Difference between revisions of "1997 USAMO Problems"
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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 | 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> | + | <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>. | 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>. |
Revision as of 19:58, 30 June 2011
Problem 1
Let be the prime numbers listed in increasing order, and let be a real number between and . For positive integer , define
where denotes the fractional part of . (The fractional part of is given by where is the greatest integer less than or equal to .) Find, with proof, all satisfying for which the sequence eventually becomes .