Draft: CaptainBlack's Problem of the Week #1 for MHB

This is the draft for the problem posted on MathHelpBoards

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Let \(x_1, ..., x_n\) be real numbers such that:

\(\sum_{i=1}^n x_i=0\) and \(\sum_{i=1}^n x_i^2=1\)

Prove that for some \( k,l \) both in \( \{1, .. , n\} \) that \(x_k x_l\le -1/n\)

CB

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Solution

Let \(a=\max(x_1, ... , x_n) \) and \( b=\min(x_1, ... , x_n) \), and put:

\[ f(x)=(x-a)(x-b)=x^2-(a+b)\; x+ab \]

Then by construction \( f(x_i)\le 0,\;i=1, ... , n \) . Now consider:

\[ \sum_{i=1}^n f(x_i)=\sum_{i=1}^n x^2 -(a+b) \sum_{i=1}^n x_i +n\;ab = 1+n\;ab \le 0\]

hence:

\[ab \le -\frac{1}{n} \]

QED