\(\frac{20102010}{20112011}=\frac{2010}{2011}\)
\(\frac{20122012}{20112011}=\frac{2012}{2011}\)
\(\left(\frac{2010}{2011}+\frac{2012}{2011}\right).x-1=2011\)
\(\frac{4022}{2011}.x-1=2011\)
\(2.x-1=2011\)
\(2.x=2011+1\)
\(2.x=2012\)
\(x=2012:2\)
\(x=1006\)