Minimum maximum birthdays in a month

There are 100 students distributed across 12 months, so $x_1 + x_2 + \dots + x_{12} = 100$.

By the Pigeonhole Principle, the maximum value $x_0 = \max(x_1, \dots, x_{12})$ must be at least the average: $x_0 \ge \left\lceil \frac{100}{12} \right\rceil = \lceil 8.33 \rceil = 9$

If $x_0 = 8$, the maximum total students would be $8 \times 12 = 96 < 100$, which is impossible. Thus, the smallest possible value of $x_0$ is 9.