State prices on a two-step tree

A stock at 50 moves up by a factor $1.2$ or down by a factor $0.8$ each period, and the interest rate is $r = 10\%$ per period. The real probability of an up move is $0.9$. Over two periods, writing $H$ for up and $T$ for down:

(a) Find the risk-neutral up probability $\tilde p$ and the Radon-Nikodym derivative $Z(\omega)$ on each of the four paths. Check that $E[Z] = 1$.

(b) Find the state price of each path and check that they sum to the price of a two-period zero-coupon bond.

(c) Price a put struck at 50 expiring after two periods. Compare it with the put's real expected payoff, discounted at $r$.