<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] Brute force says that $(\sqrt{2011}-1)^2\leq n\leq (\sqrt{2011}+1)^2$. So you can use a calculator to find the lower and upper bound integer values: $1923\leq n\leq 
2101$, do there are $2101-1923+1$ integer values $n$.
Algebraically, we see that $(\sqrt{2011}-1)^2 = 2012-2\sqrt{2011}$ and $(\sqrt{2011}+1)^2=2012+2\sqrt{2011}$. So a non-calculator value is $1+2\lfloor2\sqrt{2011}\rfloor$.
These both give the value $179$.
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