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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?

### 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 $\sqrt{2}$?

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:

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