Proving there exists no arbitrage opportunities given 3 states and 2 assets
First, you are in fact given $p$. You can think of return as a security that costs 1 in current period and pays off $R$ in the next period. The price vector and the payoff matrix are thus
$$
p = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \  X = \begin{bmatrix} 1.1 &amp; 1.0 &amp; 0.9 \\ 1.0 &amp; 1.0 &amp; 1.0 \end{bmatrix}
$$ 
and we need to find positive vector $q$ of state prices such that $p = Xq$. Because the system is undetermined, there can be many such vectors, but as long as some of them are positive, we can be sure there's no arbitrage. To actually do the computation, one way would be to treat $q_3$ as a parameter, solve for $q_1, q_2$ as functions of $q_3$, then try to find $q_3 &gt; 0$ such that implied values of $q_1,q_2$ are positive. But in this case it's easy to see that $q = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ does the job.