AMC10 2011 B

AMC10 2011 B · Q11

AMC10 2011 B · Q11. It mainly tests Pigeonhole principle, Casework.

There are 52 people in a room. What is the largest value of n such that the statement “At least n people in this room have birthdays falling in the same month” is always true?

房间里有 52 个人。最大的 n 值是多少，使得“房间里至少有 n 个人生日在同一个月”这个陈述总是成立？

(A) 2 2

(B) 3 3

(D) 5 5

(E) 12 12

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): If no more than 4 people have birthdays in any month, then at most 48 people would be accounted for. Therefore the statement is true for $n = 5$. The statement is false for $n \ge 6$ if, for example, 5 people have birthdays in each of the first 4 months of the year, and 4 people have birthdays in each of the last 8 months, for a total of $5 \cdot 4 + 4 \cdot 8 = 52$ people.

答案（D）：如果任何一个月过生日的人不超过 4 个，那么最多可以统计到 48 个人。因此该命题在 $n = 5$ 时为真。该命题在 $n \ge 6$ 时为假，例如：一年中前 4 个月每个月有 5 个人过生日，后 8 个月每个月有 4 个人过生日，总人数为 $5 \cdot 4 + 4 \cdot 8 = 52$ 人。

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