## Task 6 - 180736

Determine all rational numbers $a$ with the following property:

The product of the number $a$ and its absolute value is equal to the sum of the number $a$ and its absolute value.
Assume a rational number $a$ has the mentioned property. Then it holds that

$$
a \cdot|a|=a+|a|
$$

For the number $a$, exactly one of the following two cases applies:

1. Case: $a \geq 0$.

Then $|a|=a$, so from (1) it follows (2) $a^{2}=2 a$. In the first case, the only possibilities are that either $a=0$ or, if $a \neq 0$, from (2) it follows that $a=2$.

2. Case: $a<0$.

Then $|a|=-a$, and it follows on the one hand that $a \cdot|a| \neq 0$, and on the other hand that $a+|a|=0$. The assumption that $a$ has the property (1) thus leads to a contradiction in the second case.

Therefore, only the numbers 0 and 2 can have the mentioned property. Indeed, it holds that $0 \cdot|0|=$ $0=0+|0|$ and $2 \cdot|2|=4=2+|2|$. Thus, the numbers 0 and 2 are the only solutions.

Solutions of the III. Round 1978 taken from [5]

### 4.21 XIX. Olympiad 1979

### 4.21.1 I. Round 1979, Class 7