Question

五个朋友参加飞镖比赛。每人向同一个圆形靶子投两支飞镖，每个人的得分是击中区域分数之和。靶子区域的分数是整数 $1$ 到 $10$ ，十次投掷击中的区域分值互不相同。得分为：Alice $16$ 分，Ben $4$ 分，Cindy $7$ 分，Dave $11$ 分，Ellen $17$ 分。谁击中了分值为 $6$ 的区域？

Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers $1$ through $10.$ Each throw hits the target in a region with a different value. The scores are: Alice $16$ points, Ben $4$ points, Cindy $7$ points, Dave $11$ points, and Ellen $17$ points. Who hits the region worth $6$ points?

$\text{Alice}$

$\text{Ben}$

$\text{Cindy}$

$\text{Dave}$

$\text{Ellen}$

答案：A

知识点：逻辑推理分类讨论

难度评级：1400

解答：

Ben 得到四分的唯一方式是 $1$ 和 $3$ ，因为他不能两次击中 $2$ 。

Cindy 的得分可以由 $1 + 6,\quad 2 + 5,\quad 3 + 4.$ 得到。Ben 已经击中了 $1$ 和 $3$ ，所以 Cindy 必须击中 $2$ 和 $5$ 。

同理，Dave 必须击中 $4$ 和 $7$ 。最后，因为 $7$ 已经被使用，Alice 被迫击中 $6$ 和 $10$ ，Ellen 击中 $8$ 和 $9$ 。

所以正确答案是 A。

The only way to get Ben's score is with a $1$ and $3$ since he can't hit $2$ twice.

Cindy can achieve her score with $1 + 6,\quad 2 + 5,\quad 3 + 4.$ Ben already hit $1$ and $3,$ so Cindy must have hit $2$ and $5.$

Similarly, Dave must have hit $4$ and $7.$ Finally, since $7$ is already used, Alice is forced to have hit $6$ and $10$ with Ellen hitting $8$ and $9.$

Thus, A is the correct answer.

2004 AMC 8 第 18 题

答案：A

难度评级：1400

解答：

其他年份的第 18 题