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

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-== Problem ==
-Let <math>p_1,p_2,p_3,...</math> be the prime numbers listed in increasing order, and let <math>x_0</math> be a real number between <math>0</math> and <math>1</math>. For positive integer <math>k</math>, define
-<math> 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} </math>
-where <math>\{x\}</math> denotes the fractional part of <math>x</math>. (The fractional part of <math>x</math> is given by <math>x-\lfloor{x}\rfloor</math> where <math>\lfloor{x}\rfloor</math> is the greatest integer less than or equal to <math>x</math>.) Find, with proof, all <math>x_0</math> satisfying <math>0<x_0<1</math> for which the sequence <math>x_0,x_1,x_2,...</math> eventually becomes <math>0</math>.
-== Solution ==

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