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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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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 $\sqrt{2}$?

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Flexi Quads

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

It's Only a Minus Sign

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Remember that the differential of x means the 'rate of change' of $x$. The equation tells us exactly what that rate of change must be at each point.

What does a positive rate of change tell us about the changes in $x$? What does a negative rate of change tell us about the changes in $x$?