AMC12 2023 A

AMC12 2023 A · Q7

AMC12 2023 A · Q7. It mainly tests Casework, Digit properties (sum of digits, divisibility tests).

A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as $20230428.$ For how many dates in $2023$ does each digit appear an even number of times in the $8$-digital display for that date?

一个数字显示屏将当前日期显示为一个由 $8$ 位数字组成的整数，包括 $4$ 位年份、紧接着的 $2$ 位月份、以及月份内的 $2$ 位日期。例如，今年树日显示为 $20230428$。在 $2023$ 年的多少个日期中，$8$ 位数字显示中每个数字出现的次数都是偶数？

(A) 5 5

(B) 6 6

(D) 8 8

(E) 9 9

Answer

Correct choice: (E)

正确答案：（E）

Solution

Do careful casework by each month. Make sure to start with 2023. In the month and the date, we need a $0$, a $3$, and two digits repeated (which has to be $1$ and $2$ after consideration). After the casework, we get $\boxed{\textbf{(E)}~9}$. For curious readers, the numbers (in chronological order) are:

按每个月仔细分类讨论。确保从 $2023$ 开始。在月份和日期中，我们需要一个 $0$、一个 $3$，以及两个重复的数字（考虑后必须是 $1$ 和 $2$）。分类讨论后，得到 $\boxed{\textbf{(E)}~9}$。对于好奇的读者，按时间顺序的数字是：

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