## Task 2.

Let $a \geqslant 2018$ be a real number. In each of 2018 jars, there is a finite number of balls, with the mass of each ball being of the form $a^{k}$, where $k \in \mathbb{Z}$. The total mass of the balls in each jar is the same. What is the minimum number of balls of the same mass among the balls in the jars?
## Solution.

The smallest number of balls of equal mass among the selected balls is 2018.

Without loss of generality, we can assume that the mass of the lightest ball is 1, simply by dividing all masses (among the selected balls) by the mass of the lightest one, thereby preserving all properties.

Assume that at most 2017 balls of equal mass appear among the selected balls. Let $n$ be a natural number such that $a^{n}$ is the mass of the heaviest ball among the selected ones. Since there are at most 2017 balls of mass $a^{n}$, there exists a container that does not contain any ball of mass $a^{n}$, i.e., all are lighter. Then, the mass of that container is at most

$$
2017 \cdot \sum_{k=0}^{n-1} a^{k}=2017 \cdot \frac{a^{n}-1}{a-1} \leqslant 2017 \cdot \frac{a^{n}-1}{2017}=a^{n}-1<a^{n}
$$

which is a contradiction.

Therefore, at least 2018 balls of equal mass appear.

If we place a ball of mass 1 in each container, we see that all containers have the same mass and that exactly 2018 balls of equal mass appear.