Difference between revisions of "2024 AMC 12B Problems/Problem 24"
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| + | ==Problem 24== | ||
| + | What is the number of ordered triples <math>(a,b,c)</math> of positive integers, with <math>a\le b\le c\le 9</math>, such that there exists a (non-degenerate) triangle <math>\triangle ABC</math> with an integer inradius for which <math>a</math>, <math>b</math>, and <math>c</math> are the lengths of the altitudes from <math>A</math> to <math>\overline{BC}</math>, <math>B</math> to <math>\overline{AC}</math>, and <math>C</math> to <math>\overline{AB}</math>, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.) | ||
| + | <math> | ||
| + | \textbf{(A) }2\qquad | ||
| + | \textbf{(B) }3\qquad | ||
| + | \textbf{(C) }4\qquad | ||
| + | \textbf{(D) }5\qquad | ||
| + | \textbf{(E) }6\qquad | ||
| + | </math> | ||
Revision as of 01:21, 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.)
