2010 AMC 10B Problem 5

First figure out which weekdays appear more often than the others.

1. $31 = 28 + 3$, and $28$ days is exactly $4$ weeks. In those $4$ weeks every weekday appears $4$ times.
2. The last $3$ days of the month (days $29, 30, 31$) fall on the same weekdays as days $1, 2, 3$. Those three weekdays get a fifth appearance. Every other weekday appears only $4$ times.
3. So Monday and Wednesday have the same count exactly when the first three days of the month include both of them or neither of them.

Now check all seven possible first days:

Three first days work: Monday, Thursday and Friday.

The answer is $\boxed{\textbf{(B)}\ 3}$.

How often each weekday shows up in a month depends only on the days left over after the full weeks. Here the $3$ leftover days get an extra appearance, so the question becomes a simple check of seven short lists. Remember that "the same number" can happen in two ways: both days get the extra appearance, or neither does.

Since $31 = 4 \cdot 7 + 3$, every weekday occurs four times, and the weekdays of days $29, 30, 31$, which are the same as days $1, 2, 3$, occur a fifth time. So the first day and the two days after it appear five times; the other four weekdays appear four times.

Monday and Wednesday have equal counts exactly when the three-day block $\{d, d+1, d+2\}$ (with $d$ the starting weekday) contains both of them or neither.

• Both: the block must be Mon, Tue, Wed, so $d =$ Monday.
• Neither: the block must avoid Monday and Wednesday. Checking each start, Thu-Fri-Sat and Fri-Sat-Sun work; Sat-Sun-Mon, Sun-Mon-Tue, Tue-Wed-Thu, and Wed-Thu-Fri each contain one of the two days.

That gives $3$ possible starting days: Monday, Thursday, Friday.

The answer is $\boxed{\textbf{(B)}\ 3}$.