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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?

# Root Hunter

##### Stage: 5 Challenge Level:

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This graph is positive for $x = 5$ and negative for $x = 3$. This means that the graph must cut the $x$ axis somewhere between $3$ and $5$.
Although in this case the result is obvious (because we have the whole graph to look at!), we can also use this idea to show that more tricky functions also have roots.

Use this idea to show that these functions possess at least one solution $f(x) = 0$:

$$f(x)=\frac{1}{x-2}+\frac{1}{x-3}$$ $$f(x)= x^x - 1.5 x$$ $$f(x)= x^{1000000}+{1000000}^x - 17$$ $$f(x)=\cos(\sin(\cos x)) - \sin(\cos(\sin x))$$

Optional extension activity: Can you make a spreadsheet that helps you find the numerical values of the roots to, say, four decimal places?