### 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 36 - 2011

##### Challenge Level:

Ans: ½

Let the radius of the circle be r. This implies that the radius of the semicircle is 2r. The area of the semi circle is $1/2 \times \pi \times(2r)^2$, which is twice the area of the small circle.

This problem is taken from the UKMT Mathematical Challenges.

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