1. Given various real numbers $x$ and $y$ such that $x / y + x = y / x + y$. What can $1 / x + 1 / y$ be equal to? Find all possible values and prove that there are no others.
1. Solution. Transform the equality from the condition: $y-x=\frac{x}{y}-\frac{y}{x}=\frac{x^{2}-y^{2}}{x y}$. Using $x \neq y$, we get $-1=\frac{x+y}{x y}=\frac{1}{x}+\frac{1}{y}$. Answer: -1.