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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?

Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral. How can you maximise its area?


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

Discrete Trends

Age 16 to 18 Challenge Level:

Show that if $n$ is a positive integer then

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

Show that $n^{1/n}\rightarrow 1$ as $n\rightarrow \infty$.

Find the maximum value of $n^{1/n}$ and prove that it is indeed the maximum.