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Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

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Three by One

There are many different methods to solve this geometrical problem - how many can you find?

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Target Six

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

Complex Sine

Age 16 to 18 Challenge Level:
One method is to use the formula $$\sin z = {1\over 2i}(e^{iz} - e^{-iz})$$ substitute $w=e^{iz}$ and solve for $w$.