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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?

Ratio Riddle

Stage: 4 Short Challenge Level: Challenge Level:1
$9 : 8$
$$\frac{b}{c} = \frac{a}{c} \times\frac{b}{a}= \frac{3}{4} \times\frac{3}{2} = \frac{9}{8}$$

This problem is taken from the UKMT Mathematical Challenges.
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