Use your inequalities!

Algebra Level 3

Find the least positive integer value of nn such that for any nn positive reals a1,a2,,ana_{1},a_{2},\ldots,a_{n}, we have

a12+1a1+a22+1a2++an2+1an2016. \dfrac{a_{1}^2+1}{a_{1}}+\dfrac{a_{2}^2+1}{a_{2}}+\cdots +\dfrac{a_{n}^2+1}{a_{n}} \geq 2016.

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