Ukrainian National Mathematical Olympiad, 3rd Round

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