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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?

Pencil Turning

Stage: 4 Short Challenge Level: Challenge Level:1

Ans 1:1

Let AB have length 3r. The distance moved by A is then the circumference of a semicircle radius 3r (3$\pi$r). C moves along a circle of radius 2r (2$\pi$r), followed by a semicircle of radius r ($\pi$r). The total distance moved by C is therefore also 3$\pi$r.

This problem is taken from the UKMT Mathematical Challenges.
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