More fun in 2016, Part 20

Algebra Level 4

$S=\sum_{1\leq i<j\leq{2016}}x_ix_j$

Find the maximum of $S$ if $\displaystyle \sum_{i=1}^{2016}x_i^2=2$ , where $x_1,\ldots ,x_{2016}$ are real numbers.

The answer is 2015.

1 solution

Alan Yan
Jan 7, 2016

$\begin{aligned} \sum_{1 \leq i < j \leq 2016}(x_i - x_j)^2 & \geq 0 \\ 2015\sum_{k = 1}^{2016}{x_k^2} & \geq 2\sum_{1 \leq i < j \leq 2016}x_ix_j\\ 2015 & \geq\sum_{1 \leq i < j \leq 2016}x_ix_j \end{aligned}$

Equality when $x_i = \frac{1}{\sqrt{1008}}$

Yes, exactly! (+1)

Typo: Equality when $x_i=\frac{1}{\sqrt{1008}}$

Try my next one ... it's a bit more interesting!

Otto Bretscher - 5 years, 5 months ago

I am not able to understand how that 2015 came in the second step.

RoYal Abhik - 5 years, 5 months ago

Every $x_i^2$ occurs in exactly 2015 of $\sum{(x_j - x_k)^2}$

Alan Yan - 5 years, 5 months ago

More fun in 2016, Part 20

The answer is 2015.

1 solution

0 pending reports