1. What is the maximum number of numbers that can be selected from the set $\{1,2, \ldots, 2019\}$ such that the product of any three of the selected numbers is not divisible by nine? Provide an example of a suitable subset and explain why it cannot have more elements.

(Aleš Kobza)
1. Among the selected numbers, there must not be any multiple of nine, and at most one of them can be divisible by three, but not by nine. Therefore, we can select all numbers that are not divisible by three and add $\mathrm{k}$ to them any number that is divisible by three, but not by nine.

Conclusion. The largest possible number of numbers that can be selected in the required way from the set $\{1,2, \ldots, 2019\}$ is thus equal to $\frac{2}{3} \cdot 2019+1=1347$.

The set $\{1,2,3,4,5,7,8,10,11, \ldots, 2015,2017,2018\}$, which has exactly 1347 elements, satisfies the problem. This set contains the number 3 as the only number divisible by three, which is not divisible by nine.

For a complete solution, award 6 points, of which 1 point for explaining that there cannot be a number divisible by 9, 1 point for explaining that there can be at most one number divisible by 3 (not divisible by 9), 3 points for estimating the maximum number of elements based on these observations, and 1 point for an example of a suitable set.