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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?

Differential Equation Matcher

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

Think about the derivative as the rate of change. Would each part of the description have a positive (increasing) or negative (decreasing) effect on the quantity $X(t)$? How would the size of the change depend on the magnitude of $X(t)$ at any given moment?