International Mathematics Competition
for University Students
2026

IMC2026: Day 2, Problem 7

Problem 7. For a continuous function \(\displaystyle f\colon[0,1]\to\RR\), let \(\displaystyle S(f)\) be the union of all straight segments in the plane joining points \(\displaystyle (x,0)\) and \(\displaystyle (f(x),1)\), where \(\displaystyle x\in [0,1]\). Let \(\displaystyle A(f)\) be the area of \(\displaystyle S(f)\). Find the infimum of \(\displaystyle A(f)\) over all continuous \(\displaystyle f\).

David Preiss, University of Warwick, UK

  Solution