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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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Binary Squares

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Summing Geometric Progressions

Stage: 4 and 5 Challenge Level: Challenge Level:1
Aswaath, from Garden International School, Kuala Lumpur, adapted Alison's method and then used a spreadsheet to calculate the sums directly to verify that Alison's method worked.

Well done to Matthew from Westlakes Academy, Andrew from West Island School, Hong Kong, Michael, and Irina from Loughborough High School for their correct solutions. You can read Michael's solution and Irina's solution.