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2002 AMC 12B Problem 8

Problem 8 of 25EasierAlgebraNumber Theory

Suppose July of year NN has five Mondays. Which of the following must occur five times in August of year N?N? (Note: Both months have 3131 days.)

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Solution

Since July has 31=47+331=4\cdot7+3 days, the weekday of July 11 occurs five times, so Monday falls on July 1,1, 2,2, or 3.3. The days that occur five times in August are those of August 1,1, 2,2, 3,3, and August 11 is three weekdays after July 1.1. Testing the three cases, the August weekdays occurring five times are Thursday through Saturday, Wednesday through Friday, and Tuesday through Thursday. Thursday appears in every case. Thus, the correct answer is D.

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Concepts: date and time · modular arithmetic · casework

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.