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 203186 or 17.
Notice none of the other 18 can be larger than 17 by how the largest element was chosen.

Posted by Gamer
on 20100924 02:47:24 