2010 AMC 12B Problem 4
Attempt Problem 4 of the 2010 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 12B solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
4.
A month with days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
Answer: B
Small hint:
so exactly three weekdays occur five times
Big hint:
Those three days are the first three days of the month; check when Monday and Wednesday match
Solution:
Since the first three days of the month each occur five times, and the other four days occur four times.
Mondays and Wednesdays are equal in number exactly when both fall in the five-time group or both fall in the four-time group.
If the first day is Monday, the five-time days are Mon, Tue, Wed (both appear five times). If the first day is Thursday or Friday, the five-time days miss both Monday and Wednesday (both appear four times). Every other starting day includes exactly one of Monday or Wednesday.
So the first day can be Monday, Thursday, or Friday, giving possibilities.
Thus, the correct answer is B.
Problem 4 in other years
- 1950 AMC 12
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