## Task 1 - 050921

Determine all rational numbers $a$ and $b$ for which $(a+b)^{3}=a^{3}+b^{3}$ holds.
The equation

$$
(a+b)^{3}=a^{3}+b^{3}
$$

is, because

$$
(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}
$$

equivalent to each of the following equations:

$$
\begin{array}{r}
a^{3}+3 a^{2} b+3 a b^{2}+b^{3}=a^{3}+b^{3} \\
3 a^{2} b+3 a b^{2}=3 a b \cdot(a+b)=0 \\
a b \cdot(a+b)=0
\end{array}
$$

Since a product is exactly zero if at least one of its factors is zero, the given equation holds exactly when at least one of the three conditions is satisfied:

a) $a=0, b$ arbitrary rational,

b) $b=0, a$ arbitrary rational,

c) $a=-b, b$ arbitrary rational.