\(P=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2012}}{2011+\dfrac{2010}{2}+\dfrac{2009}{3}+...+\dfrac{1}{2011}}\\ =\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2012}}{\left(\dfrac{2010}{2}+1\right)+\left(\dfrac{2009}{3}+1\right)+...+\left(\dfrac{1}{2011}+1\right)+1}\\ =\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2012}}{\dfrac{2012}{2}+\dfrac{2012}{3}+\dfrac{2012}{4}+....+\dfrac{2012}{2011}+\dfrac{2012}{2012}}\\ =\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2012}}{2012\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2011}+\dfrac{1}{2012}\right)}\\ =\dfrac{1}{2012}\)
