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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?


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

We have $y^{2}=x(2-x)$.

$y^{2}\ge0$ for all real $y$, hence $x(2-x)\ge0$.

Hence $0\le x\le2$.

In fact we can rewrite the equation as $(x-1)^{2}+y^{2}=1$

So this is a circle of radius $1$ with centre at $(1,0)$.

This problem is taken from the UKMT Mathematical Challenges.
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