= Solution
For projectile velocity $v$ and a resting target, let $M=m_x+m_X$, $\mu=m_xm_X/M$, and $V=m_xv/M$. Subtract the <kinetic energy> of the centre of mass from the laboratory <kinetic energy>:
$$
T^*=\tfrac12m_xv^2-\tfrac12MV^2
=T_x\frac{m_X}{M}=\boxed{\frac{\mu}{m_x}T_x}.
$$
In the <centre-of-momentum frame> the two momenta are opposite with equal magnitude, so
$$
T^*=\frac{p^{*2}}{2m_x}+\frac{p^{*2}}{2m_X}=\frac{p^{*2}}{2\mu},
\qquad\boxed{p^*=\sqrt{2\mu T^*}}.
$$
For masses quoted in $\mathrm{MeV}/c^2$, use $p^*c=\sqrt{2(\mu c^2)T^*}$. These are the nonrelativistic relations stipulated for this question; no relativistic correction should be silently mixed into the numerical results.
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