International Mathematics Competition
for University Students
2016

IMC2016: Day 2, Problem 7

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

Hint: Apply the condition for special (or for all) pairs $(x,y)$ and integrate it.

  Solution