MathNet 2009-part-04 — 175 problems (EN–VI) Question 38

Let $x$ be a real number such that $x + \frac{1}{x} + 1$ is a positive integer. Prove that $x^2 + \frac{1}{x^2} + 1$ is a positive integer divisible by $x + \frac{1}{x} + 1$.