Let A be a $\displaystyle k$-element subset of {$\displaystyle 1,2,3,......,16$}. It is known that every two subsets of A have distinct sum of their elements and for any $\displaystyle (k + 1)$ element subset B of {$\displaystyle 1,2,3,......16$}, containing A, there exist subsets of B with equal sums.
a) Prove that $\displaystyle k \le 5$
b) For different subsets A with the given property find the maximum and minimum possible values of the sum of the elements of A.