2002 AMC 12B Problems/Problem 8

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Let $M$ be a 31-day month and $W$ a day of the week. We can easily see that $W$ occurs five times in $M$ if and only if one of the first three days of $M$ falls on a $W$ . This is because the 5th occurrence of $W$ is 28 days after the first one, so the only possibilities for their dates are $(1,29)$ , $(2,30)$ , and $(3,31)$ .

We now know that one of July 1, July 2 and July 3 was a Monday.

In these three cases, August 1 is a Thursday, a Wednesday, and a Tuesday.

The answer is a day of the week that is guaranteed to fall on one of August 1, August 2, and August 3. We can easily see that the only such day of the week is $\boxed{\mathrm{Thursday}}$ (August 1 in case 1, August 2 in case 2, and August 3 in case 3) $\Longrightarrow \mathrm{(D)}.$

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2002 AMC 12B Problems/Problem 8

From AoPSWiki