Difference between revisions of "2002 AMC 12B Problems/Problem 8"
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| − | + | {{duplicate|[[2002 AMC 12B Problems|2002 AMC 12B #8]] and [[2002 AMC 10B Problems|2002 AMC 10B #8]]}} | |
| + | == Problem == | ||
| + | Suppose July of year <math>N</math> has five Mondays. Which of the following must occurs five times in the August of year <math>N</math>? (Note: Both months have <math>31</math> days.) | ||
| − | + | <math>\textrm{(A)}\ \text{Monday} \qquad \textrm{(B)}\ \text{Tuesday} \qquad \textrm{(C)}\ \text{Wednesday} \qquad \textrm{(D)}\ \text{Thursday} \qquad \textrm{(E)}\ \text{Friday}</math> | |
| − | + | == Solution == | |
| + | If there are five Mondays, there are only three possibilities for their dates: <math>(1,8,15,22,29)</math>, <math>(2,9,16,23,30)</math>, and <math>(3,10,17,24,31)</math>. | ||
| − | + | In the first case August starts on a Thursday, and there are five Thursdays, Fridays, and Saturdays in August. | |
| − | + | In the second case August starts on a Wednesday, and there are five Wednesdays, Thursdays, and Fridays in August. | |
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| − | + | In the third case August starts on a Tuesday, and there are five Tuesdays, Wednesdays, and Thursdays in August. | |
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| + | The only day of the week that is guaranteed to appear five times is therefore <math>\boxed{\textrm{(D)}\ \text{Thursday}}</math>. | ||
| + | |||
| + | == See Also == | ||
| + | {{AMC10 box|year=2002|ab=B|num-b=7|num-a=9}} | ||
| + | {{AMC12 box|year=2002|ab=B|num-b=7|num-a=9}} | ||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Revision as of 02:52, 7 January 2021
- The following problem is from both the 2002 AMC 12B #8 and 2002 AMC 10B #8, so both problems redirect to this page.
Problem
Suppose July of year 
has five Mondays. Which of the following must occurs five times in the August of year ? (Note: Both months have 
days.)
Solution
If there are five Mondays, there are only three possibilities for their dates: 
, 
, and 
.
In the first case August starts on a Thursday, and there are five Thursdays, Fridays, and Saturdays in August.
In the second case August starts on a Wednesday, and there are five Wednesdays, Thursdays, and Fridays in August.
In the third case August starts on a Tuesday, and there are five Tuesdays, Wednesdays, and Thursdays in August.
The only day of the week that is guaranteed to appear five times is therefore 
.
See Also
| 2002 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
| 2002 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 7 |
Followed by Problem 9 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
