Question

一个房间里有 $52$ 人。使“这个房间中至少有 $n$ 人的生日在同一个月份”这一陈述总为真的最大 $n$ 是多少？

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?

$2$

$3$

$4$

$5$

$12$

答案：D

知识点：抽屉原理

难度评级：1070

小提示：

对 $12$ 个月使用抽屉原理。

Use the pigeonhole principle with $12$ months

大提示：

尝试把 $52$ 个生日尽量平均分配。

Try distributing $52$ birthdays as evenly as possible

解答：

$n\geq 6$ 不一定成立，因为可以让 $5$ 人的生日落在前 $4$ 个月中的每个月，而其余八个月各有 $4$ 人生日。

但一定有某个月至少 $5$ 人生日，因为平均每月人数为 $\frac{52}{12}$ ，大于 $4$ 。

所以正确答案是 D。

It isn’t necessarily true for $n\geq 6$ as we could have $5$ people born in the first $4$ months and $4$ people born in the subsequent months.

However, one month must be greater than or equal to $5$ as the average of the number of people born in each month is $\frac{52}{12}$ which is greater than $4,$ and some month must be above average.

Thus, the correct answer is D .

2011 AMC 10B 第 11 题

答案：D

难度评级：1070

小提示：

大提示：

解答：

其他年份的第 11 题