Problems

In certain kingdom there are a lot of cities, it is known that all the distances between the cities are distinct. One morning one plane flew out of each city to the nearest city. Could it happen that in one city landed more than \(5\) planes?

Between two mirrors \(AB\) and \(AC\), forming a sharp angle two points \(D\) and \(E\) are located. In what direction should one shine a ray of light from the point \(D\) in such a way that it would reflect off both mirrors and hit the point \(E\)?
If a ray of light comes towards a surface under a certain angle, it is reflected with the same angle as on the picture.

\(ABC\) is a triangle. The circumscribed circle is the circle that touches all three vertices of the triangle \(ABC\). It is also the smallest circle lying entirely outside the triangle. The center of the circumscribed circle is \(D\).

The inscribed circle is the circle which touches all three sides of the triangle \(ABC\). It is also the largest circle lying entirely inside the triangle. The center of the inscribed circle is \(E\).

The points \(D\) and \(E\) are symmetric with respect to the segment \(AC\). Find the angles of the triangle \(ABC\).