HMMT November, 2014

Algebra Difficulty 5.0 AIME, harder Prove it United States

Problem:
Let $x$ be a complex number such that $x + x^{-1}$ is a root of the polynomial $p(t) = t^{3} + t^{2} - 2 t - 1$ . Find all possible values of $x^{7} + x^{-7}$ .

Solution

Solution:
Since $x + x^{-1}$ is a root,
$\begin{aligned} 0 & = \left(x + x^{-1}\right)^{3} + \left(x + x^{-1}\right)^{2} - 2\left(x + x^{-1}\right) - 1 \\ & = x^{3} + x^{-3} + 3x + 3x^{-1} + x^{2} + 2 + x^{-2} - 2x - 2x^{-1} - 1 \\ & = x^{3} + x^{-3} + x^{2} + x^{-2} + x + x^{-1} + 1 \\ & = x^{-3}\left(1 + x + x^{2} + \cdots + x^{6}\right). \end{aligned}$
Since $x \neq 0$ , the above equality holds only if $x$ is a primitive seventh root of unity, i.e. $x^{7} = 1$ and $x \neq 1$ . Therefore, the only possible value of $x^{7} + x^{-7}$ is $1 + 1 = 2$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.