Solve the equation $\frac{x-49}{50}+\frac{x-50}{49}=\frac{49}{x-50}+\frac{50}{x-49}$.
Let $u=\frac{x-49}{50}, v=\frac{x-50}{49}$. Then the equation becomes $u+v=1 / u+1 / v$. Multiplying both sides by $uv$, moving all terms to one side, and factoring, we get $(u+v)(uv-1)=0$, from which it follows that either $u+v=0$ or $uv=1$. Substituting the expressions for $u$ and $v$, in the first case we get $x=4901 / 99$, and in the second case $x=99$ or $x=0$.

## Answer

$x=0, 4901 / 99$ or 99.