Problems

Age
Difficulty
Found: 2017

Find the locus of the points \(X\) such that the tangents drawn from \(X\) to a given circle have a given length.

The point \(A\) is fixed on a circle. Find the locus of the point \(X\) which divides the chords that end at point \(A\) in a \(1:2\) ratio, starting from the point \(A\).

Construct the triangle \(ABC\) along the side \(a\), the height \(h_a\) and the angle \(A\).

Construct a straight line passing through a given point and tangent to a given circle.

Three segments whose lengths are equal to \(a, b\) and \(c\) are given. Construct a segment of length: a) \(ab/c\); b) \(\sqrt {ab}\).

In a triangle, the lengths of two of the sides are 3.14 and 0.67. Find the length of the third side if it is known that it is an integer.

The opposite sides of a convex hexagon are pairwise equal and parallel. Prove that it has a centre of symmetry.