Question

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 how many dates in $2023$ will each digit appear an even number of times in the digital display for that date?

$5$

$6$

$7$

$8$

$9$

Solution:

The year $2023$ already gives two $2$ s (even), one $0,$ and one $3.$ So to make every digit even overall, the four digits of $MM$ and $DD$ have to supply one more $0,$ one more $3,$ and then two digits equal to each other, all while keeping the count of $2$ s even. Run through the legal months and days of $2023$ under that rule and exactly $9$ dates survive. Thus, E is the correct answer.

2023 AMC 10A Problem 9

Solution:

Problem 9 in Other Years