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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

If $n$ is a positive integer then the units digit of $66^{n}$ is $6$. So when a power of $66$ is divided by $2$, the units digit of the quotient is either $3$ or $8$. Now $66^{66}$ is clearly a multiple of $4$, so $\frac{1}{2}\left(66^{66}\right)$ is even and therefore has units digit $8$ rather than $3$.

This problem is taken from the UKMT Mathematical Challenges.
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