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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.

At a Glance

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?

Burning Down

Age 14 to 16 Challenge Level:

candles

Two different candles are lit. They burn at different rates and one is 3 cm longer than the other.

The longer one was lit at 5.30 pm and the shorter one at 7 pm.

At 9.30 pm they were both the same length.

The longer one, burned out at 11.30 pm and the shorter one burned out at 11 pm.

How long was each candle originally?