Prove that under homothety, a circle transforms into a circle.
Two circles touch at point \(K\). The line passing through point \(K\) intersects these circles at points \(A\) and \(B\). Prove that the tangents to the circles drawn through points \(A\) and \(B\) are parallel.
Two circles \(c\) and \(d\) are tangent at point \(B\). Two straight lines intersecting the first circle at points \(D\) and \(E\) and the second circle at points \(G\) and \(F\) are drawn through the point \(B\). Prove that \(ED\) is parallel to \(FG\).
Prove that the points symmetric to an arbitrary point relative to the midpoints of the sides of a square are vertices of some square.
The points \(A\) and \(B\) and the line \(l\) are given on a plane. On which trajectory does the intersection point of the medians of the triangles \(ABC\) move, if the point \(C\) moves along the line \(l\)?
There are 5 points inside an equilateral triangle with side of length 1. Prove that the distance between some two of them is less than 0.5.
A \(3\times 4\) rectangle contains 6 points. Prove that amongst them there will be two points, such that the distance between them is no greater than \(\sqrt5\).
A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).
What is the minimum number of points necessary to mark inside a convex \(n\)-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?
A plane contains \(n\) straight lines, of which no two are parallel. Prove that some of the angles will be smaller than \(180^\circ/n\).