Answer
The proof is below.
Work Step by Step
We know the following equation:
$x= x_0 + v_{x0}t$
We also know:
$ R = \frac{v_0^2sin(2\theta_0)}{g}$
We know that $\Delta x $ and R are the same value, so:
$ v_{x0}t=\frac{v_0^2sin(2\theta_0)}{g}$
$ v_{0}cos\theta t=\frac{2v_0^2sin(\theta_0)(cos\theta_0)}{g}$
$ t = 2\frac{v_0sin\theta}{g}$
Here, we see that to land at the same time, $v_0sin\theta$ would have to be the same. Above, we see that to land at the same point, $v_0^2sin(2\theta_0)$ would have to be the same. These will never be equal, so the balls will never land in the same place.
