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 example, Arbor Day this year is displayed as $20230428.$ For how many dates in $2023$ will each digit appear an even number of times in the $8$ -digit display for that date?

$5$

$6$

$7$

$8$

$9$

Answer: E

Concepts:digits parity casework

Difficulty rating: 1410

Solution:

The year $2023$ already gives two $2$ s (even), one $0$ , and one $3$ . So to make every digit occur an even number of times, the four digits of $MM$ and $DD$ must supply one more $0$ , one more $3$ , and two equal digits, while keeping the number of $2$ s even. Checking the legal dates gives $01/13,01/31,02/23$ , $03/11,03/22,10/13$ , and $10/31,11/03,11/30$ , exactly $9$ dates. Thus, E is the correct answer.

2023 AMC 10A Problem 9

Answer: E

Difficulty rating: 1410

Solution:

Problem 9 in Other Years