Determine all possible triplet(s) (a, b, c) of positive integers, with a ≤ b ≤ c, that satisfy this equation.

a*b+b*c+c*a - (a+b+c) = 21

1) A cannot be greater than 2.

Let f(a,b,c) = a*b+b*c+c*a - (a+b+c)

f(3,3,3) = 18, so (3,3,3) is not a solution,

and

f(3,3,4) = 23, which is already too big. Since the products increase faster than the sum, a must be 1 or 2

2) If a = 1, then solving for c in terms of b gives c = 22/b

The only solutions where a = 1 are (1,1,22) and (1,2,11)

3) If a = 2, then solving for c in terms of b

gives c = (24/(b+1)) - 1.

The only solutions where a = **2** are (2,2,7) and (2,3,5).

*(Edited to fix copy-and-paste error. Thanks Dej Mar!)*

*Edited on ***June 8, 2010, 9:36 am**