Question 62
Nonnegative real numbers $p_1, p_2, \dots, p_n$ and $q_1, q_2, \dots, q_n$ are given such that
$$
p_1 + p_2 + \dots + p_n = q_1 + q_2 + \dots + q_n.
$$
Among all matrices with nonnegative real entries for which sum of entries of the $i$th row is $p_i$ and sum of entries of the $j$th column is $q_j$, find the maximum value that the trace of the matrix can have.
$$
p_1 + p_2 + \dots + p_n = q_1 + q_2 + \dots + q_n.
$$
Among all matrices with nonnegative real entries for which sum of entries of the $i$th row is $p_i$ and sum of entries of the $j$th column is $q_j$, find the maximum value that the trace of the matrix can have.