Question 102
Equation $\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_m} - \frac{1}{y_1} - \frac{1}{y_2} - \dots - \frac{1}{y_n} = \frac{577}{408}$ has an infinite number of natural solutions $x_1, x_2, \dots, x_m, y_1, y_2, \dots, y_n$ for some non-negative integers $m, n$. Prove that an equation $\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_k} - \frac{1}{y_1} - \frac{1}{y_2} - \dots - \frac{1}{y_l} = \frac{577}{408}$ has a solution in natural numbers $x_1, x_2, \dots, x_k, y_1, y_2, \dots, y_l$ for some non-negative integers $k \lt m$ and $l \lt n$.
(Olexander Golovanov)
(Olexander Golovanov)