I did the calculations, and actually both current answers seem wrong... can someone confirm or point towards any mistake in the following reasoning?
What I claim to show is that: you can use a standard stock price tree (i.e. the terminal values of the stock price do not involve the dividend yield and write $S_0 u^2$, $S_0 ul$ and $S_0 l^2$), but your risk-neutral probabilities will depend on the dividend yield $Y$.
In Exercise A, this can be seen by writing the replicating portfolio: $\Pi = \Delta S + B$, which by the end of a period should satisfy
$$\Pi_u = \Delta S_u + \beta (1 + R) + {\color{red}{\Delta S_u Y}} = V_u$$
$$\Pi_d = \Delta S_d + \beta (1 + R) + {\color{red}{\Delta S_d Y}} = V_d$$
Where the additional terms come from the proportional dividend payments that are reinvested in the cash account at the end of each period. These equations are equivalent to
$$\Delta S_u (1 + Y) + \beta (1 + R) = V_u$$
$$\Delta S_d (1 + Y)+ \beta (1 + R) = V_d$$
Solving them yields to (it is as if $u$ and $d$ in the standard formulae were replaced by $u(1+Y)$ and $d(1+Y)$
$$ q_u = \frac{ \frac{1+R}{1+Y} - d}{u - d} $$
$$ q_l = 1 - q_u $$
This result seems consistent with many papers. Even Wikipedia exhibits something similar with
$$ q_u = \frac{e^{(r-q)\Delta t} - d}{u -d} $$
Plus, it is straightforward to show that these probabilities provide the right forward price if you evaluate $F(0,t)=E[S(t)]=qS_u+(1-q)S_d=S_0(1+R)/(1+Y)$
Yet, because $(1+R)/(1+Y)$ here is smaller than $d$ on the second period, I doubt that this problem is feasible at all.
In Exercise B, using the same rationale I end up with
$$\Delta S_u + \beta (1 + R) + {\color{red}{\Delta S_u Y}} = V_u$$
$$\Delta S_d + \beta (1 + R) = V_d$$
and
$$ q_u = \frac{ 1+R - d}{u(1+Y) - d} $$
$$ q_l = 1 - q_u $$
and now the problem indeed seems feasible (no negative probabilities). But this is never a question of the strike contrary to what the OP's professor claims.