MathLabs

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 2 of 6: Reduce to nonnegative variables
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bi=∣ai∣:∑bi2=n,∑i<j1n−aiaj≤∑i<j1n−bibj.b_i=|a_i|:\quad\sum b_i^2=n,\quad\sum_{i<j}\frac1{n-a_i a_j}\le\sum_{i<j}\frac1{n-b_i b_j}.
Detailed analysis

Since a_i a_j≤b_i b_j and x↦1/(n-x) is increasing on the relevant range, replacing every variable by its absolute value can only increase the left side. It is therefore enough to assume all variables are nonnegative.