MHB Can You Solve the 1999 Putnam Competition Problem A-5?

[Ackbach]

Here is this week's POTW:

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Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $2017,$ then
\[|p(0)|\leq C \int_{-1}^1 |p(x)|\,dx.\]

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[Ackbach]

Re: Problem Of The Week # 253 - Feb 16, 2017

This was essentially Problem A-5 in the 1999 William Lowell Putnam Mathematical Competition.

Congratulations to Opalg for his correct solution, which follows. Also, honorable mention to Kiwi.

Spoiler

Let $\mathcal P$ be the vector space of all polynomials of degree at most 2017. Then $\mathcal P$ is finite-dimensional (in fact, its dimension is 2018).
For $p(x) \in \mathcal P$, define $$\|p(x)\|_1 = \int_{-1}^1|p(x)|\,dx$$ and $$\|p(x)\|_\infty = \max_{-1\leqslant x \leqslant1}|p(x)|$$. Then $\|\,.\,\|_1$ and $\|\,.\,\|_\infty$ are both norms on $\mathcal P$. But there is a theorem that any two norms on a finite-dimensional vector space are equivalent. In other words, there exists a constant $C$ such that $\|p(x)\|_\infty \leqslant C\|p(x)\|_1$ for all $p(x) \in \mathcal P$. It follows that $|p(0)| \leqslant \|p(x)\|_\infty \leqslant C\|p(x)\|_1$.