2011 AMC 10B 第 11 题

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

对

12

个月使用抽屉原理。

Use the pigeonhole principle with

12

months

尝试把

52

个生日尽量平均分配。

Try distributing

52

birthdays as evenly as possible

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 .

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