Max pain as a median

(a) Prove the corollary of Proposition 9A.3: if $c_K=p_K=m_K$ for all $K$, every max pain strike is a median of the distribution $\{m_K\}$ over strikes. (b) Suppose instead $p_K=\alpha c_K$ for all $K$ with $\alpha>0$. Show that max pain is an $\frac{\alpha}{1+\alpha}$-quantile of the call OI distribution over strikes: $\sum_{K<x^\ast}c_K\le\frac{\alpha}{1+\alpha}C\le\sum_{K\le x^\ast}c_K$ with $C=\sum_Kc_K$. Which way does max pain move as the PCR $\alpha$ rises? (c) Use (a)–(b) to explain why max pain tends to sit near the recent price of the underlying.