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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)?

Sixty-seven Squared

Age 16 to 18 Challenge Level:

Evaluate the following:

$67^2$
$667^2$
$6667^2$
$66667^2$

What do you notice?

Can you convince someone what the answer would be to (a million sixes followed by a 7) 2 ?