# Viete relations problem

[SOLVED] Viete relations problem

## Homework Statement

Find all real numbers r for which there is at least one triple (x,y,z) of nonzero real numbers such that

$$x^2 y + yz^2 + z^2 x = xy^2 + yz^2 + zx^2 = rxyz$$

## Homework Equations

http://en.wikipedia.org/wiki/Viète's_formulas

## The Attempt at a Solution

This is equivalent to finding the possible values of r+s+t = 1/r + 1/s + 1/t where r,s,t are real but I don't see how that leads to a solution.

Fix r and assume that x,y,z exist. Let f(t) = t^3 + at^2 + bt+c be the monic polynomial with
x,y,z as its zeros. By assumption c is not zero. Its not hard to show that ab = (3+2r)c and a^3 = x^3+ y^3+z^3 + (3+2r)c using Viete's relations. But I am not sure what to do with those or how to get any sort of condition on r.

Just to be sure, did you type out the equations correctly? Or is it supposed to be $\sum x^2 y = rxyz$ instead?
Just to be sure, did you type out the equations correctly? Or is it supposed to be $\sum x^2 y = rxyz$ instead?