<s>[INST] Integer solutions for $n$ for $|{\sqrt{n} - \sqrt{2011}}| < 1$ $$|{\sqrt{n} - \sqrt{2011}}| < 1$$
What is the number of positive integer $n$ values, which satisfy the above inequality.
My effort:
$ 
({\sqrt{n} - \sqrt{2011}})^2 < 1 \\n + 2011 -2\sqrt{2011n} < 1\\ n+2010<2\sqrt{2011n}\\ n^2+2 \times 2010 \times n +2010^2<4 \times 2011n  \\n^2 -4024n +2010^2 < 0 $  
But it seems this won't lead me for desired answer.
 [/INST] $\sqrt{n}\in(\sqrt{2011}-1,\sqrt{2011}+1)\iff n\in ( (\sqrt{2011}-1)^2,(\sqrt{2011}+1)^2)=(2012-2\sqrt{2011},2012+2\sqrt{2011})$
There are $2\lfloor2\sqrt{2011} \rfloor + 1$ integers in this range.
We approximate $\sqrt{2011}$ as $44.8$ and we find the answer is $2\cdot 89 + 1=179$.
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