Draw a circle $k$ around the center of a regular hexagon, with a radius no larger than that of the circle tangent to the sides of the hexagon, then draw a circle around each vertex of the hexagon that touches $k$ and has a diameter no larger than the side of the hexagon. The 7 circular segments together cover a part of the hexagon. What is the radius of the circle $k$ if the area of the covered part is to be as small as possible, or as large as possible?
Let the side of the hexagon be denoted by $a$, and the radius of the circle $k$ by $r$. According to this, the radius of the circles circumscribed around the vertices is $a-r$, and according to the condition, $2(a-r) \leqq a$, from which $r \geqq \frac{1}{2} a$ follows. On the other hand, $r$ is less than or equal to the radius of the circle inscribed in the hexagon, $a \sqrt{3} / 2$.

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Let the area of the covered part of the hexagon be denoted by $T$. This consists of seven parts: the circle $k$ and 6 congruent $120^{\circ}$-sectors, whose centers are the vertices of the hexagon and whose radii are $a-r$. Since $a-r \leqq \frac{a}{2}$, the sectors do not overlap, and thus

$$
T=r^{2} \pi+6 \frac{(a-r)^{2} \pi}{3}=3 \pi\left(r-\frac{2}{3} a\right)^{2}+\frac{2 \pi}{3} a^{2}
$$

This quadratic function has a minimum when $r=2 a / 3$, and this value falls within the desired interval.

On the other hand, if the value of $r$ runs from $a / 2$ to $2 a / 3$, $T$ decreases, and if it runs from $2 a / 3$ to the other limit, $a \sqrt{3} / 2$, $T$ increases. Thus, $T$ takes its maximum value at one of the extreme points. Since

$$
\left|\frac{a \sqrt{3}}{2}-\frac{2}{3} a\right|>\left|\frac{a}{2}-\frac{2}{3} a\right|
$$

this point is $r=a \sqrt{3} / 2$.

Therefore, the area of the covered part is minimal when $r=2 a / 3$, that is, when the radius of the circle drawn around the center is two-thirds of the side of the hexagon, and it is maximal when $k$ is the inscribed circle of the hexagon.

Remark. Several of our solvers noticed that the problem in a more general form had already appeared in our journal (1976, Volume 52, Issue 2, Page 63, F. 2004). There, the problem was to construct the circle for a regular $n$-gon under similar conditions, for which the covered part of the area is minimal or maximal. The solution given there, in certain aspects, is more than what we intended for the case $n=6$, and the solution to our problem can be read from there. However, we did not accept those solutions that only stated that the solution to the problem can be found here and there, but did not answer the question of the problem (i.e., what is the radius of $k$ when the covered area is minimal or maximal). This is because it is not mentioned in the cited solution.