1. On the board, various natural numbers are written. Their arithmetic mean is a decimal number whose decimal part is exactly 0.2016. What is the smallest value this mean can have?
1. Let $s$ be the sum of the numbers on the board, $n$ be their count, and $p$ be the integer part of the arithmetic mean. Then the following equality holds:

$$
\frac{s}{n}=p+\frac{2016}{10000}=p+\frac{126}{625}
$$

Multiplying through, we get:

$$
625(s-p n)=126 n \text {. }
$$

From this, we see that 625 divides the left side, so it must also divide the right side. However, the numbers 126 and 625 are coprime, so $625 \mid n$. Therefore, $n \geqq 625$.

Next, we use the distinctness of the numbers on the board to determine that:

$$
p=\frac{s}{n}-\frac{126}{625} \geqq \frac{1+2+\ldots+n}{n}-\frac{126}{625}=\frac{n(n+1) / 2}{n}-\frac{126}{625} \geqq \frac{625+1}{2}-\frac{126}{625}>312
$$

The integer $p$ is therefore at least 313, and the value of the mean is at least 313.2016.

It remains to find a set of numbers whose arithmetic mean exactly equals this value. By choosing the 625-element set $\{1,2, \ldots, 624,751\}$, we get the arithmetic mean:

$$
\frac{1+2+\ldots+624+751}{625}=\frac{312 \cdot 625+751}{625}=313+\frac{126}{625}=313.2016 .
$$

Answer: The smallest possible value of the mean is 313.2016.

Note. The example of a set of numbers with an arithmetic mean of 313.2016 is not unique. Let's show how to find one (the one mentioned above) and what all the others look like.

From the derived estimate:

$$
p \geqq \frac{n+1}{2}-\frac{126}{625}
$$

it follows that the value $p=313$ is possible only for $n=625$, in which case substituting both values of $p$ and $n$ into the equation $625(s-p n)=$ $=126 n$ gives $s=126+625 \cdot 313$. This sum of elements characterizes all the sought (known to be 625-element) sets of natural numbers. Since $625 \cdot 313$ is the sum of the numbers in the set $\{1,2, \ldots, 625\}$, a suitable set can be obtained from it, for example, by increasing its largest element 625 by 126 to the value $625+126=751$. Let's add that to achieve the goal, we can also perform such an operation on any element $i$ of the mentioned set for $i \geqq 500$ (remember that the numbers on the board were distinct), or appropriately increase two or more numbers simultaneously (for example, we can get a suitable set $\{1,2, \ldots, 499,501, \ldots, 625,626\}$).

For a complete solution, award 6 points. Award 2 points for constructing a set with a mean equal to 313.2016, and the remaining 4 points for proving that a lower mean cannot be achieved. Of the 4 points allocated for the second part, award 2 points for proving that the number of numbers on the board is at least 625 (it is sufficient to show the relation $625 \mid n$), 1 point for using the distinctness of the numbers to estimate the mean by $(n+1) / 2$, and 1 point for combining both results. If the solver does not mention the coprimality of the numbers 126 and 625, do not deduct points. Also, do not penalize if the set constructed by the solver is incorrect only due to a minor arithmetic error (for example, the number 751 in our example is replaced by the number 750).