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How many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths \$a\$, \$b\$, and \$c\$ such that \$a\$, \$b\$, \$c\$ is an arithmetic sequence?

Guest Sep 10, 2017
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How many distinct, non-equilateral 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 :)

Melody  Sep 10, 2017

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