12. Retro Collection (recommended from 8th grade, 2 points). Vitya collects toy cars from the "Retro" series. The problem is that the total number of different models in the series is unknown - it is the biggest commercial secret, but it is known that different cars are produced in the same edition, and therefore it can be assumed that all models are uniformly and randomly distributed across different online stores.

On different websites, Vitya found several offers, but upon closer inspection, it turned out that among the offered cars, there were only 12 different ones. Vitya is almost convinced that there are only 12 cars in total and that further search is futile. But who knows?

How many more offers from other stores should Vitya consider to be convinced that there are only 12 models in the series? Vitya considers himself convinced of something if the probability of this event is higher than 0.99.
Solution. Let there be a series of $n \geq 13$ cars. The probability that among the next $k$ offers there will only be the 12 models that Vitya has already seen is

$$
\left(\frac{12}{n}\right)^{k} \leq\left(\frac{12}{13}\right)^{k}
$$

We form the inequality: $\left(\frac{12}{13}\right)^{k}\log _{12 / 13} 0.01=\frac{\ln 100}{\ln 13-\ln 12}=57.53 \ldots
$$

The smallest integer value of $k$ is 58.

Answer: 58.

Comment. The smallest integer solution to the inequality can be found without logarithms by trial and error using a calculator or computer.