Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m
m (minor fix)
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
−
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>.
+
{{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.)
  
−
We now know that one of July 1, July 2 and July 3 was a Monday.
+
<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>
  
−
In these three cases, August 1 is a Thursday, a Wednesday, and a Tuesday.
+
== 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>.  
  
−
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>
+
In the first case August starts on a Thursday, and there are five Thursdays, Fridays, and Saturdays in August.  
  
−
==See also==
+
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 <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}}
 
{{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 $N$ 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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2002_AMC_12B_Problems/Problem_8&oldid=141657"