Consider the first equality,

A + BC = B+CA

Solving for C, we find C = (A-B)/(A-B).

Since (1,1,1) is not a solution, it must be the case that A=B=C

Then A^2 +A - 1980 = 0

The square root of 1980 is about 44.5,

so (A+45)(A-44) = 0

Solutions are (-45, -45, -45) and (44,44,44)