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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?

Circle Time

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

The triangle that joins up the centres of the circles has sides of length 3 cm, 4 cm, 5 cm so must be a right angled triangle by the converse of Pythagoras' Theorem. Therefore the length of the longer arc of the circle $C_1$ is $\frac{3}{4} \times 2\pi \times 1 = \frac{3}{2} \pi$ cm.

This problem is taken from the UKMT Mathematical Challenges.
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