The problem was here.
In brief
$x_i$ are $n$ real numbers such that
$$ \sum _{k=1}^{n} x_i = 0$$
$$ \sum_{k=1}^{n} x_i^2 = 1$$
Show that for some $i,j$,
$$ x_i x_j \le -\frac{1}{n}$$
Solution
The official solution is quite neat.
wlog, assume $x_1 \le x_2 \le \dots \le x_n$.
Now
$$ 0 \le \sum_{k=1}^n (x_k - x_1)(x_n - x_k) = -nx_1x_n - 1$$
(The equality is just gotten from expanding out and using the given identities).
The inequality now follows immediately.
In brief
$x_i$ are $n$ real numbers such that
$$ \sum _{k=1}^{n} x_i = 0$$
$$ \sum_{k=1}^{n} x_i^2 = 1$$
Show that for some $i,j$,
$$ x_i x_j \le -\frac{1}{n}$$
Solution
The official solution is quite neat.
wlog, assume $x_1 \le x_2 \le \dots \le x_n$.
Now
$$ 0 \le \sum_{k=1}^n (x_k - x_1)(x_n - x_k) = -nx_1x_n - 1$$
(The equality is just gotten from expanding out and using the given identities).
The inequality now follows immediately.
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