Every month had four Mondays, five Sundays, and one Friday the 13th.

Which day of the week was New Year's Day that year? What day of the week will New Year's Day be in the following year?

(M. Volfová)
Because it was the 13th on Friday, it must have been the 15th on Sunday and the 16th on Monday. Therefore, the dates falling on Sundays are 1st, 8th, 15th, 22nd, 29th, and the dates falling on Mondays are 2nd, 9th, 16th, 23rd, and possibly 30th. Since the month in question had five Sundays and four Mondays, it had exactly 29 days, which means it was February in a leap year.

From the above, we know that February 1st was a Sunday, so January 31st was a Saturday. The other Saturdays in January were 24th, 17th, 10th, and 3rd. Therefore, New Year's Day in the year in question was on a Thursday.

A leap year has 366 days, which is 52 full weeks plus two extra days $(366=52 \cdot 7+2)$. Therefore, New Year's Day in the following year will be on a Saturday.

Scoring suggestion. 3 points for determining that it was February in a leap year; 2 points for determining that New Year's Day in the year in question was on a Thursday; 1 point for determining that New Year's Day in the following year will be on a Saturday.