Question 105
Find all polynomials $P(x)$ with real coefficients such that for all nonzero real numbers $x, y, z$ satisfying $2xyz = x + y + z$,
$$
\frac{P(x)}{yz} + \frac{P(y)}{zx} + \frac{P(z)}{xy} = P(x - y) + P(y - z) + P(z - x)
$$
$$
\frac{P(x)}{yz} + \frac{P(y)}{zx} + \frac{P(z)}{xy} = P(x - y) + P(y - z) + P(z - x)
$$