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|{x^n}-a|<=1/ 2(p+q)

Source: Mathcenter Contest / Oly - Thai Forum 2010 R2 p2 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 15, 2022
fractional partnumber theoryalgebra

Problem Statement

A positive rational number xx is called banzai if the following conditions are met: ∙\bullet x=\frac{p}{q}>1 where p,qp,q are comprime natural numbers ∙\bullet exist constants α,N\alpha,N such that for all integers n≥Nn\geq N,∣{ xn}−α∣≤12(p+q).\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}. Find the total number of banzai numbers.
Note:{ x}\left\{\,x\right\} means fractional part of xx
(tatari/nightmare)
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