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?

Do you think it was the intention (though not stated) that these be 19 DISTINCT integers, i.e. disallowing any duplication ?

My first thought here was that the current scenario resembled a dart board and its standard and nonduplicated perimeter numbering (though with 20 integers). Several years ago, in another venue, we debated optimal arrangements of the values 1..20 for various conditions.