International Mathematics Competition
for University Students
2026

IMC2026: Day 2, Problem 9

Problem 9. Let \(\displaystyle a_1,a_2,\ldots\) be an infinite sequence of positive real numbers satisfying

\(\displaystyle a_1 + a_2 + \cdots + a_{2n-1} = a_n^2 \)

for all positive integers \(\displaystyle n\). Prove that \(\displaystyle a_n \geq 2n-1\) for all positive integers \(\displaystyle n\).

Ilya I. Bogdanov, MIPT, Moscow and Aleksandr Kuznetsov, SPbU, Saint Petersburg

  Solution