How many $n$-digit numbers are there? How many digits are needed to write them down?
The smallest $n$-digit number is $10^{n-1}$, the smallest $n+1$-digit number: $10^{n}$. Therefore, the number of $n$-digit numbers is: $10^{n} - 10^{n-1} = 9 \times 10^{n-1}$. To describe them, $n \cdot 9 \cdot 10^{n-1}$ digits are needed.

(Alfréd Haar, Budapest.)

The problem was also solved by: Brachfeld S., Déri Zs., Deutsch I., Erdélyi I., Harsányi Z., Hirschfeld Gy., Kelemen M., Kertész G., Kohn L., Kund B., Kürti I., Messer P., Popoviciu M., Simonyi Gy., Sonnenfeld J., Söpkéz Gy., Stolzer J., Veress G.