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Difference between revisions of "2015 AMC 12B Problems/Problem 10"

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How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?
 
How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?
  
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<math>\textbf{(A)}\; ? \qquad\textbf{(B)}\; ? \qquad\textbf{(C)}\; ? \qquad\textbf{(D)}\; ? \qquad\textbf{(E)}\; ?</math>
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<math>\textbf{(A)}\; 3 \qquad\textbf{(B)}\; 4 \qquad\textbf{(C)}\; 5 \qquad\textbf{(D)}\; 6 \qquad\textbf{(E)}\; 7</math>
  
 
==Solution==
 
==Solution==
 +
Since we want non-congruent triangles that are neither isosceles nor equilateral, we can just list side lengths <math>(a,b,c)</math> with <math>a<b<c</math>. Furthermore, "positive area" tells us that <math>c < a + b</math> and the perimeter constraints means <math>a+b+c < 15</math>.
  
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There are no triangles when <math>a = 1</math> because then <math>c</math> must be less than <math>b+1</math>, implying that <math>b \geq c</math>, contrary to <math>b < c</math>.
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When <math>a=2</math>, similar to above, <math>c</math> must be less than <math>b+2</math>, so this leaves the only possibility <math>c = b+1</math>. This gives 3 triangles <math>(2,3,4), (2,4,5), (2,5,6)</math> within our perimeter constraint.
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When <math>a=3</math>, <math>c</math> can be <math>b+1</math> or <math>b+2</math>, which gives triangles <math>(3,4,5), (3,4,6), (3,5,6)</math>. Note that <math>(3,4,5)</math> is a right triangle, so we get rid of it and we get only 2 triangles.
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All in all, this gives us <math>3+2 = \boxed{\textbf{(C)}\; 5}</math> triangles.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2015|ab=B|num-a=11|num-b=9}}
 
{{AMC12 box|year=2015|ab=B|num-a=11|num-b=9}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 15:10, 5 March 2015

Problem

How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?

$\textbf{(A)}\; 3 \qquad\textbf{(B)}\; 4 \qquad\textbf{(C)}\; 5 \qquad\textbf{(D)}\; 6 \qquad\textbf{(E)}\; 7$

Solution

Since we want non-congruent triangles that are neither isosceles nor equilateral, we can just list side lengths $(a,b,c)$ with $a<b<c$. Furthermore, "positive area" tells us that $c < a + b$ and the perimeter constraints means $a+b+c < 15$.

There are no triangles when $a = 1$ because then $c$ must be less than $b+1$, implying that $b \geq c$, contrary to $b < c$.

When $a=2$, similar to above, $c$ must be less than $b+2$, so this leaves the only possibility $c = b+1$. This gives 3 triangles $(2,3,4), (2,4,5), (2,5,6)$ within our perimeter constraint.

When $a=3$, $c$ can be $b+1$ or $b+2$, which gives triangles $(3,4,5), (3,4,6), (3,5,6)$. Note that $(3,4,5)$ is a right triangle, so we get rid of it and we get only 2 triangles.

All in all, this gives us $3+2 = \boxed{\textbf{(C)}\; 5}$ triangles.

See Also

2015 AMC 12B (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 12 Problems and Solutions

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