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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.

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Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Penny Farthing

Stage: 4 Short Challenge Level: Challenge Level:1

As the ratio of the radii is $3\colon 4$ the number of revolutions made by the larger wheel is $$120000\times\frac{3}{4}= 90000\;.$$

This problem is taken from the UKMT Mathematical Challenges.
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