IMC 2016 Problem 7

Today, Ivan the Confessor prefers continuous functions $f:[0,1]\to\mathbb{R}$ satisfying $f(x)+f(y)\geq |x-y|$ for all pairs $x,y\in [0,1]$ . Find the minimum of $\int_0^1 f$ over all preferred functions.

(Proposed by Fedor Petrov, St. Petersburg State University)