This is an old revision of the document!

Problem of the Week

Problem 3 (due Monday, March 11)

Let $p(x)=cx^n+c_1x^{n-1}+\ldots$ be a polynomial of degree $n$ with real coefficients and the leading coefficient $c\neq 0$. Prove that at least one of the numbers $|p(0)|, |p(1)|, \ldots, |p(n)|$ is greater or equal than $\displaystyle \frac{|c|n!}{2^n}$. Prove furthermore that this bound is best possible.

Overview

Every other Monday (starting 01/22/24), we will post a problem to engage our mathematical community in the problem solving activity and to enjoy mathematics outside of the classroom. Students (both undergraduate and graduate) are particularly encouraged to participate as there is no better way to practice math than working on challenging problems. If you have a solution and want to be a part of it, e-mail your solution to Marcin Mazur (mazur@math.binghamton.edu) by the due date. We will post our solutions as well as novel solutions from the participants and record the names of those who've got the most number of solutions throughout each semester.

When you submit your solutions, please provide a detailed reasoning rather than just an answer. Also, please include some short info about yourself for our records.