Difference between revisions of "2024 AMC 12B Problems/Problem 24"
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<cmath>\frac{R}{a}=\frac{BC}{AB+BC+AC}</cmath> | <cmath>\frac{R}{a}=\frac{BC}{AB+BC+AC}</cmath> | ||
Hence, | Hence, | ||
− | \begin{align} | + | \begin{align}\label{e1} |
\frac{1}{R}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} | \frac{1}{R}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} | ||
\end{align} | \end{align} | ||
− | 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 | + | 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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+ | <math>\textbf{Case 1: R=1}</math> |
Revision as of 01:47, 14 November 2024
Problem 24
What is the number of ordered triples of positive integers, with , such that there exists a (non-degenerate) triangle with an integer inradius for which , , and are the lengths of the altitudes from to , to , and to , respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)
Solution
First we derive the relationship between the inradius of a triangle , and its three altitudes .
Using an area argument, we can get the following well known result
where are the side lengths of , and is the triangle's area. Substituting into the above we get
Similarly, we can get
Hence,
Note that there exists a unique, non-degenerate triangle with altitudes if and only if are the side lengths of a non-degenerate triangle. With this in mind, it remains to find all positive integer solutions to the above such that from a non-degenerate triangle, and . We do this by doing casework on the value of .