3. The attempt at a solution
I got parts (a) and (b) easily enough by evaluating 4-vector products in the observer's rest frame,
(a) [tex]\gamma = -U^{\mu} V_{\mu} \gg 1[/tex]
(b) [tex]E = U^{\mu}P_{\mu}[/tex]
The problem is with part (c). We're supposed to express P' in terms of U, V, and P, but it doesn't seem to be possible without knowing the angle at which the scattered photon exits (or the angle at which the charged particle exits). Am I missing something, or is this actually impossible?

Momentum, I know... P + mV = P' + mV', where V' is the final four-velocity of the charged particle. But I still don't see how that helps... I don't know V'.

I thought about that, but it'd be just expressing one unknown in terms of another :/ Anyway, it doesn't matter now, the assignment was due earlier today. In the end even our professor did admit that it was impossible.