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

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 $N(N-1) = 42$ 。这给出因而于是 $N^2 -N +0.25 = 42.25$ $(N-0.5)^2 = 6.5^2.$ $N-0.5 = 6.5$ $N = 7$

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 .

2012 AMC 8 第 14 题

答案：B

难度评级：1100

视频讲解：

文字解答：

其他年份的第 14 题