Difference between revisions of "2003 IMO Problems/Problem 5"
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==Solution== | ==Solution== | ||
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\begin{align*}\left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2 &=\left(2\sum_{1\le i\le j\le n}(x_j-x_i)\right)^2 \ | \begin{align*}\left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2 &=\left(2\sum_{1\le i\le j\le n}(x_j-x_i)\right)^2 \ | ||
&= \left((2n-2)x_n+(2n-6)x_{n-1}+\dots +(2-2n)x_1\right)^2 \ | &= \left((2n-2)x_n+(2n-6)x_{n-1}+\dots +(2-2n)x_1\right)^2 \ |
Latest revision as of 03:08, 26 March 2024
Problem
Let be a positive integer and let be real numbers. Prove that
with equality if and only if form an arithmetic sequence.
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
have
See Also
2003 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |