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Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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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?

Perception Versus Reality

Stage: 4 and 5 Challenge Level: Challenge Level:1
Well done to Martha from Kings Norton Girls' School who created a video response to this problem:

I created a video about the distribution of religions in England based on this task, however changed the question about 'how it should be' to 'how it was 50 years ago' as this was more suitable.

You can watch Martha's video on YouTube.