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 3 of 6: Estimate one pair
Detailed analysis
For nonnegative a_i,a_j, use a_i a_j≤(a_i+a_j)^2/4 and n-a_i a_j≥n-(a_i^2+a_j^2)/2. Combining these gives the displayed estimate.