Difference between revisions of "1997 USAMO Problems/Problem 1"

Revision as of 12:33, 4 July 2013

Problem

Let $p_1,p_2,p_3,...$ 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

$x_{k}=\begin{cases}0&\text{ if }x_{k-1}=0\\ \left\{\frac{p_{k}}{x_{k-1}}\right\}&\text{ if }x_{k-1}\ne0\end{cases}$

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 $x_0,x_1,x_2,...$ eventually becomes $0$ .

Solution

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Revision as of 16:06, 12 April 2012 (view source) 1=2 (talk \| contribs) (added solution tag & category) ← Older edit		Revision as of 12:33, 4 July 2013 (view source) Nathan wailes (talk \| contribs) Newer edit →
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	[[Category:Olympiad Number Theory Problems]]		[[Category:Olympiad Number Theory Problems]]
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1997 USAMO (Problems • Resources)
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Difference between revisions of "1997 USAMO Problems/Problem 1"

Revision as of 12:33, 4 July 2013

Problem

Solution

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