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Algebra Level 5

{a1+a2+a3+a4++a10=4b1+b2+b3+b4++b10=1\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 a1,a2,a3,a4,,a10a_{1}, a_{2}, a_{3} , a_{4}, \ldots ,a_{10} and b1,b2,b3,b4,,b10b_{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? (a12+b12+a22+b22++a102+b102)2\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}

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