Question 20
Given real numbers $a_i$, $b_i$ and positive reals $c_i$, $d_i$, let $e_{ij} = (a_i + b_j) / (c_i + d_j)$. Let $M_i = \max_{0 \leq j \leq n} e_{ij}$, $m_j = \min_{1 \leq i \leq n} e_{ij}$. Show that we can find an $e_{ij}$ with $1 \leq i, j \leq n$ such that $e_{ij} = M_i = m_j$.