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 "2012 AMC 10A Problems/Problem 10"

(→‎Problem 10)
(→‎Solution 3)
 
(19 intermediate revisions by 7 users not shown)
Line 1: Line 1:
−
== Problem 10 ==
+
{{duplicate|[[2012 AMC 12A Problems|2012 AMC 12A #7]] and [[2012 AMC 10A Problems|2012 AMC 10A #10]]}}
 +
 
 +
== Problem ==
  
 
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
 
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
  
 
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 </math>
 
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 </math>
−
== Solution ==
+
[[Category: Introductory Geometry Problems]]
−
When you say the smallest sector is A and the common difference is D they you have, adding the angels together, A+A+D+A+2D....+A+11D. This gives 12A+66D. 12A+66D must equal 360 degrees. 12A+66D=360. This gives A+11D=60. To get the smallest A you must have the largest D. D=5 so A+55=60.This means A is 5.
+
 
 +
==Solution 1==
 +
 
 +
Let <math>a_1</math> be the first term of the arithmetic progression and <math>a_{12}</math> be the last term of the arithmetic progression. From the formula of the sum of an arithmetic progression (or arithmetic series), we have <math>12*\frac{a_1+a_{12}}{2}=360</math>, which leads us to <math>a_1 + a_{12} = 60</math>. <math>a_{12}</math>, the largest term of the progression, can also be expressed as <math>a_1+11d</math>, where <math>d</math> is the common difference. Since each angle measure must be an integer, <math>d</math> must also be an integer. We can isolate <math>d</math> by subtracting <math>a_1</math> from <math>a_{12}</math> like so: <math>a_{12}-a_1=a_1+11d-a_1=11d</math>. Since <math>d</math> is an integer, the difference between the first and last terms, <math>11d</math>, must be divisible by <math>11.</math> Since the total difference must be less than <math>60</math>, we can start checking multiples of <math>11</math> less than <math>60</math> for the total difference between <math>a_1</math> and <math>a_{12}</math>. We start with the largest multiple, because the maximum difference will result in the minimum value of the first term. If  the difference is <math>55</math>, <math>a_1=\frac{60-55}{2}=2.5</math>, which is not an integer, nor is it one of the five options given. If the difference is <math>44</math>, <math>a_1=\frac{60-44}{2}</math>, or <math>\boxed{\textbf{(C)}\ 8}</math>
 +
 
 +
-Solution by Rhiju
 +
 
 +
== Solution 2==
 +
 
 +
If we let <math>a</math> be the smallest sector angle and <math>r</math> be the difference between consecutive sector angles, then we have the angles <math>a, a+r, a+2r, \cdots. a+11r</math>. Use the formula for the sum of an arithmetic sequence and set it equal to 360, the number of degrees in a circle.
 +
 
 +
<cmath>\begin{align*} \frac{a+a+11r}{2}\cdot 12 &= 360\\
 +
2a+11r &= 60\\
 +
a &= \frac{60-11r}{2} \end{align*}</cmath>
 +
 
 +
All sector angles are integers so <math>r</math> must be a multiple of 2. Plug in even integers for <math>r</math> starting from 2 to minimize <math>a.</math> We find this value to be 4 and the minimum value of <math>a</math> to be <math>\frac{60-11(4)}{2} = \boxed{\textbf{(C)}\ 8}</math>
 +
 
 +
== Solution 3==
 +
Starting with the smallest term, <math>a - 5x \cdots a, a + x \cdots a + 6x</math> where <math>a</math> is the sixth term and <math>x</math> is the difference. The sum becomes <math>12a + 6x = 360</math> since there are <math>360</math> degrees in the central angle of the circle. The only condition left is that the smallest term in greater than zero. Therefore, <math>a - 5x > 0</math>.
 +
<cmath>2a + x = 60</cmath>
 +
<cmath>x = 60 - 2a</cmath>
 +
<cmath>a - 5(60 - 2a) > 0</cmath>
 +
<cmath>11a > 300</cmath>
 +
Since <math>a</math> is an integer, it must be <math>28</math>, and therefore, <math>x</math> is <math>4</math>. <math>a - 5x</math> is <math>\boxed{\textbf{(C)}\ 8}</math>
 +
 
 +
==Video Solution (CREATIVE THINKING)==
 +
https://youtu.be/1FyU20KeEKU
 +
 
 +
~Education, the Study of Everything
 +
 
 +
== See Also ==
 +
 
 +
{{AMC10 box|year=2012|ab=A|num-b=9|num-a=11}}
 +
{{AMC12 box|year=2012|ab=A|num-b=6|num-a=8}}
 +
{{MAA Notice}}

Latest revision as of 12:54, 1 July 2023

The following problem is from both the 2012 AMC 12A #7 and 2012 AMC 10A #10, so both problems redirect to this page.

Problem

Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12$

Solution 1

Let $a_1$ be the first term of the arithmetic progression and $a_{12}$ be the last term of the arithmetic progression. From the formula of the sum of an arithmetic progression (or arithmetic series), we have $12*\frac{a_1+a_{12}}{2}=360$, which leads us to $a_1 + a_{12} = 60$. $a_{12}$, the largest term of the progression, can also be expressed as $a_1+11d$, where $d$ is the common difference. Since each angle measure must be an integer, $d$ must also be an integer. We can isolate $d$ by subtracting $a_1$ from $a_{12}$ like so: $a_{12}-a_1=a_1+11d-a_1=11d$. Since $d$ is an integer, the difference between the first and last terms, $11d$, must be divisible by $11.$ Since the total difference must be less than $60$, we can start checking multiples of $11$ less than $60$ for the total difference between $a_1$ and $a_{12}$. We start with the largest multiple, because the maximum difference will result in the minimum value of the first term. If the difference is $55$, $a_1=\frac{60-55}{2}=2.5$, which is not an integer, nor is it one of the five options given. If the difference is $44$, $a_1=\frac{60-44}{2}$, or $\boxed{\textbf{(C)}\ 8}$

-Solution by Rhiju

Solution 2

If we let $a$ be the smallest sector angle and $r$ be the difference between consecutive sector angles, then we have the angles $a, a+r, a+2r, \cdots. a+11r$. Use the formula for the sum of an arithmetic sequence and set it equal to 360, the number of degrees in a circle.

\begin{align*} \frac{a+a+11r}{2}\cdot 12 &= 360\\ 2a+11r &= 60\\ a &= \frac{60-11r}{2} \end{align*}

All sector angles are integers so $r$ must be a multiple of 2. Plug in even integers for $r$ starting from 2 to minimize $a.$ We find this value to be 4 and the minimum value of $a$ to be $\frac{60-11(4)}{2} = \boxed{\textbf{(C)}\ 8}$

Solution 3

Starting with the smallest term, $a - 5x \cdots a, a + x \cdots a + 6x$ where $a$ is the sixth term and $x$ is the difference. The sum becomes $12a + 6x = 360$ since there are $360$ degrees in the central angle of the circle. The only condition left is that the smallest term in greater than zero. Therefore, $a - 5x > 0$. \[2a + x = 60\] \[x = 60 - 2a\] \[a - 5(60 - 2a) > 0\] \[11a > 300\] Since $a$ is an integer, it must be $28$, and therefore, $x$ is $4$. $a - 5x$ is $\boxed{\textbf{(C)}\ 8}$

Video Solution (CREATIVE THINKING)

https://youtu.be/1FyU20KeEKU

~Education, the Study of Everything

See Also

2012 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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
2012 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
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=2012_AMC_10A_Problems/Problem_10&oldid=195019"