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