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Difference between revisions of "2002 AMC 12B Problems/Problem 8"

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Let <math>M</math> be a 31-day month and <math>W</math> a day of the week. We can easily see that <math>W</math> occurs five times in <math>M</math> if and only if one of the first three days of <math>M</math> falls on a <math>W</math>. This is because the 5th occurrence of <math>W</math> is 28 days after the first one, so the only possibilities for their dates are <math>(1,29)</math>, <math>(2,30)</math>, and <math>(3,31)</math>.
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{{duplicate|[[2002 AMC 12B Problems|2002 AMC 12B #8]] and [[2002 AMC 10B Problems|2002 AMC 10B #8]]}}
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== Problem ==
  
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We now know that one of July 1, July 2 and July 3 was a Monday.
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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.)
  
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In these three cases, August 1 is a Thursday, a Wednesday, and a Tuesday.
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<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>
  
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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 <math>\boxed{\mathrm{Thursday}}</math> (August 1 in case 1, August 2 in case 2, and August 3 in case 3) <math>\Longrightarrow \mathrm{(D)}.</math>
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==Solution==
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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>.
  
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==See also==
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In the first case August starts on a Thursday, and there are five Thursdays, Fridays, and Saturdays in August.
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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>.
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==See Also==
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{{AMC10 box|year=2002|ab=B|num-b=7|num-a=9}}
 
{{AMC12 box|year=2002|ab=B|num-b=7|num-a=9}}
 
{{AMC12 box|year=2002|ab=B|num-b=7|num-a=9}}
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[[Category:Introductory Algebra Problems]]

Revision as of 16:49, 28 July 2011

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 $N$ has five Mondays. Which of the following must occurs five times in the August of year $N$? (Note: Both months have $31$ days.)

$\textrm{(A)}\ \text{Monday} \qquad \textrm{(B)}\ \text{Tuesday} \qquad \textrm{(C)}\ \text{Wednesday} \qquad \textrm{(D)}\ \text{Thursday} \qquad \textrm{(E)}\ \text{Friday}$

Solution

If there are five Mondays, there are only three possibilities for their dates: $(1,8,15,22,29)$, $(2,9,16,23,30)$, and $(3,10,17,24,31)$.

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 $\boxed{\textrm{(D)}\ \text{Thursday}}$.

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
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