AMC 8 2004 Problem 18
AMC8 2004 年第 18 题
特殊数与不定方程★★★☆☆
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?
五个朋友参加掷镖比赛。每个人都有两个飞镖向同一个圆形目标投掷，每个人的得分是被他击中的目标区域分值的总和。各个目标区域的分值是 1 到 10 的整数。每次投掷都击中到不同分值的目标区域。他们的分数是：Alice 得 16 分、Ben 得 4 分、Cindy 得 7 分、Dave 得 11 分，Ellen 得 17 分。谁击中了分值为 6 分的目标区域？
- A.
- B.
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正确答案 · Correct Answer
A
解析 · Solution
Solution 1
The only way to get Ben's score is with 1+3=4. Cindy's score can be made of 3+4 or 2+5, but since Ben already hit the 3, Cindy hit 2+5=7. Similarly, Dave's darts were in the region 4+7=11. Lastly, because there is no 7 left, must have hit the regions 6+10=16 and Ellen 8+9=17.
Taking the 2 largest scores to narrow out the points we get Ellen with either 10,7 or 9,8 and Alice with either 10,6 or 9,7, the 2 pairs that work are Alice= 10,6 Ellen= 9,8, therefore Alice scored the 6. This way, we don't need to check anyway if anyone else got 6.
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