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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?

Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

Unit Interval

Age 16 to 18 Short Challenge Level:
Notice that showing $x+y< 1+xy$ is equivalent to showing that $xy -x -y +1 > 0$