51st Ukrainian National Mathematical Olympiad, 3rd Round

We use the following formulas: $$\frac{1}{n}=\frac{n-1}{n(n-1)}=\frac{1}{n-1}-\frac{1}{n(n-1)},\frac{1}{n}=\frac{n+1}{n(n+1)}=\frac{1}{n+1}+\frac{1}{n(n+1)}$$ Using them we arrive at: $$\begin{gathered} \frac{1}{2011}=\frac{1}{2010}-\frac{1}{2010\cdot 2011}=\frac{1}{2011}+\frac{1}{2011\cdot 2012}-\frac{1}{2010\cdot 2011+1} \\ -\frac{1}{2010\cdot 2011(2010\cdot 2011+1)}=\frac{1}{a+1}+\frac{1}{b+1}-\frac{1}{c+1}-\frac{1}{d+1} \end{gathered}$$ We add 1 to the first two fractions and subtract one from the last two, we get the following representation: $$\begin{gathered} \frac{1}{2011}=\frac{2010\cdot 2011\cdot 2012-1}{2011\cdot 2011\cdot 2012}+\frac{2010\cdot 2011}{2010\cdot 2011+1}+ \\ +\frac{2010\cdot 2011(2010\cdot 2011+1)-1}{2010\cdot 2011(2010\cdot 2011+1)}=\frac{a}{a+1}+\frac{b}{b+1}-\frac{c}{c+1}-\frac{d}{d+1} \end{gathered}$$