## Task 2 - 210732

Determine all pairs $(x ; y)$ of rational numbers with the property that the sum $x+y$ is the same number as the product $x \cdot y$ and also the same number as the quotient $x: y$!
If a pair $(x ; y)$ of rational numbers satisfies the conditions of the problem, then

(1) $x+y=x \cdot y$ and

(2) $x+y=x: y$.

From (1) and (2) it follows that (3) $x \cdot y=x: y$.

If $x=0$, then according to (1) $y=0$ as well, which contradicts the existence of $x: y$. Therefore, one can divide (3) by $x$ and obtain $y=\frac{1}{y}$.

The only rational numbers that are equal to their reciprocal are the numbers +1 and -1. If $y=+1$, then (1) would lead to the contradiction $x+1=x$. Therefore, $y=-1$ and thus, according to (1), $x-1=x$, which means $x=\frac{1}{2}$. Hence, only the pair $\left(\frac{1}{2} ;-1\right)$ can satisfy the conditions of the problem.