Quick question on Fermi Golden Rule

Main Question or Discussion Point

Adopted from my lecture notes, found it a little fishy:

Shouldn't $\frac{dp}{dE} = \frac{E}{p}$ given that $p = \sqrt{E^2 - m^2}$. Then the relation should be instead:

$$\frac{dp}{dE} = \frac{E}{p} = \frac{E}{\sqrt{E^2 - m^2}}$$

Yes, you are right but the problem tells you that B and C are massless. This means that the magnitude of their momenta is equal to their energies. In particular, conservation of energy tells you that $E_B=E_C=E_A/2$ and so $p_B=p_C=E_A/2$, giving $dp_B/dE=1/2$.
Yes, you are right but the problem tells you that B and C are massless. This means that the magnitude of their momenta is equal to their energies. In particular, conservation of energy tells you that $E_B=E_C=E_A/2$ and so $p_B=p_C=E_A/2$, giving $dp_B/dE=1/2$.
And I suppose $c=1$?