2023 AMC 12B 第 16 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。
16.
在 Coinland，有三种硬币，面值分别为 ，，和 。无法凑出的最大金额的 各位数字之和是多少？
In Coinland, there are three types of coins, each worth and What is the sum of the digits of the maximum amount of money that is impossible to have?
答案：D
小提示：
列出哪些金额能由 凑出；寻找最大的不能凑出的金额
List which amounts can be formed from look for the largest that cannot
大提示：
一旦六个连续金额都能凑出，再添加更多 就能凑出之后所有金额
Once six consecutive amounts are all attainable, adding more ’s reaches everything beyond
解答：
金额 都能凑出（例如 ，，，，，）。之后不断加上 元硬币，就能凑出所有更大的金额。检查更小的数， 无法凑出：若不用 元硬币，若干 元和 元硬币的总额为偶数；若用一枚 元硬币，剩余的 不能写成 ；两枚 元硬币已经超过 。所以最大无法凑出的金额是 ，其数位和为 。
因此，正确答案是 D。
The amounts are all attainable (for instance ). Adding ’s then reaches every larger amount. Checking below, is impossible: without a -coin a sum of ’s and ’s is even, while with one -coin the remaining cannot be written as two -coins already exceed So the largest impossible amount is whose digit sum is
Thus, the correct answer is D.
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