Given a 12-member set of consecutive integers (1,2,3…11,12).
Partition it into 3 subsets, each with a distinct number of members ,
e.g . A=(1,2); B=(3,5,11,12) & C=(4,6,7,8,9,10).
Evaluate the sums of the members for each of the subsets, in our example:
S_a=3 ; S_b=31 & S_c=44;

Now evaluate:
F1 (product of the three sums) = S_a * S_b * S_c, i.e. 3*31*44 = 4092 in our example.

F2= S_a * S_b + S_c; equals to 93+44=137 in our case.

F3= d(S_a) +d(S_b) + d(S_c); where d(N) is the number of N's divisors.
So in our case F3=2+2+6 = 10.

The 3 independent tasks :

Executing the above procedures find the partitions of the main set, that provide maximum values for F1, F2 & F3.

(In reply to Are null sets allowed? by Steve Herman)

The text does not preclude. empty sets.

Posted by Ady TZIDON on 2014-08-21 15:54:12