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Kindly asking to find the set of possible solutions if they exist of the equation $$\lfloor x+\lfloor x+\lfloor x\rfloor\rfloor\rfloor+3\lfloor x\rfloor=18$$ Of course I have 4 intervals as choices:

  1. $[\frac{5}{2}, \frac{7}{2})$
  2. $[3, 5)\smallsetminus \{4\}$
  3. $[3, 4)$
  4. $\emptyset$

$\lfloor x\rfloor$ is largest integer not greater than $x$. Thank you very much.

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It seems to me that $x=3$ is a solution, while $x=5/2$ is not. This should exclude 1 and 4. – Siminore Jun 24 '12 at 12:48
up vote 7 down vote accepted

Hint: show that if $n$ is an integer, and $r$ is any real number, then $$\lfloor r+n\rfloor=\lfloor r\rfloor+n.$$ Apply this to the expression $\lfloor x+\lfloor x+\lfloor x\rfloor\rfloor\rfloor$.

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I should have considered that [r+n]=[r]+n. Thank you – Ned Dabby Jun 24 '12 at 13:01

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