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Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

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Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

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Quartics

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Discrete Trends

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

As $n$ is an integer try finding $n^{1\over n}$ for some small values of $n$. What do you find? If you think you might have found the maximum value then you'll need to use the first part of the question to prove it really is the maximum. As the problem is about discrete (whole number) values you can find a solution without calculus. To show that

$$n^{1/n} < 1 + \sqrt {{2\over {n-1}}}$$

write $n^{1/n} = 1 + \delta$ and use the Binomial Theorem. If $n> 1$ then $\delta> 0$. Throw away all but one term of the Binomial expansion to get the inequality.