\[\boxed{\text{512.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\]
\[Пусть\ \text{x\ }\frac{км}{ч} - скорость\ \]
\[первого\ поезда,\ y - второго.\]
\[Составим\ систему\ уравнений:\]
\[\left\{ \begin{matrix} x + y = \frac{270}{3}\text{\ \ \ \ \ \ \ \ \ \ } \\ \frac{270}{x} = \frac{270}{y} + 1\frac{21}{60} \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x + y = 90\ \ \ \ \ \ \ \ \\ \frac{270}{x} - \frac{270}{y} = \frac{81}{60} \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 90 - x\ \ \ \ \ \ \ \ \ \ \ \ \\ \frac{10 \cdot (y - x)}{\text{xy}} = \frac{1}{20} \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 90 - x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 200 \cdot (90 - 2x) = 90x - x^{2} \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 90 - x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 490x + 18000 = 0 \\ \end{matrix} \right.\ \]
\[D = 245^{2} - 18000 = 42025 =\]
\[= 205^{2}\]
\[x_{1,2} = 245 \pm 205;\]
\[\left\{ \begin{matrix} x_{1} = 40 \\ y_{1} = 50 \\ \end{matrix} \right.\ \ \ \ \ \ или\ \]
\[\ \left\{ \begin{matrix} x_{2} = 450\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y_{2} = 90 - 450 < 0 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow не\ подходят.\]
\[Ответ:40\ \frac{км}{ч}\ и\ 50\ \frac{км}{ч}.\]