<s>[INST] Minimizing problem. The real numbers $x_1,$ $x_2,$ $\dots,$ $x_{10}$ satisfy
$$\sqrt{x_1 - 1^2} + 2 \sqrt{x_2 - 2^2} + 3 \sqrt{x_3 - 3^2} + \dots + 10 \sqrt{x_{10} - 10^2} = \frac{x_1 + x_2 + \dots + x_{10}}{2}.$$Find the minimum value of $x_1 + x_2 + x_3 + \dots + x_{10}.$

I tried using Cauchy to get $$\left(1^2(x_1-1^2)+2^2(x_2-2^2)+\cdots+10^2(x_{10}-10^2)\right)(1^2+\cdots+1^2)\geq \left(\sqrt{x_1 - 1^2} + 2 \sqrt{x_2 - 2^2} + 3 \sqrt{x_3 - 3^2} + \dots + 10 \sqrt{x_{10} - 10^2}\right)^2,$$ but this doesn't really do much. I'm not really sure what else to do... It would be great if anyone could help!
Thanks in advance!!
 [/INST] \begin{align}
& \text{Since $\sqrt{x_i-i^2}$ is real number, $x_i \geq i^2. $} \\
\text{let } & \sum_{x=1}^{10} x_i = S. \\
\therefore \; & S \geq \sum_{i=1}^{10} i^2 = 385. \\
& \Bigg( \sum_{i=1}^{10} x_i-i^2\Bigg)\Bigg(\sum_{i=1}^{10} i^2 \Bigg) \geq \Bigg( \sum_{i=1}^{10} i\sqrt{x_i-i^2} \Bigg). & \text{(C.S. Inequality.)} \\
\therefore \; & (S-385)(385) \geq \Bigg( \dfrac {S}{2} \Bigg)^2 \\
\Rightarrow \; & 385S-385^2 \geq \dfrac {S^2}{4}. \\
\therefore \; & S^2-4\cdot385\cdot S + 4 \cdot385^2 \leq 0. \\
\ \\
\therefore \; & (S-770)^2 \leq 0, S = 770.
\end{align}
Is it just me or the answer is this...? I thought I could answer this...
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