Question 37
Prove that three distinct non-zero integers $a$, $b$, $c$ satisfy the equation
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = 3
$$
if and only if $a$, $b$, $c$ are given by
$$
a = kuv^2, \quad b = -ku^2(u+v), \quad c = kv(u+v)^2,
$$
(up to cyclic permutations) for some integers $u$, $v$, $k$.
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = 3
$$
if and only if $a$, $b$, $c$ are given by
$$
a = kuv^2, \quad b = -ku^2(u+v), \quad c = kv(u+v)^2,
$$
(up to cyclic permutations) for some integers $u$, $v$, $k$.