a/

\(\dfrac{x+3}{2011}+\dfrac{x+1}{2013}\ge\dfrac{x+10}{2004}+\dfrac{x+13}{2001}\)

\(\Leftrightarrow\dfrac{x+2014-2011}{2011}+\dfrac{x+2014-2013}{2013}\ge\dfrac{x+2014-2004}{2004}+\dfrac{x+2014-2001}{2001}\)

\(\Leftrightarrow-1+\dfrac{x+2014}{2011}-1+\dfrac{x+2014}{2013}\ge-1+\dfrac{x+2014}{2004}-1+\dfrac{x+2014}{2001}\)

\(\Leftrightarrow\dfrac{x+2014}{2011}+\dfrac{x+2014}{2013}-2\ge\dfrac{x+2014}{2004}+\dfrac{x+2014}{2001}-2\)

\(\Leftrightarrow\left(x+2014\right)\left(\dfrac{1}{2011}+\dfrac{1}{2013}\right)\ge\left(x+2014\right)\left(\dfrac{1}{2004}+\dfrac{1}{2001}\right)\)

\(\Leftrightarrow\dfrac{1}{2011}+\dfrac{1}{2013}>\dfrac{1}{2004}+\dfrac{1}{2001}\) hoặc \(\left(x+2014\right)\left(\dfrac{1}{2011}+\dfrac{1}{2013}\right)\ge\left(x+2014\right)\left(\dfrac{1}{2004}+\dfrac{1}{2001}\right)\)

(với mọi x>0) \(\Leftrightarrow x=2014\)