2010 AIME II 第 4 题

Question

Dave 到达一座机场，机场有十二个登机口排成一直线，相邻登机口之间正好相距

100

英尺。他的出发登机口随机分配。在该登机口等待后，Dave 被告知出发登机口改到了另一个不同的登机口，这个新登机口也随机分配。设 Dave 走到新登机口的距离为

400

英尺或更少的概率为分数

\frac{m}{n}

，其中

m

和

n

是互质的正整数。求

m + n

。

Dave arrives at an airport which has twelve gates arranged in a straight line with exactly

100

feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks

400

feet or less to the new gate be a fraction

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m + n.

答案

答案：52

知识点：基本概率数对计数分类讨论

难度评级：2170

小提示：

走

400

英尺或更少意味着两个登机口编号相差至多

4

；数一数不同登机口的有序对。

Walking

400

feet or less means the gate numbers differ by at most

4;

count ordered pairs of distinct gates

大提示：

靠近两端的登机口近邻较少：计数为

4, 5, 6, 7, 8, 8, 8, 8, 7, 6, 5, 4

。

Gates near the ends have fewer close neighbors: the counts are

4, 5, 6, 7, 8, 8, 8, 8, 7, 6, 5, 4

解答：

将登机口编号为

1

到

12

。所有

12 \cdot 11 = 132

个不同（原登机口、新登机口）的有序对等可能，且 Dave 走

400

英尺或更少，当且仅当登机口编号相差至多

4

。

登机口

i

有

\min(i + 4, 12) - \max(i - 4, 1)

个合格的新登机口：登机口

1

和

12

各有

4

个，登机口

2

和

11

各有

5

个，登机口

3

和

10

各有

6

个，登机口

4

和

9

各有

7

个，登机口

5

到

8

各有

8

个。总数为

2(4 + 5 + 6 + 7) + 4 \cdot 8 = 76

。

概率为

\frac{76}{132} = \frac{19}{33}

，所以

m + n = 19 + 33 = 52

。

Number the gates

1

through

12.

All

12 \cdot 11 = 132

ordered pairs of distinct (old, new) gates are equally likely, and Dave walks

400

feet or less exactly when the gate numbers differ by at most

4.

A gate

i

has

\min(i + 4, 12) - \max(i - 4, 1)

qualifying new gates: gates

1

and

12

have

4

each, gates

2

and

11

have

5,

gates

3

and

10

have

6,

gates

4

and

9

have

7,

and gates

5

through

8

have

8

each. The total is

2(4 + 5 + 6 + 7) + 4 \cdot 8 = 76.

The probability is

\frac{76}{132} = \frac{19}{33},

so

m + n = 19 + 33 = 52.

2010 AIME II 第 4 题

答案：52

难度评级：2170

小提示：

大提示：

解答：

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