Three equilateral triangles ABC, AYX and XZB are drawn with the
point X a moveable point on AB. The points P, Q and R are the
centres of the three triangles. What can you say about triangle
PQR?

P is the midpoint of an edge of a cube and Q divides another edge
in the ratio 1 to 4. Find the ratio of the volumes of the two
pieces of the cube cut by a plane through PQ and a vertex.

Chord

Age 16 to 18 Challenge Level:

Three circles all of radius 2 cm touch as shown in the
diagram.

A straight line $AG$ is tangential to the third circle, meeting
the middle circle at $H$ and $I$.