International Mathematics Competition
for University Students
2020

IMC2020: Day 1, Problem 4

Problem 4. A polynomial \(\displaystyle p\) with real coefficients satisfies the equation \(\displaystyle p(x+1)-p(x)=x^{100}\) for all \(\displaystyle x\in\mathbb{R}\). Prove that \(\displaystyle p(1-t)\geqslant p(t)\) for \(\displaystyle 0\leqslant t\leqslant 1/2\).

Daniil Klyuev, St. Petersburg State University

Hint: Let \(\displaystyle p_n\) be a polynomial such that \(\displaystyle p_n(x+1)-p_n(x)=x^n\). Find a recurrence relation.

  Solution