Here's to a prolific winter-break USAMO marathon

by shiningsunnyday, Dec 10, 2016, 10:52 AM

1997 USAMO P1 wrote:
Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between $0$ and $1.$ For positive integer $k$, define
\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases}  \]where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes $0.$

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Still have two weeks until winter break for me :(

Go go go
This post has been edited 1 time. Last edited by shiningsunnyday, Dec 10, 2016, 11:24 PM

by MathAwesome123, Dec 10, 2016, 2:29 PM

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Still have two essays due for English until winter break for me :noo:

I have 3 more exams. go go go
This post has been edited 1 time. Last edited by shiningsunnyday, Dec 11, 2016, 5:01 AM

by zephyrcrush78, Dec 11, 2016, 4:56 AM

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And 5 more finals :dry:

by zephyrcrush78, Dec 11, 2016, 6:08 AM

To share with readers my favorite problem I came across today :) (Shout for contrib.)

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