IMC 2022 · Day 1 · P1

Hint: Apply the AM–GM inequality.

By the AM–GM inequlity we have $$f(x) + f(1 - x) \ge 2 \sqrt{f(x) f(1 - x)} = 2.$$ By integrating in the interval $[0, \frac12]$ we get $$\int_0^1 f(x)\,dx = \int_0^{\frac12} f(x)\,dx + \int_0^{\frac12} f(1 - x)\,dx = \int_0^{\frac12} \bigl( f(x) + f(1 - x) \bigr)\,dx \ge \int_0^{\frac12} 2\,dx = 1.$$

From the condition, we have $$\int_0^1 f(x)\,dx = \int_0^1 f(1 - x)\,dx = \int_0^1 \frac{1}{f(x)}\,dx$$ and hence, using the positivity of $f$, the claim follows since $$\left( \int_0^1 f(x)\,dx \right)^2 = \int_0^1 f(x)\,dx \cdot \int_0^1 \frac{1}{f(x)}\,dx \ge \left( \int_0^1 1\,dx \right)^2 \ge 1$$ by the Cauchy-Schwarz inequality.