Problem:
What is the number of ordered triples (a,b,c) of positive integers, with a≤b≤c≤9, such that there exists a (non-degenerate) triangle △ABC with an integer inradius for which a,b, and c are the lengths of the altitudes from A to BC,B to AC, and C to AB, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)
Answer Choices:
A. 2
B. 3
C. 4
D. 5
E. 6
Solution:
Let x,y, and z be the lengths of BC,AC, and AB, respectively. Let r be the inradius of △ABC. Then
Area(△ABC)=21xa=21yb=21zc=21(x+y+z)r
Therefore x=a2Area(△ABC),y=b2Area(△ABC), and z=c2Area(△ABC), so
implying r≤3. It is then possible to find the solutions (a,b,c) by examining cases based on the value of r.
item If r=1, then, because a≤b≤c, either a=2 or a=3. If a=2, then b1+c1=21. Then, because b≤c, either b=3 or b=4. If b=3, then c=6; and if b=4, then c=4. If a=3, then it must be that b=c=3. So the solutions in this case are (2,3,6),(2,4,4), and (3,3,3).
If r=2, then, because a≤b≤c, it follows that a=3,a=4,a=5, or a=6. If a=3, then b1+c1=61, which has no solutions because 91+91>61. If a=4, then b1+c1=41. In this case, b=c=8 is the only solution. If a=5, then b1+c1=103, which gives no solutions. If a=6, then it follows that b=c=6. So the solutions in this case are (4,8,8) and (6,6,6).
If r=3, then (a,b,c)=(9,9,9) is the only solution because if a<9, then c>9.
Finally, the altitude lengths must be checked to ensure that these lengths give dimensions for a valid triangle. In the (2,3,6) case, the side lengths of the triangles become (3t,2t,t) for some t, which does not form a triangle. In the (2,4,4) and (4,8,8) cases, the side lengths of the triangles become (2t,t,t) for some t, which again does not form a triangle. The rest of the cases, namely (3,3,3), (6,6,6), and (9,9,9), do produce valid triangles because if all of the altitudes have length h, then it is possible to form an equilateral triangle with side length 323h. Thus there are (B)3 ordered triples satisfying the given conditions.