RU-ALLRUSSIAN 1975 Problem 15

Given real numbers $\mathbf{a}_{\mathrm{i}}$ , $\mathbf{b}_{\mathrm{i}}$ and positive reals $\mathbf{c}_{\mathrm{i}}$ , $\mathbf{d}_{\mathrm{i}}$ , let $\mathbf{e}_{\mathrm{ij}} = (\mathbf{a}_{\mathrm{i}} + \mathbf{b}_{\mathrm{j}}) / (\mathbf{c}_{\mathrm{i}} + \mathbf{d}_{\mathrm{j}})$ . Let $\mathbf{M}_{\mathrm{i}} = \max_{0\leq \mathrm{j}\leq \mathrm{n}}\mathbf{e}_{\mathrm{ij}}$ , $\mathbf{m}_{\mathrm{j}} = \min_{1\leq \mathrm{i}\leq \mathrm{n}}\mathbf{e}_{\mathrm{ij}}$ . Show that we can find an $\mathbf{e}_{\mathrm{ij}}$ with $1\leq \mathrm{i},\mathrm{j}\leq \mathrm{n}$ such that $\mathbf{e}_{\mathrm{ij}} = \mathbf{M}_{\mathrm{i}} = \mathbf{m}_{\mathrm{j}}$ .