### Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

### Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

### Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

# Sliced

##### Age 14 to 16Challenge Level

Slice the tetrahedron in two with a cut through the line $b$ and one of the equal edges. The perpendiculars are very important because of what you need to find the volume of a pyramid. You now have two identical pyramids like this: