We define the distance between two circles $\omega ,\omega '$by the length of the common external tangent of the circles and show it by $d(\omega , \omega ')$. If two circles doesn't have a common external tangent then the distance between them is undefined. A point is also a circle with radius $0$ and the distance between two cirlces can be zero. (a) Centroid. $n$ circles $\omega_1,\dots, \omega_n$ are fixed on the plane. Prove that there exists a unique circle $\overline \omega$ such that for each circle $\omega$ on the plane the square of distance between $\omega$ and $\overline \omega$ minus the sum of squares of distances of $\omega$ from each of the $\omega_i$s $1\leq i \leq n$ is constant, in other words:\[d(\omega,\overline \omega)^2-\frac{1}{n}{\sum_{i=1}}^n d(\omega_i,\omega)^2= constant\] (b) Perpendicular Bisector. Suppose that the circle $\omega$ has the same distance from $\omega_1,\omega_2$. Consider $\omega_3$ a circle tangent to both of the common external tangents of $\omega_1,\omega_2$. Prove that the distance of $\omega$ from centroid of $\omega_1 , \omega_2$ is not more than the distance of $\omega$ and $\omega_3$. (If the distances are all defined) (c) Circumcentre. Let $C$ be the set of all circles that each of them has the same distance from fixed circles $\omega_1,\omega_2,\omega_3$. Prove that there exists a point on the plane which is the external homothety center of each two elements of $C$. (d) Regular Tetrahedron. Does there exist 4 circles on the plane which the distance between each two of them equals to $1$? Time allowed for this problem was 150 minutes.