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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?
Target Six
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
8 Methods for Three by One
This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?
Age 16 to 18
Challenge Level
Draw a simple diagram which makes it clear that the following inequality holds for any complex numbers $z$ and $w$ $$|z-w|\geq |z| -|w|$$
Now draw another simple diagram which makes it clear that the following inequality holds for any real numbers $\alpha> \beta$
$$|e^{i\alpha}-e^{i\beta}|\leq \alpha - \beta$$
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.