Question

Among 100 students, $x_1$ have birthdays in January, $x_2$ have birthdays in February, and so on. If $x_0 = \max(x_1, x_2, \dots, x_{12})$, then the smallest possible value of $x_0$ is

Accepted Answer

D. 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: $$x_0 \ge \left\lceil \frac{100}{12} ightceil = \lceil 8.33 ceil = 9$$ If $x_0 = 8$, the maximum total students would be $8 	imes 12 = 96 < 100$, which is impossible. Thus, the smallest possible value of $x_0$ is 9.

Minimum maximum birthdays in a month

Explanation

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