What is the solution to Problem of the Week #207?

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    2016
In summary, the conversation was about the importance of setting and maintaining boundaries in relationships. The speakers discussed how boundaries help maintain healthy relationships and prevent the development of toxic dynamics. They also emphasized the need for effective communication and compromise when setting boundaries.
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Ackbach
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Here is this week's POTW:

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A repunit is a positive integer whose digits in base 10 are all ones. Find all polynomials $f$ with real coefficients such that if $n$ is a repunit, then so is $f(n)$.

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Re: Problem Of The Week # 207 - March 15, 2016

No one solved this week's POTW, which is from the 2007 Putnam archive. The solution, attributed to Kiran Kedlaya and his associates, follows:

Note that $n$ is a repunit if and only if $9n+1=10^m$ for some power of $10$ greater than 1. Consequently, if we set
$$g(n)=9f\left(\frac{n-1}{9}\right)+1,$$
then $f$ takes repunits to repunits if and only if $g$ takes powers of $10$ greater than $1$ to powers of $10$ greater than $1$. We will show that the only such functions $g$ are those of the form $g(n)=10^cn^d$ for $d\ge 0,\; c\ge 1-d$ (all of which work), which will mean that the desired polynomials $f$ are those of the form
$$f(n)=\frac19 (10^c(9n+1)^d-1)$$
for the same $c,d$.

It is convenient to allow "powers of $10$" to be of the form $10^k$ for any integer $k$. With this convention, it suffices to check that the polynomials $g$ taking powers of $10$ greater than $1$ to powers of $10$ are of the form $10^cn^d$ for any integers $c,d$ with $d\ge 0$.

Suppose that the leading term of $g(x)$ is $ax^d$, and note that $a>0$. As $x\to\infty$, we have $g(x)/x^d\to a;$ however, for $x$ a power of $10$ greater than $1$, $g(x)/x^d$ is a power of $10$. The set of powers of $10$ has no positive limit point, so $g(x)/x^d$ must be equal to $a$ for $x=10^k$ with $k$ sufficiently large, and we must have $a=10^c$ for some $c$. The polynomial $g(x)-10^cx^d$ has infinitely many roots, so must be identically zero.
 

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What is the Problem of the Week #207?

The Problem of the Week #207 is a weekly challenge presented by a scientific organization or publication. It typically involves a real-world problem that requires critical thinking and problem-solving skills to find a solution.

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The purpose of the Problem of the Week #207 is to promote critical thinking and problem-solving skills among scientists and individuals interested in science. It also serves as a fun and challenging way to engage with current scientific issues.

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