Problem:
A digital display shows the current date as an -digit integer consisting of a -digit year, followed by a -digit month, followed by a -digit date within the month. For example, Arbor Day this year is displayed as . For how many dates in does each digit appear an even number of times in the -digit display for that date?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because the first four digits of the display must be 2023, the last four digits must contain one 0 , one 3 , and two of the same digit. These digits cannot contain three 0 s or three 3 s , so the only possible repeated digits among them are 1 and 2.
If the repeated digit is 1 , the possible dates are , and . If the repeated digit is 2 , the possible dates are and . In all there are dates in 2023 for which each digit appear an even number of times.
The problems on this page are the property of the MAA's American Mathematics Competitions