How many distinct, nonequilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ such that $a$, $b$, $c$ is an arithmetic sequence?
How many distinct, nonequilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ such that $a$, $b$, $c$ is an arithmetic sequence?
let the sides by a, a+d, a+2d where a and d are both positive integers and d>0
\(a+a+d+a+2d=60\\ 3a+3d=60\\ a+d=20\\ \)
d  a  a+d  a+2d  Is the sum of the two little numbers bigger than the 3rd one. If not then it cannot be a triangle

19  1  20  39  no 
18  2  20  38  no 
17  3  
16  4  
15  5  n  
14  6  20  34  no 
13  7  n  
12  8  n  
11  9  20  31  no 
10  10  20  30  no 
9  11  20  yes 1  
8  12  20  28  yes 2 
7  13  20  27  y 3 
6  14  20  26  4 
5  15  20  25  5 
4  16  20  24  6 
3  17  20  23  7 
2  18  20  22  8 
1  19  20  21  9 
9 of the combinations work :)