19 numbers are written on a circumference of a circle, in any order.
Their sum is 203 and the biggest number is X.
Any 3 adjacent numbers sum up to 31 or more.
What is the maximal possible value of X ?
If this number is selected, what can be said about the other 18?
One can separate the 19 elements into a largest one, and then partition the rest into 6 triplets of 3 adjacent numbers.
Each triplet must sum to at least 31, and thus under any configuration, all six triplets must together sum to at least 6*31 or 186, thus the largest element can be at most 203-186 or 17.
Notice none of the other 18 can be larger than 17 by how the largest element was chosen.
Posted by Gamer
on 2010-09-24 02:47:24