2012 AMC 8 第 14 题

Question

在中学橄榄球联盟 BIG N 中，每支球队与其他每支球队恰好比赛一次。如果

2012

赛季共进行了

21

场联盟比赛，那么 BIG N 联盟有多少支球队？

In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of

21

conference games were played during the

2012

season, how many teams were members of the BIG N conference?

6

7

8

9

10

答案：B

知识点：组合三角形数

难度评级：1100

小提示：

列出每增加一支球队会新增多少场比赛。

List how many games are added when each new team joins.

大提示：

n

支球队时，比赛场数为

\frac{n(n-1)}{2}

。

With

n

teams, the number of games is

\frac{n(n-1)}{2}

.

视频讲解：

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文字解答：

若有

N

支球队，每支球队要与

N-1

支球队比赛。不过，每场比赛涉及

2

支球队，所以把

N

与

N-1

相乘会把每场比赛数两次。因此

N(N-1) = 42

。这给出

N^2 -N +0.25 = 42.25

从而

(N-0.5)^2 = 6.5^2\text{，}

于是

N-0.5 = 6.5

N = 7

所以正确答案是 B。

Each of the

N

teams plays

N-1

games. However,

2

teams play each game, so multiplying

N

and

N-1

would be twice the number of games. Therefore, we know

N(N-1) = 42.

This leads to

N^2 -N +0.25 = 42.25

which implies

(N-0.5)^2 = 6.5^2.

In turn, this suggests:

N-0.5 = 6.5

N = 7

Thus, the answer is B .