### 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.

### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

### 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?

# Big Fibonacci

##### Stage: 4 Short Challenge Level:

Let the first two terms of the sequence be $a$ and $b$ respectively.

Then the next three terms are $a+b$, $a+2b$, $2a+3b$. So $2a+3b = 2004$.

For $a$ to be as large as possible, we need $b$ to be as small as possible, consistent with both being positive integers.

If $b=1$ then $2a=2001$, but $a$ is an integer, so $b\not=1$.

However, if $b=2$ then $2a=1998$, so the maximum possible value of $a$ is $999$, giving us the first five terms:
999, 2, 1001, 1003, 2004

This problem is taken from the UKMT Mathematical Challenges.
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