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 4 of 6: Apply Cauchy-Schwarz

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\frac{(a_i+a_j)^2}{n-(a_i^2+a_j^2)/2}\le\frac{a_i^2}{n-a_i^2}+\frac{a_j^2}{n-a_j^2}.

Detailed analysis

Cauchy-Schwarz applied to the two positive weights n-a_i^2 and n-a_j^2 gives the second displayed inequality. If some a_i^2=n, the square-sum condition makes every other variable zero and the required inequality is immediate; otherwise all these weights are positive.