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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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Kissing

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

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Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

Weekly Challenge 2: Quad Solve

Stage: 5 Short Challenge Level: Challenge Level:2 Challenge Level:2
To do this problem you will need to know that $a^0 = 1$ when $a\neq 0$ and $1^b=1$ for any $b$ along with other power manipulations which are needed from the beginning of C1.
 
Note that $0^0$ is not defined as a number.
 
You will also need to know the formula for the solution of a quadratic equation.
 
If you are studying complex numbers then you can also bring those ideas into the problem if desired.