### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

### From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

### Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

# Circuit Training

##### Stage: 4 Challenge Level:

Why do runners always run around the track in an anticlockwise direction?

Do they develop a clinical lean?

Mike and Monisha meet at the race track, which is $400 \; \text{m}$ round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise, normally Mike can complete a circuit of the track in two thirds of the time Monisha takes.

They agree to stop when they both meet at the start again. Where will they meet on their way around and will they ever meet at the start again? If so, after how many circuits?