BALTIC-WAY 2015 Problem 19

Three pairwise distinct positive integers $a, b, c$, with $\operatorname{gcd}(a, b, c)=1$, satisfy

$$a\left|(b-c)^{2}, b\right|(c-a)^{2} \text { and } c \mid(a-b)^{2} \text {. }$$

Prove that there does not exist a non-degenerate triangle with side lengths $a, b, c$.