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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?

### 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 $\sqrt{2}$?

# It's Only a Minus Sign

##### Age 16 to 18 Challenge Level:

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 $x$?