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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?

Maximised Area

Stage: 4 Short Challenge Level: Challenge Level:1
Let the large circles have radius R. $$ Area A = 2 x \pi x ({1 \over 2}R)^2 = 1/2 \pi R^2 $$ $$ Area B = 3 x {1 \over 2} x R^2 x {\sqrt 3\over 2} = {3\over 4} \sqrt 3 R^2$$ $$ Area C = (2 R^2) - \pi R^2 = (4 - \pi)R^2 $$ $$ Area D = 2 x {1 \over 2} x R^2 = R^2 $$ $$ Area E = \pi R^2 (\sqrt 2 R)^2 = (\pi - 2)R^2 $$

This problem is taken from the UKMT Mathematical Challenges.
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