no less than $201+200+199+\ldots+152=353 \cdot 25=8825$. Therefore, with 8824 guests, the director can free up a room.

Now let's provide an example proving that with 8825 or more guests, there exists a distribution in which it is impossible to free up a room in the specified manner.

Order the rooms by increasing capacity. Let there be 76 people in the first fifty rooms, and in the room with capacity $k$ for $151 \leqslant k \leqslant 200$, there are $k-75$ people. Let's calculate the number of people living in the hotel:

\[
\begin{aligned}
& 76 \cdot 50 + (76 + 77 + 78 + \ldots + 125) = \\
& \quad = 3800 + 201 \cdot 25 = 3800 + 5025 = 8825
\end{aligned}
\]

Consider any two arbitrary rooms with capacities $a < b$. If the number of people in the room with capacity $a$ is $a-75$ and the number of people in the room with capacity $b$ is $b-75$, then the total number of people in these two rooms is $(a-75) + (b-75) = a + b - 150$. Since $a + b \geq 301$, we have $a + b - 150 \geq 151$, which is greater than the capacity of the room with capacity $a$. Therefore, it is impossible to move all the people from these two rooms into the room with capacity $a$.

If the number of people in the hotel is greater than 8825, then it is sufficient to house the remaining people alternately in any rooms where there are still free spaces.