## Task 4 - 030924

Give all pairs of real numbers whose sum, product, and quotient are equal to each other!
If $(a, b)$ is a pair of real numbers for which

$$
a+b=a \cdot b \quad(1) \quad a \cdot b=\frac{a}{b}
$$

holds, then it follows from (2) that $b \neq 0$, and subsequently from (1) that $a \neq 0$, and from (2) that $b^{2}=1$. Therefore, $b_{1}=+1$ or $b_{2}=-1$.

The first case leads to $a+1=a$ and thus to a contradiction.

The second case yields $a-1=-a$ and thus $a=\frac{1}{2}$ as the only possible solution.

By substituting into (1) and (2), one can show that $\left(\frac{1}{2},-1\right)$ is a solution and thus the only pair of real numbers that satisfies all conditions of the problem.