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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.

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The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Contact

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Bus Stop

Age 14 to 16 Challenge Level:




Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other.

At each mile along the road are milestones. The buses' speeds are constant and in the ratio 5 to 4; the one leaving Shipton is faster (It has a bigger engine!). The buses travel to and fro between the towns .

They meet for the second time at the 145th milestone and for the third time at the 201st milestone. What milestones are at Shipton and Veston?