<s>[INST] Prove that $\sqrt{2011} + \sqrt{2013} + \sqrt{2015} + \sqrt{2017} < 4 \sqrt{2014}$ How to prove that $\sqrt{2011} + \sqrt{2013} + \sqrt{2015} + \sqrt{2017} < 4\sqrt{2014}$ without using calculator?
 [/INST] Hint:
$$\sqrt{k-r}+\sqrt{k+r}<2\sqrt{k},\ \ \ \forall k\ge 0, k\ge |r|, r\neq 0$$
Since:
$$\iff 2k+2\sqrt{k^2-r^2}<4k\iff 2\sqrt{k^2-r^2}<2k\iff k^2-r^2<k^2\iff -r^2<0$$
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