### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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?

### 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.

# Weekly Problem 38 - 2011

##### Challenge Level:

Ans: 28%

The area of the larger circle is 25$\pi$ cm $^2$.
The area of the smallest circle is 9$\pi$ cm $^2$.
The area of the middle circle is 16$\pi$ cm $^2$.
Therefore the area of the ring is (16$\pi$ - 9 $\pi$) cm $^2$ i.e. 7\ $\pi$ cm $^2$.

Therefore the percentage shaded is 28%.

This problem is taken from the UKMT Mathematical Challenges.

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