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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 $\sqrt{2}$?

Differential Equation Matcher

Age 16 to 18 Challenge Level:

Freddie from Almond Hill school noticed that a decreasing quantity requires a negative first derivative, which allowed him correctly to match the equation for radioactive decay. Can you use these sorts of ideas to match all of the equations?

Pete Pederson from the Acadia Summer AP Calculus AB Workshop was the first to correctly identify the matches. Can you see how he did it?

A - W
B - X
C - V
D - Y
E - Z