Đặt \(A=\frac{\frac{2000}{11}+\frac{2000}{12}+...+\frac{2000}{100}}{\frac{1}{99}+\frac{2}{98}+\frac{3}{97}+...+\frac{98}{2}+\frac{99}{1}}\)
\(\Rightarrow A=\frac{2000.\left(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}\right)}{\left(1+\frac{1}{99}\right)+\left(1+\frac{2}{98}\right)+...+\left(1+\frac{98}{2}\right)+1}\)
\(\Rightarrow A=\frac{2000.\left(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}\right)}{\frac{100}{99}+\frac{100}{98}+...+\frac{100}{2}+\frac{100}{100}}\)
\(\Rightarrow A=\frac{2000.\left(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}\right)}{100.\left(\frac{1}{99}+\frac{1}{98}+...+\frac{1}{2}+\frac{1}{100}\right)}\)
\(\Rightarrow A=\frac{20.\left(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}\right)}{\frac{1}{99}+\frac{1}{98}+...+\frac{1}{2}+\frac{1}{100}}\)
\(\Rightarrow A=\frac{\frac{20}{11}+\frac{20}{12}+..+\frac{20}{100}}{\frac{1}{99}+\frac{1}{98}+..+\frac{1}{2}+\frac{1}{100}}\)
