You may also like

problem icon

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?

problem icon

Three by One

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

problem icon

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:
The formula $$\sin z = {1\over 2\pi}(e^{iz} - e^{-iz})$$ can be verified by showing that the series expansion for $\sin z$, that is $$\sin z = z - z^3/3! + z^5/5! - z^7/7! + ... $$ can be obtained using the series expansions $$e^{iz} = 1 + iz + (iz)^2/2! + ...$$ and $$e^{-iz} = 1 -iz + (iz)^2/2! + ...$$