### 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?

### 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?

### Good Approximations

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

# Weekly Challenge 39: Symmetrically So

##### Stage: 5 Challenge Level:

Make a substitution to find two exact real solutions to the equation $(x + 3)^4 + (x + 5) ^4 = 20.$
Did you know ... ?

Frequently mathematicians spend their time stuck wondering how to solve equations or problems. One way of cracking a tough problem is to make a transformation to turn it into a more familiar form which allows the solution to proceed. Finding good substitutions or transformations is one of the more creative aspects of mathematics.