Unable to determine price ratio for a pure exchange economy
Your analysis seems to be spot on, except for some minor typos:
$$
x_A^* = 100 \frac{p_x + p_y}{p_x + p_y} = 100,
$$
because the value of A's endowment is $100 \cdot (p_x + p_y)$. And you could include the case of
$$
x_B^* \in [0,50]  \mbox{ if } p_x = p_y.
$$

As you state at the end the equilibrium price vector is not unique, there is equilibrium for all $\frac{p_x}{p_y} &lt; 1$, because with this price vector we have
$$
(x_A^*,y_A^*) = (100,100), \hskip 20pt (x_B^*,y_B^*) = (50,0) 
$$
and the market is then indeed in equilibrium as
$$
x_A^* + x_B^* = x_A + x_B, \hskip 20pt y_A^* + y_B^* = y_A + y_B.
$$
The price ratio $\frac{p_x}{p_y} = 1$ also results in equilibrium, because one of the optimal consumptions of B is still $(x_B^*,y_B^*) = (50,0)$.