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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:1

The terms of the sequence are

$$6, 3, 14, 7, 34, 17, 84, 42, 21, 104, 52, 26, 13, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, \dots$$

As can be seen, there will now be no other terms in the sequence other than $4$, $2$, and $1$. It can also be seen that the only values of $n$ for which the $n$th term equals $n$ are $13$ and $16$.

This problem is taken from the UKMT Mathematical Challenges.
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