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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 $\sqrt{2}$?

Root Hunter

Age 16 to 18
Challenge Level


Why do this problem?

This gives an interesting challenge in which students develop their understanding of functions and skills at approximation and estimation.


Possible approach

Students should be encouraged to try out function evaluation for various choices of numbers. Remember, that the key point is whether the function is positive or negative: we don't need to evaluate the exact values. Students should be encouraged to focus on whether the function is positive or negative, rather than computing the exact values.



Key Questions



What do we know about the values of a function either side of a solution?

Can you think of functions for which this sort of approach might not work?

Does this method tell us anything about the number of roots of an equation?

Possible extension


Could students create an algorithm (i.e. recipe or clear sequence of steps) to solve this problem for other functions? Can they clearly express their algorithm so that someone else could apply it for a function of their choice?

Note that at university this sort of idea is extended in courses on Analysis, and this result is called the Intermediate Value Theorem. It is actually a very useful and powerful mathematical idea.


Possible support

Suggest that students try key values of $0.5, 1, 1.5$ and so on. Give them calculators.Suggest that they tabulate the results of their calculations.