Converging Product

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

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?

Binary Squares

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Golden Fibs

Stage: 5 Challenge Level:

The Fibonacci recurrence relation gives a relationship between the first three terms of the sequence and, if it is also a geometric sequence with common ratio $r$, then this relation gives a quadratic equation in $r$. The question asks for both the result and also its converse to be proved.