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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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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?

Circled Corners

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

The three angles of the triangle add to $180^{\circ}$, so the combined area of the three sectors of the circles that are inside the triangle add up to half a circle with area:
 
$\frac{1}{2} \times \pi \times \times 2^2 = \frac {4\pi}{2}= 2\pi$. 
 
So the grey area is $(80- 2\pi) cm^2$.
 
 

This problem is taken from the UKMT Mathematical Challenges.

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