\(A=\frac{2010}{2}+\frac{2010}{6}+...+\frac{2010}{9900}\)
\(=2010.\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{9900}\right)\)
\(=2010.\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{99.100}\right)\)
\(=2010.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\right)=2010.\left(1-\frac{1}{100}\right)=2010.\frac{99}{100}\)
\(=\frac{19899}{10}\)