SIMON-MARAIS 2022 B2

Let $a, b,$ and $c$ be real numbers, and let $P$ be the polynomial

$P(x) = x^6 + ax^4 + bx^2 + c.$

Suppose that there is a unique circle $\Gamma$ in the complex plane such that all of the roots of $P$ lie on $\Gamma$ . Prove that $b^3 = a^3c$ .

A model read this text from the scan of the paper by OCR, with vision/local (GLM-OCR-8bit). No person has checked it against the paper yet.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.