8,9  Solve the equation: $\frac{x-49}{50}+\frac{x-50}{49}=\frac{49}{x-50}+\frac{50}{x-49} .$
Let's bring the expressions on the left and right sides of the equation to a common denominator. We get:

\[
\frac{99 x^{2}-49^{2}-50^{2}}{50 \cdot 49}=\frac{99 x^{2}-49^{2}-50^{2}}{(x-50)(x-49)}
\]

This equality will be true in two cases.

1) The numerators of the obtained fractions are equal to zero, that is, \(99 x=49^{2}+50^{2}\). Then \(x=\frac{49^{2}+50^{2}}{99}\).
2) The denominators of the obtained fractions are equal, that is, \(50 \cdot 49=(x-50)(x-49)\). Two solutions - \(x=0\) and \(x=99\) are obvious. Since this is a quadratic equation, there are no other solutions.

## Answer

\(0, 99, 4901 / 99\).