International Mathematics Competition
for University Students
2026

IMC2026: Day 1, Problem 5

Problem 5. Prove that there exists a constant \(\displaystyle C>0\) such that for every pair \(\displaystyle A,B\) of positive integers, there is a real polynomial \(\displaystyle p(x)\) with

\(\displaystyle p(0)^2 > \sum_{i=1}^A p(-i)^2 + \sum_{i=1}^B p(i)^2 \quad\text{and}\quad \deg p < C\sqrt{AB}. \)

Géza Kós, Loránd Eötvös University, Budapest

  Solution