Number of Tuesdays in five consecutive calendar years taken together is exactly 260. Which day of the week was 1st January of the first of these five years?
I don't know at all how to do this question. I know that in leap year there are 2 odd days and in ordinary year there is 1 odd day.
Since $5*52=260$, the question is equivalent to asking what day of the week the first year must begin on if none of five consecutive years has 53 Tuesdays.
A year will have 53 Tuesdays if the dominical letter is F, FE, or GF.
Examining the possibilities for the dominical letters (including the cases involving non-leap century years) gives the following:
For the cases that do not involve non-leap century years, one can see that the first year must be a non-leap year beginning on a Wednesday.
For the cases that involve non-leap century years, the above $15$ cases are the only ones that one need to check for because the Gregorian calendar repeats every $400$ years. Of those $15$ cases, $4$ of them do not contain a year with $53$ Tuesdays. The years $2098, 2200,$ and $2296$ begin on Wednesdays, while the year $2099$ begins on a Thursday.
So, the answer is that the first year must usually begin on a Wednesday. Once every $400$ years, there is a solution beginning with a year that begins on a Thursday (namely, the years $2099$ to $2103$).