Difference between revisions of "2006 AMC 10A Problems/Problem 19"
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The sum of the angles of a triangle is <math>180</math> degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of all of the terms, making it <math>\frac{180}{3} = 60</math> degrees. The minimum possible value for the smallest angle is <math>1</math> and the highest possible is <math>59</math> (since the numbers are distinct), so there are <math>\boxed{\textbf{(C) }59}</math> possibilities. | The sum of the angles of a triangle is <math>180</math> degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of all of the terms, making it <math>\frac{180}{3} = 60</math> degrees. The minimum possible value for the smallest angle is <math>1</math> and the highest possible is <math>59</math> (since the numbers are distinct), so there are <math>\boxed{\textbf{(C) }59}</math> possibilities. | ||
| − | ==Solution 2(Stars and Bars)== | + | ==Solution 2 (Stars and Bars)== |
Let the first angle be <math>x</math>, and the common difference be <math>d</math>. The arithmetic progression can now be expressed as <math>x + (x + d) + (x + 2d) = 180</math>. Simplifiying, <math>x + d = 60</math>. Now, using stars and bars, we have <math>\binom{61}{1} = 61</math>. | Let the first angle be <math>x</math>, and the common difference be <math>d</math>. The arithmetic progression can now be expressed as <math>x + (x + d) + (x + 2d) = 180</math>. Simplifiying, <math>x + d = 60</math>. Now, using stars and bars, we have <math>\binom{61}{1} = 61</math>. | ||
| − | However, we must subtract the two cases in which either <math>x</math> or <math>d</math> equal <math>0</math>, so we have <math>61 - 2</math> = <math>C</math>. | + | However, we must subtract the two cases in which either <math>x</math> or <math>d</math> equal <math>0</math>, so we have <math>61 - 2</math> = <math>\boxed{\textbf{(C) }59}</math>. |
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| + | ==Solution 3 (Quick Summation)== | ||
| + | Consider that we have <math>(a+n)+(a+n+1)+(a+n+2)=180 \Longleftrightarrow 3a+3(n+1)=180 \Longleftrightarrow a=59-n</math>, where <math>n \geq 0</math> and <math>n</math> is an integer. Since <math>a \neq 0</math>, <math>n=0,1,2,3,\cdots, 58</math> which is <math>\boxed{\textbf{(C) }59}</math> solutions. | ||
| + | |||
| + | ~~QuantumPsiInverted | ||
== See also == | == See also == | ||
Latest revision as of 03:10, 22 June 2023
Contents
Problem
How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?
Solution
The sum of the angles of a triangle is 
degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of all of the terms, making it 
degrees. The minimum possible value for the smallest angle is 
and the highest possible is 
(since the numbers are distinct), so there are 
possibilities.
Solution 2 (Stars and Bars)
Let the first angle be 
, and the common difference be 
. The arithmetic progression can now be expressed as 
. Simplifiying, 
. Now, using stars and bars, we have 
.
However, we must subtract the two cases in which either or equal 
, so we have 
= .
Solution 3 (Quick Summation)
Consider that we have 
, where 
and 
is an integer. Since 
, 
which is solutions.
~~QuantumPsiInverted
See also
| 2006 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
