the maximum value of $\displaystyle \sum_{n=2}^{101} 1729x_1x_n$

Question

Let $x_1,x_2,...,x_{101}$ are positive real number satisfying the following equation

$$x_1^2 +x^2_2+...+x_{101}^2=1$$

Find the maximum value of $\displaystyle \sum_{n=2}^{101} 1729x_1x_n$

All help would be appreciated.

Answer 1

Consider two vectors in $\Bbb R^{100}$: $x=(x_1,\dots,x_1)$ with $\|x\|=|x_1|\sqrt{100}$ and $y=(x_2,\dots,x_{101})$ with $\|y\|=\sqrt{x_2^2+\dots+x_{101}^2}=\sqrt{1-x_1^2}$. By Schwarz Inequality we have $$\left|\sum_{i=2}^{102}1729x_1x_i\right|=1729|\langle x,y\rangle|\le 1729\|x\|\cdot\|y\|=1729\sqrt{100}|x_1|\cdot\sqrt{1-x_1^2}.$$ Now find the maximum to this function of one variable. This task is rather standard, so I finish my solution here. In the final step take into account that the equality in Schwarz inequality is attained for linearly dependent vectors. Hence it is really enough to maximize this function on the right.