Problems

Let \(BCDE\) be an inscribed quadrilateral. Show that \(\angle BCD + \angle BED = 180^{\circ}\).

In a circle with the centre \(A\) the angles \(\angle CBD\) and \(\angle EFG\) are equal. Prove that the segments \(CD\) and \(EG\) are equal.

On the diagram below find the value of the angles \(\angle CFD\) and \(\angle CGD\) using angles \(\angle CBD\) and \(\angle BDE\).

Point \(A\) is the centre of a circle. Points \(B,C,D,E\) lie on this circle. Lines \(BC\) and \(DE\) cross at \(F\). Angles \(\angle BAD =\delta\), \(\angle CAE = \gamma\). Express the angle \(DFB\) in terms of \(\gamma\) and \(\delta\).

The triangle \(ABC\) is inscribed into the circle with centre \(E\), the line \(AD\) is perpendicular to \(BC\). Prove that the angles \(\angle BAD\) and \(\angle CAE\) are equal.

On the diagram below \(BC\) is the tangent line to a circle with the centre \(A\), and it is known that the angle \(\angle ABC = 90^{\circ}\). Prove that the angles \(\angle DEB\) and \(\angle DBC\) are equal.

The triangle \(BCD\) is inscribed in a circle with the centre \(A\). The point \(E\) is chosen as the midpoint of the arc \(CD\), which does not contain \(B\), the point \(F\) is the centre of the circle inscribed into \(BCD\). Prove that \(EC = EF = ED\).

Alex got a new set of crayons. They started colouring various things. First, Alex coloured a line under the following condition: each point on a line is coloured either red or blue. Show that there are three different points \(A,B,C\) on the line of the same colour such that \(AB = BC\).

As the next step Alex has coloured a plane. They want to colour the entire plane in such a way that each straight line can only have points of three or fewer different colours. Show that they can use however many different colours they want and still be able to achieve this goal.

Alex continues colouring the plane. They decided to use \(9\) different colours, but they want to colour the entire infinite plane in such a way, that if two points are in distance \(1\), they must be different colours. Show that it is always possible.