Problem Of The Week # 288 - Nov 09, 2017

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  • #1
Ackbach
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Here is this week's POTW:

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Let $f$ be a polynomial with positive integer coefficients. Prove that if $n$ is a positive integer, then $f(n)$ divides $f(f(n)+1)$ if and only if $n=1$. Assume $f$ is non-constant.

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  • #2
Congratulations to castor28 for his correct solution to the Putnam Archive Problem B-1 from 2007, which follows:

[sp]Since $f$ has integer coefficients, we have $f(k+1)\equiv f(1)\pmod{k}$ for any integer $k$. Taking $k=f(n)$, we get $f(f(n)+1)\equiv f(1)\pmod{f(n)}$. This means that we must prove that, if $n>0$, $f(n)\mid f(1)$ if and only if $n = 1$.

The "if" part is obvious. For the converse, note that, since $f$ has positive coefficients and is not constant, $f(x)$ is a strictly increasing function of $x$ for $x > 0$. In particular, if $n>1$, we have $f(n) > f(1) > 0$, and $f(n)$ cannot divide $f(1)$.[/sp]
 

FAQ: Problem Of The Week # 288 - Nov 09, 2017

1. What is the problem being addressed in Problem of the Week #288?

The problem being addressed in Problem of the Week #288 is a mathematical problem involving finding the minimum number of moves needed to solve a puzzle with three different types of pieces.

2. What is the significance of this problem?

This problem is significant because it challenges individuals to think critically and use problem-solving skills to find the most efficient solution. It also has real-world applications in areas such as computer science and game design.

3. How can I approach solving this problem?

There are a few different approaches to solving this problem, but one possible method is to break down the puzzle into smaller, more manageable parts and then work backwards to find the most efficient solution.

4. Are there any known solutions to this problem?

Yes, there are known solutions to this problem. However, the challenge lies in finding the most efficient solution, which may not be the same as the known solutions.

5. Can this problem be applied to other situations or puzzles?

Yes, the principles and techniques used to solve this problem can be applied to other similar situations or puzzles. It is a good exercise in critical thinking and problem-solving that can be useful in various fields.

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