2022 AMC 8 第 21 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
21.
Steph 在一场比赛上半场投进 球，共出手 次；下半场投进 球，共出手 次。Candace 上半场出手 次，下半场出手 次。在每个半场中，Steph 的命中率都高于 Candace。令人惊讶的是，她们最终总命中率相同。Candace 下半场比上半场多进了多少球？
Steph scored baskets out of attempts in the first half of a game, and baskets out of attempts in the second half. Candace took attempts in the first half and attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?
小提示：
Steph 总共投进 球，共出手 次。
Steph made baskets in attempts overall
大提示：
Candace 也必须总共投进 球。
Candace must also have made baskets total
解答：
Steph 总共进了 球，出手 次。Candace 也出手了 次，且两人的总命中率相同，所以 Candace 也进了 球。
设 Candace 上半场进 球，下半场进 球，则 。
Steph 上半场命中率为 ，所以 ，得到 。Steph 下半场命中率为 ，所以 。
因此 且 。又 ，唯一可能是 、，所以 。
正确答案是 C。
Steph made baskets in attempts. Candace also took attempts, and their overall percentages were the same, so Candace also made baskets.
Let be Candace’s first-half baskets and be her second-half baskets. Then .
Steph’s first-half percentage was , so , giving . Steph’s second-half percentage was , so .
Thus and . Since , the only possibility is and , so .
Thus, the correct answer is C.
其他年份的第 21 题
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
- 1995 AMC 8
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- 1997 AMC 8
- 1998 AMC 8
- 1999 AMC 8
- 2000 AMC 8
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