= Solution
Data processing states $D(\Phi(\rho)\|\Phi(\sigma))\leq D(\rho\|\sigma)$ for every quantum channel $\Phi$. Joint convexity states
$$
D\left(\sum_ip_i\rho_i\middle\|\sum_ip_i\sigma_i\right)
\leq\sum_ip_iD(\rho_i\|\sigma_i).
$$
Strong subadditivity states $S(ABC)+S(B)\leq S(AB)+S(BC)$.
For strong subadditivity, apply data processing under $\operatorname{Tr}_C$ to $\rho_{ABC}$ and $I_A/d_A\otimes\rho_{BC}$. Expanding both relative entropies cancels $\log d_A$ and gives precisely the stated inequality. For joint convexity, use flagged states $\widehat\rho=\sum_ip_i|i\rangle\langle i|\otimes\rho_i$ and $\widehat\sigma=\sum_ip_i|i\rangle\langle i|\otimes\sigma_i$. Part a gives $D(\widehat\rho\|\widehat\sigma)=\sum_ip_iD(\rho_i\|\sigma_i)$; discarding the flag and applying data processing gives <joint convexity of quantum relative entropy>.
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