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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

Why do this problem?
It gives practice in working with inequalities.

As we know $n$ is a positive integer learners can investigate $n^{{1\over n}}$ for for different values of $n$ and make conjectures about where the maximum value occurs.

Possible approach

You need to find a local maximum for a small value of $n$ and then prove that this is the only maximum value. Clearly it is impossible to check all values of $n$. One method of proving the result uses the Binomial theorem.