2023 AMC 10A Problem 9

Problem 9 of 25EasierNumber Theory

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?

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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,

10/31,

11/03,

and

11/30,

exactly

9

dates. Thus, E is the correct answer.