Task 1. The number of weeks with an odd date in a month is an odd number, and the number of all other days in that month, which have an odd date, is an even number. On which day of the week is the 29th of that month? Can that month be March?
Solution. It is clear that the first Sunday of the month must be a number from 1 to 7. Moreover, an odd number of odd dates that are Sundays are possible if the Sunday is the 1st or 3rd of the month. If the first Sunday of the month is on the 3rd, then the following Sundays are $10, 17, 24$, and 31. This means all other odd dates of the month are $1, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25$, 27, and 29, i.e., 13, which is not an even number. Therefore, the desired month must start on a Sunday, i.e., Sundays are: $1, 8, 15, 22$, and 29, and all other odd dates are: $3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25$, and 27, which are a total of 12. Because of this, the month must start on a Sunday and have 30 days or be February in a leap year. Finally, the 29th of that month is a Sunday, and that month cannot be March because March has 31 days.