THE ANSWER:
THURSDAY.Â
The only way for a 31-day month (like December) to have exactly 4 Mondays and 4 Fridays is if the first day of the month is on Tuesday, and the last day of the month is on Thursday.
So, to answer the question, this coming December, New Year’s Eve will occur on Thursday, December 31, 2009.
Friday, January 1 will be the first day of 2010.