MathDB
Math Prize 2011 Problem 12

Source:

September 19, 2011
floor functionlogarithms

Problem Statement

If xx is a real number, let ⌊x⌋\lfloor x \rfloor be the greatest integer that is less than or equal to xx. If nn is a positive integer, let S(n)S(n) be defined by S(n)=⌊n10⌊log⁡n⌋⌋+10(n−10⌊log⁡n⌋⋅⌊n10⌊log⁡n⌋⌋) . S(n) = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor + 10 \left( n - 10^{\lfloor \log n \rfloor} \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor \right) \, . (All the logarithms are base 10.) How many integers nn from 1 to 2011 (inclusive) satisfy S(S(n))=nS(S(n)) = n?
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