Problem 5
Let n be an integer at least 2. Prove that if real numbers a_1, a_2, ..., a_n satisfy a_1^2+a_2^2+...+a_n^2=n, then the sum over 1≤i<j≤n of 1/(n-a_i a_j) is at most n/2.
Step 1 of 6: Ensure positive denominators
Detailed analysis
The square-sum condition bounds each pair sum by n. Thus every denominator is positive, so all later comparisons are legitimate.