a. Ta có \(\sqrt{2016}+\sqrt{2015}>\sqrt{2015}+\sqrt{2014}\to\frac{1}{\sqrt{2016}+\sqrt{2015}}

\(A-B=\sqrt{2009}-\sqrt{2007}+\sqrt{2010}-\sqrt{2008}+\sqrt{2011}-\sqrt{2015}\)

\(=\frac{2}{\sqrt{2009}+\sqrt{2007}}+\frac{2}{\sqrt{2010}+\sqrt{2008}}-\frac{4}{\sqrt{2011}+\sqrt{2015}}\)

Ta có \(\left\{{}\begin{matrix}\sqrt{2009}+\sqrt{2007}< \sqrt{2011}+\sqrt{2015}\\\sqrt{2010}+\sqrt{2008}< \sqrt{2011}+\sqrt{2015}\end{matrix}\right.\)

\(\Rightarrow\frac{2}{\sqrt{2009}+\sqrt{2007}}+\frac{2}{\sqrt{2010}+\sqrt{2008}}>\frac{2}{\sqrt{2011}+\sqrt{2015}}+\frac{2}{\sqrt{2011}+\sqrt{2015}}=\frac{4}{\sqrt{2011}+\sqrt{2015}}\)

\(\Rightarrow\frac{2}{\sqrt{2009}+\sqrt{2007}}+\frac{2}{\sqrt{2010}+\sqrt{2008}}-\frac{4}{\sqrt{2011}+\sqrt{2015}}>0\)

\(\Rightarrow A-B>0\Rightarrow A>B\)

Easy

Ta có:

\(\sqrt{2006}-\sqrt{2005}=\frac{2006-2005}{\sqrt{2006}+\sqrt{2005}}=\frac{1}{\sqrt{2006}+\sqrt{2005}}\)

Tương tự cũng có: \(\frac{1}{\sqrt{2007}+\sqrt{2008}}\)

Dễ thấy: \(\sqrt{2005}+\sqrt{2006}< \sqrt{2007}+\sqrt{2008}\)

\(\Rightarrow\frac{1}{\sqrt{2006}+\sqrt{2005}}>\frac{1}{\sqrt{2007}+\sqrt{2008}}\)

Easy

Ta có:

\(\sqrt{2006}-\sqrt{2005}=\frac{2006-2005}{\sqrt{2006}+\sqrt{2005}}=\frac{1}{\sqrt{2006}+\sqrt{2005}}\)

Tương tự cũng có: \(\frac{1}{\sqrt{2007}+\sqrt{2008}}\)

Dễ thấy: \(\sqrt{2005}+\sqrt{2006}< \sqrt{2007}+\sqrt{2008}\)

\(\Rightarrow\frac{1}{\sqrt{2006}+\sqrt{2005}}>\frac{1}{\sqrt{2007}+\sqrt{2008}}\)

\(\sqrt{2009}-\sqrt{2008}

\(\frac{1}{\sqrt{2009}-\sqrt{2008}}=\frac{\sqrt{2009}+\sqrt{2008}}{\left(\sqrt{2009}+\sqrt{2008}\right)\left(\sqrt{2009}-\sqrt{2008}\right)}=\frac{\sqrt{2009}+\sqrt{2008}}{2009-2008}=\sqrt{2009}+\sqrt{2008}\)

CMTT : \(\frac{1}{\sqrt{2008}-\sqrt{2007}}=\sqrt{2008}+\sqrt{2007}\)

Vì \(\sqrt{2009}+\sqrt{2008}>\sqrt{2008}+\sqrt{2007}\)

=> \(\frac{1}{\sqrt{2009}-\sqrt{2008}}\sqrt{2008}-\sqrt{2007}\)