Question 72
Suppose all of the $200$ integers lying in between (and including) $1$ and $200$ are written on a blackboard. Suppose we choose exactly $100$ of these numbers and circle each one of them. By the *score* of such a choice, we mean the square of the difference between the sum of the circled numbers and the sum of the non-circled numbers. What is the average of the scores over all possible choices for $100$ numbers?