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Algebra Level 5

\[\large \begin{cases}{a_{1} + a_{2} + a_{3} + a_{4} + \ldots + a_{10} = 4} \\ {b_{1} + b_{2} + b_{3} + b_{4} + \ldots + b_{10} = -1}\end{cases} \]

Let \(a_{1}, a_{2}, a_{3} , a_{4}, \ldots ,a_{10}\) and \(b_{1}, b_{2}, b_{3} , b_{4}, \ldots ,b_{10}\) be real numbers such that they satisfy the system of equations above.

What is the minimum positive value of the expression below? \[\left(\sqrt{a_{1}^{2} + b_{1}^{2}} + \sqrt{a_{2}^{2} + b_{2}^{2}} + \ldots + \sqrt{a_{10}^{2} + b_{10}^{2}}\right)^{2}\]

Also try this.
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