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?

The problem did not specify whole numbers which would seem to be more complicated.  Here is a simple solution using rational numbers.

203/19 is the average number.  3 times this is 609/19 = 32+1/19.  This is more than 31 so they can be big enough.

For 3 numbers to sum to 31 their average must be 31/3 = 10+1/3.

I propose starting with 19 equal numbers, removing a little bit from 18 of the numbers to reduce them to 31/3, then adding this to the last.  203/19 - 31/3 = 20/57 is removed from each.
18*20/57 = 120/19 is added to 203/19 = 17.

So if the largest number is 17 all the others could be 10+1/3 .

Posted by Jer on 2010-09-23 17:16:04