Each of A, B, C and D is a positive integer, with A > B, such that: A+B = C - 11*D.

Determine (with proof) the quadruplet (A, B, C, D) that generates the minimum value of A+B+C+D. What quadruplet generates the next smallest value of A+B+C+D?

Clearly we want each to be as small as possible, especially D since increasing D by 1 will increase C by 11.

So let D=1.  The smallest A and B can be are 2 and 1.
This gives 2+1=14-11*1 and the minimum A+B+C+D=2+1+14+1=18

Clearly increasing A and/or B will cause C to increase, which will increase the total.  C can be increased by multiples of 11 which can leave A+B unchanged by increasing D, but that again increases the total.  That should suffice as a proof.

The next smallest is 3+2=16-11*1 with a total of 22.

 

Posted by Jer on 2011-04-06 15:09:44