2010 AMC 8 第 7 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。
7.
只使用一美分、五美分、十美分和二十五美分硬币，弗雷迪最少需要多少枚硬币，才能支付小于一美元的任意金额？
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?
小提示：
能支付从零到 美分的所有金额后，二十五美分硬币可以把范围扩展到 美分
After making every amount through cents, quarters can extend the range to cents.
大提示：
先考虑支付 、、 美分等小金额所必须的硬币
First force the coins needed to make small amounts like , , and cents.
解答：
要支付 美分，至少需要四枚一美分硬币。有了这四枚后，为了能支付 美分，下一枚硬币的面值至多为 美分，最好选一枚五美分硬币。这五枚硬币总值只有 美分。下一枚硬币的面值至多为 美分；即使选一枚十美分硬币，总值也只有 美分。因此还需要一枚面值至多为 美分的硬币，所以在不产生金额缺口的情况下使用二十五美分硬币之前，至少需要七枚硬币。
若总共只有九枚硬币，在必需的七枚之后最多只剩两枚。前七枚的总值至多为 美分，再加两枚二十五美分硬币也只有 美分，所以九枚硬币不能支付从一美分到 美分的每个金额。
十枚硬币确实可以做到：四枚一美分硬币、一枚五美分硬币、两枚十美分硬币和三枚二十五美分硬币，可以支付从 到 美分的每个金额。因此最少需要 枚。
所以正确答案是 B。
At least four pennies are necessary to pay cents. After those four pennies, the next coin must be worth at most cents so that cents can be paid; a nickel is the best choice. Those five coins total only cents. The next coin must be worth at most cents, and even a dime brings the total to only cents. One more coin worth at most cents is therefore necessary, so at least seven coins are needed before quarters can be used without leaving a gap.
With only nine coins total, at most two coins could remain after those required seven. The first seven can total at most cents, and two more quarters would bring the total to only cents, so nine coins cannot pay every amount through cents.
Ten coins do work: four pennies, one nickel, two dimes, and three quarters can make every amount from through cents. Therefore the minimum is .
Therefore, the answer is B .
其他年份的第 7 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
- 1996 AMC 8
- 1997 AMC 8
- 1998 AMC 8
- 1999 AMC 8
- 2000 AMC 8
- 2001 AMC 8
- 2002 AMC 8
- 2003 AMC 8
- 2004 AMC 8
- 2005 AMC 8
- 2006 AMC 8
- 2007 AMC 8
- 2008 AMC 8
- 2009 AMC 8
- 2011 AMC 8
- 2012 AMC 8
- 2013 AMC 8
- 2014 AMC 8
- 2015 AMC 8
- 2016 AMC 8
- 2017 AMC 8
- 2018 AMC 8
- 2019 AMC 8
- 2020 AMC 8
- 2022 AMC 8
- 2023 AMC 8
- 2024 AMC 8
- 2025 AMC 8
- 2026 AMC 8