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Difference between revisions of "2024 AMC 12B Problems/Problem 24"

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Note that there exists a unique, non-degenerate triangle with altitudes <math>a, b, c</math> if and only if <math>\frac{1}{a}, \frac{1}{b}, \frac{1}{c}</math> are the side lengths of a non-degenerate triangle. With this in mind, it remains to find all positive integer solutions <math>(R, a, b, c)</math> to the above such that <math>\frac{1}{a}, \frac{1}{b}, \frac{1}{c}</math> from a non-degenerate triangle, and <math>a\le b\le c\le 9</math>. We do this by doing casework on the value of <math>R</math>.
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Note that there exists a unique, non-degenerate triangle with altitudes <math>a, b, c</math> if and only if <math>\frac{1}{a}, \frac{1}{b}, \frac{1}{c}</math> are the side lengths of a non-degenerate triangle. With this in mind, it remains to find all positive integer solutions <math>(R, a, b, c)</math> to the above such that <math>\frac{1}{a}, \frac{1}{b}, \frac{1}{c}</math> from a non-degenerate triangle, and <math>a\le b\le c\le 9</math>. We do this by doing casework on the value of <math>R</math>. Since <math>R</math> is a positive integer <math>R\ge 1</math>. Since <math>a\le b\le c\le 9</math>, <math>\frac{1}{R}\ge \frac{1}{3}</math>, so <math>R\le3</math>. The only possible values for <math>R</math> are 1, 2, 3.
  
 
<math>\textbf{Case 1: R=1}</math>
 
<math>\textbf{Case 1: R=1}</math>

Revision as of 01:49, 14 November 2024

Problem 24

What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a$, $b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)

$\textbf{(A) }2\qquad \textbf{(B) }3\qquad \textbf{(C) }4\qquad \textbf{(D) }5\qquad \textbf{(E) }6\qquad$

Solution

First we derive the relationship between the inradius of a triangle $R$, and its three altitudes $a, b, c$. Using an area argument, we can get the following well known result \[\left(\frac{AB+BC+AC}{2}\right)R=A\] where $AB, BC, AC$ are the side lengths of $\triangle ABC$, and $A$ is the triangle's area. Substituting $A=\frac{1}{2}\cdot AB\cdot c$ into the above we get \[\frac{R}{c}=\frac{AB}{AB+BC+AC}\] Similarly, we can get \[\frac{R}{b}=\frac{AC}{AB+BC+AC}\] \[\frac{R}{a}=\frac{BC}{AB+BC+AC}\] Hence, \begin{align}\label{e1} \frac{1}{R}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \end{align}

Note that there exists a unique, non-degenerate triangle with altitudes $a, b, c$ if and only if $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are the side lengths of a non-degenerate triangle. With this in mind, it remains to find all positive integer solutions $(R, a, b, c)$ to the above such that $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ from a non-degenerate triangle, and $a\le b\le c\le 9$. We do this by doing casework on the value of $R$. Since $R$ is a positive integer $R\ge 1$. Since $a\le b\le c\le 9$, $\frac{1}{R}\ge \frac{1}{3}$, so $R\le3$. The only possible values for $R$ are 1, 2, 3.

$\textbf{Case 1: R=1}$

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