12. (NET 1) ${ }^{\mathrm{IMO2}}$ Find two positive integers $a, b$ such that none of the numbers $a, b, a+b$ is divisible by 7 and $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$.
12. By the binomial formula we have
$$
\begin{aligned}
(a+b)^{7}-a^{7}-b^{7} & =7 a b\left[\left(a^{5}+b^{5}\right)+3 a b\left(a^{3}+b^{3}\right)+5 a^{2} b^{2}(a+b)\right] \\
& =7 a b(a+b)\left(a^{2}+a b+b^{2}\right)^{2} .
\end{aligned}
$$

Thus it will be enough to find $a$ and $b$ such that $7 \nmid a, b$ and $7^{3} \mid a^{2}+a b+b^{2}$. Such numbers must satisfy $(a+b)^{2}>a^{2}+a b+b^{2} \geq 7^{3}=343$, implying $a+b \geq 19$. Trying $a=1$ we easily find the example $(a, b)=(1,18)$.