MathDB
Fractional part of x function

Source: 2012 AMC 12A Problem #25

February 8, 2012
functionAMC

Problem Statement

Let f(x)=2{x}1f(x)=|2\{x\} -1| where {x}\{x\} denotes the fractional part of xx. The number nn is the smallest positive integer such that the equation nf(xf(x))=xnf(xf(x)) = x has at least 20122012 real solutions xx. What is nn?
<spanclass=latexbold>Note:</span><span class='latex-bold'>Note:</span> the fractional part of xx is a real number y={x}y= \{x\}, such that 0y<1 0 \le y < 1 and xyx-y is an integer.
<spanclass=latexbold>(A)</span> 30<spanclass=latexbold>(B)</span> 31<spanclass=latexbold>(C)</span> 32<spanclass=latexbold>(D)</span> 62<spanclass=latexbold>(E)</span> 64 <span class='latex-bold'>(A)</span>\ 30\qquad<span class='latex-bold'>(B)</span>\ 31\qquad<span class='latex-bold'>(C)</span>\ 32\qquad<span class='latex-bold'>(D)</span>\ 62\qquad<span class='latex-bold'>(E)</span>\ 64