Problem 5...................................(just to reach minimum character limit)

Part (a) is simple to disprove. Just arrange the numbers 64, 63,...,57 along the diagonal in descending order (i.e. 64 in a_1,1, 63 in a_2,2,..., 57 in a_8,8). That way there will be no interesting number.

Part (b) Suppose p and q are two such numbers that are interesting in row r and s resp. WLOG p>q

 

Consider p^’ in row s and in the same col as p. Then q>p’>p which is a contradiction. Hence at most 1 interesting number can be there.