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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 $\sqrt{2}$?

# Unit Interval

##### Age 16 to 18 ShortChallenge Level

Take any two numbers between $0$ and $1$. Prove that the sum of the numbers is always less than one plus their product. That is, if $0< x< 1$ and $0< y< 1$ then prove $$x+y< 1+xy$$
Did you know ... ?

Pure inequalities such as this one are often used in the analysis of far more difficult mathematics problems: whilst the inequalities might be simple to prove in themselves, they can be surprisingly useful as tools.