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

  • MHB
  • Thread starter Ackbach
  • Start date
  • Tags
    2017
In summary, the 1999 Putnam Competition Problem A-5 is a challenging math problem featured in an annual undergraduate mathematics competition. It requires a combination of algebraic manipulation and creative thinking to find the maximum value of a specific function. Tips for solving the problem include breaking it down into smaller steps, using algebraic techniques, and working with others. This problem is significant in showcasing the difficulty and importance of problem-solving skills in mathematics competitions.
  • #1
Ackbach
Gold Member
MHB
4,155
92
Here is this week's POTW:

-----

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

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
  • #2
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.

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).[/FONT]
For $p(x) \in \mathcal P$, define \(\displaystyle \|p(x)\|_1 = \int_{-1}^1|p(x)|\,dx\) and \(\displaystyle \|p(x)\|_\infty = \max_{-1\leqslant x \leqslant1}|p(x)|\). 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 $|p(0)| \leqslant \|p(x)\|_\infty \leqslant C\|p(x)\|_1$.
 

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

What is the 1999 Putnam Competition Problem A-5?

The 1999 Putnam Competition Problem A-5 is a math problem that was featured in the 1999 William Lowell Putnam Mathematical Competition, an annual undergraduate mathematics competition held in the United States and Canada.

How difficult is the 1999 Putnam Competition Problem A-5?

The difficulty of the 1999 Putnam Competition Problem A-5 varies depending on the person attempting to solve it. It was considered one of the more challenging problems in the competition, but some individuals may find the problem easier than others.

What is the objective of the 1999 Putnam Competition Problem A-5?

The objective of the 1999 Putnam Competition Problem A-5 is to find the maximum value of a specific function, given certain conditions. The problem requires a combination of algebraic manipulation and creative thinking to solve.

Are there any tips for solving the 1999 Putnam Competition Problem A-5?

Some tips for solving the 1999 Putnam Competition Problem A-5 include breaking down the problem into smaller, more manageable steps, using algebraic techniques to simplify the given conditions, and considering different approaches to the problem. It is also helpful to work with others and discuss different strategies for solving the problem.

What is the significance of the 1999 Putnam Competition Problem A-5?

The 1999 Putnam Competition Problem A-5 is significant because it showcases the level of difficulty and creativity required in mathematics competitions. It also serves as a reminder of the importance of perseverance and problem-solving skills in the field of mathematics.

Similar threads

Replies
1
Views
2K
Replies
1
Views
2K
Replies
1
Views
1K
Replies
1
Views
2K
Replies
1
Views
2K
Replies
1
Views
1K
Back
Top