2014/057) given p+q+r+s = 63 find the maximum of pq+qr+rs

pq + qr + rs = pq + qr + rs + sp - sp = (p+r)(q+s) - sp

now we need to maximize (p+r)(q+s) and minimise sp as s and p are in different expressions

 p+r and q + s should be as close as possible as p+r + q + s = 63

so p+ q = 32 and r+ s = 31 or vice versa and p = s = 1

so p = 1 , q = 31, r = 30, s = 1 or p =1, q = 30, r = 31, s = 1
 
you can find some discussion at http://mathhelpboards.com/challenge-questions-puzzles-28/find-maximum-sum-11061.html#post51454