Problems

A white plane is arbitrarily sprinkled with black ink. Prove that for any positive \(l\) there exists a line segment of length \(l\) with both ends of the same colour.

A Cartesian plane is coloured in in two colours. Prove that there will be two points on the plane that are a distance of 1 apart and are the same colour.

Prove that the bisectors of a triangle intersect at one point.

Let \(ABC\) and \(A_1B_1C_1\) be two triangles with the following properties: \(AB = A_1B_1\), \(AC = A_1C_1\), and angles \(\angle BAC = \angle B_1A_1C_1\). Then the triangles \(ABC\) and \(A_1B_1C_1\) are congruent. This is usually known as the “side-angle-side" criterion for congruence.

In the triangle \(\triangle ABC\) the sides \(AC\) and \(BC\) are equal. Prove that the angles \(\angle CAB\) and \(\angle CBA\) are equal.

Let \(ABC\) be a triangle with given angles \(\angle BAC\) and \(\angle ABC\). What is the value of the angle \(\angle BCD\) in terms of \(\angle BAC\) and \(\angle ABC\)?

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.

Let \(\triangle ABC\) and \(\triangle DEF\) be triangles such that the following angles are equal: \(\angle ABC = \angle DEF\) and \(\angle ACB = \angle DFE\). Prove that \(\triangle ABC\) and \(\triangle DEF\) are similar triangles.

The medians \(AD\) and \(BE\) of the triangle \(\triangle ABC\) intersect at the point \(F\). Prove that \(\triangle AFB\) and \(\triangle DFE\) are similar. What is their similarity coefficient?

In a triangle \(\triangle ABC\), the angle \(\angle B = 90^{\circ}\) . The altitude from point \(B\) intersects \(AC\) at \(D\). We know the lengths \(|AD|=9\) and \(|CD|=25\). What is the length \(|BD|\)?