International Mathematics Competition
for University Students
2026

IMC2026: Day 1, Problem 4

Problem 4. Let \(\displaystyle x_{1} > 0\). Define the sequence \(\displaystyle \{x_{n}\}\) by the recurrence

\(\displaystyle x_{n+1}=\arctan\left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right) \text{ for all }n\geq 1. \)

Find \(\displaystyle \lim\limits_{n \to \infty} x_{n} \sqrt{\ln n}\), where \(\displaystyle \ln x\) denotes the natural logarithm of \(\displaystyle x\).

Wanlong Han, Henan, China

  Solution