IMC 2022 · Problem 1

Let $f \colon [0, 1] \to (0, \infty)$ be an integrable function such that $f(x) \cdot f(1 - x) = 1$ for all $x \in [0, 1]$. Prove that

$$\int_0^1 f(x)\,\mathrm{d}x \ge 1.$$