Open Contests

Multiplying the given equation by $abc$ we get $bc+ca+ab=0$.

If $a$, $b$, $c$ were all odd, then $bc$, $ca$ and $ab$ were also odd and their sum could not be $0$.

If one of the numbers $a$, $b$, $c$ was even and the others were odd, then two of the numbers $bc$, $ca$ and $ab$ were even and one odd, which also would not add up to $0$.

Hence at least two of the numbers $a$, $b$, $c$ are even, which satisfy the conditions.