$1$ Question: Switching Costs

Consider two airlines, A and B$,$ flying the same route and engaging in a single$-$period Cournot competition. Industry demand $($the demand from the entire population of customers$)$ is $q=100-p,$ and each firm has a constant unit cost of $10($total cost is $10$ times the quantity produced$).$ Suppose in the past $50\%($randomly determined$)$ of the consumers had flown A and the other $50\%$ flown B$.$ Now each firm announces that it will offer a "frequent$-$flyer" discount of $10$ to anyone who flew on its flight before.

Furthermore, suppose that firm B suffers an unexpected capacity constraint so that its quantity is fixed at a low level of $q_B=15.$ Firm A does not have such a constraint and is free to choose any $q_A.$

For all $q_{A}15,$ find the market prices $($before discounts$)p_A$ and $p_B$ for the two firms' products as a function of $q_A.$