Find all real solutions of the equation:
$$[x]+[2x]+[4x]+[8x]+[16x]+[32x]=12345$$ Where $[.]$ is greatest integer fuction.
2026-03-29 19:58:23.1774814303
Find all real solutions to $[x]+[2x]+[4x]+[8x]+[16x]+[32x]=12345$
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Hint: Write $x$ in binary as $n + 0.b_1b_2b_3b_4b_5b_6\ldots$, where $b_i \in \{0,1\}$ are binary digits, and $n$ is an integer. You should be able to compute the expression, and it should only depend on $n, b_1, b_2, b_3, b_4, b_5$. Then set it equal to $12345$.