Given \(n \gt 2\) and reals \(x_1 \le x2 \le \ldots \le x_n\), show that $$ \left({\textstyle\sum_{\displaystyle i,\;j}}\left|x_i-x_j\right|\right)^2\;\leq\;\frac23\left(n^2\;-\;1\right){\textstyle\sum_{i,\;j}}\;\left(x_i\;-\;x_j\right)^2. $$ Show that we have equality iff the sequence is an arithmetic progression.
| Attributes | Олімпіадна |
|---|---|
| Source | International Mathematical Olympiad |
| Year | 2003 |
| Number | 5 |
| Difficulty | 10.0 |
| Themes |