An Etude on a Putnam Problem

摘要

We discuss a generalization of Problem A6 from the 2016 Putnam Competition, which asks to find the smallest constant C_n such that for any polynomial f of degree n which has a root in the interval [0,1], the integral of |f| on [0, 1] is bounded above by C_n times the largest value of |f| on [0,1]. Using several basic facts about integrals, polynomials, and linear algebra, we present a line of thought which naturally leads to the discovery of the answer and to several other interesting results.