1999 AMC 8 第 15 题
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15.
平原镇的自行车牌照每个含有三个字母。第一个字母从集合 中选，第二个从 中选，第三个从 中选。
当平原镇需要更多牌照时，他们增加了两个新字母。这两个新字母可以都加入同一个集合，也可以分别加入两个不同集合。加入两个字母后，最多能多制作多少个额外牌照？
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set the second from and the third from
When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of additional license plates that can be made by adding two letters?
小提示：
原来的牌照数是
The old number of plates is
大提示：
要让乘积最大，应让三个因子尽量接近平衡
To maximize a product, make the three factors as balanced as possible
解答：
目前一共可以制作 个牌照。
如果两个字母都加入第一个集合，那么可以制作 个牌照。
如果两个字母都加入第二个集合，那么有 个牌照。
如果两个字母都加入第三个集合，那么有 种选择。
如果一个字母加入第一个集合，另一个加入第二个集合，那么有 个牌照。
如果另一个字母改为加入第三个集合，那么可以得到 个牌照。
最后，如果两个字母分别加入第二个和第三个集合，那么有 个牌照。
最大可以达到 个牌照，因此额外增加 个。
所以正确答案是 D。
There are currently license plates that can be made.
If both letters are added to the first set, then there are possible plates.
If they are both added to the second, there are plates.
If they are added to the third, there are choices.
If one is added to the first set and the other to the second set, there are plates.
If the other is added to the third set, we get possible plates.
Finally, if the letters are added to the second and third sets, there are plates.
We see that is the greatest number of plates that we can achieve. This is an additional plates.
Thus, D is the correct answer.
其他年份的第 15 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
- 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
- 2010 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