Problem 1. Anya calls a date beautiful if all 6 digits of its notation are different. For example, 19.04.23 is a beautiful date, while 19.02.23 and 01.06.23 are not. How many beautiful dates are there in 2023? 3 $[4$ points] (M. Evdokimov)
Answer: 30.

Solution: The digits 2 and 3 are already used in the year number, so we need to consider only the months 01, 04, 05, 06, 07, 08, 09, and 10. Each of these month numbers contains a 0, so in a beautiful date, there will be no day number starting with 0, 2, or 3, and there will also be no days 10, 11, 12, and 13—only the 6 days from 14 to 19 remain. However, in each of these months, a beautiful date starts with 1, and only 6 months, from 04 to 09, are suitable. It remains to note that for each suitable month, exactly one day ending in the same digit will not be beautiful—thus, in each of the 6 months, there are 5 beautiful dates, and in total, in 2023, there are 30.