(NIMO 2014) Looks like Cauchy-Schwarz!

Algebra Level 5

For all positive integers $$k$$, define $$f(k)=k^2+k+1$$. Compute the largest positive integer $$n$$ such that $2015f(1^2)f(2^2)\cdots f(n^2)\geq \Big(f(1)f(2)\cdots f(n)\Big)^2.$

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