International Mathematics Competition
for University Students
2021

IMC2021: Day 1, Problem 3

Problem 3. We say that a positive real number \(\displaystyle d\) is good if there exists an infinite sequence \(\displaystyle a_1,a_2,a_3,\ldots \in (0,d)\) such that for each \(\displaystyle n\), the points \(\displaystyle a_1,\dots,a_n\) partition the interval \(\displaystyle [0,d]\) into segments of length at most \(\displaystyle 1/n\) each. Find

\(\displaystyle \sup\Big\{d\ \big|\ d \text{ is good}\Big\}. \)

Josef Tkadlec

Hint: To get an upper bound, use that some of the gaps after \(\displaystyle n\) steps are still intact some steps later.

  Solution