Problem 4. Determine the non-zero numbers $a$ and $b$ for which

$$
a+b=a b=\frac{a}{b} .
$$
Solution. From $a b=\frac{a}{b}$, it follows that $b^{2}=1$, hence $b=1$ or $b=-1$. If $b=1$, then from the equation $a+b=a b$ it follows that $a=a+1$, which is not possible. If $b=-1$, then $a-1=-a$ from which it follows $a=\frac{1}{2}$.