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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?


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?

Mixed up Mixture

Age 14 to 16 Challenge Level:

First divide the numbers into groups according to whether they are concentrations, pure numbers, or volumes.

Secondly, group the concentrations into pairs, along with the required dilution factor. Then, divide 5ml by the dilution factor to additionally find the volume added.

The numbers should be in groups of 4 now, with three additional outlying numbers.

Thirdly, think which of the components added would be measured as a number per ml, rather than a number of moles per ml. This will assign one group of 4 numbers.

Fourthly, think which of the remaining items has a large molecular weight, and so will only have a low molar concentration added. Assign this group of 4 numbers.

The final two groups are hard to assign: research what the intracellular concentration of sodium ions is, since the NaCl is added to keep the solution isotonic.

The data should be all assigned now, and the sum of the added volumes should be equal to one of the leftover numbers. What might the final number represent?