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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?

Annulus Area

Stage: 4 Short Challenge Level: Challenge Level:1

Ans: 28%

The area of the larger circle is 25$ \pi $ cm $^2$.
The area of the smallest circle is 9$ \pi $ cm $^2$.
The area of the middle circle is 16$ \pi $ cm $ ^2 $.
Therefore the area of the ring is (16$ \pi $ - 9 $ \pi $) cm $ ^2 $ i.e. 7 $ \pi $ cm $^2 $.

Therefore the percentage shaded is 28%.

This problem is taken from the UKMT Mathematical Challenges.
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