Floor Functions With Unlucky 13

Algebra Level 4

\[\begin{eqnarray} \text{ Equation 1 }: &&\quad \lfloor x \rfloor + \lfloor 2 x \rfloor + \lfloor 4x \rfloor = 13 \\ \text{ Equation 2 }: &&\quad \lfloor x \rfloor + \lfloor 2 x \rfloor + \lfloor 4x \rfloor+ \lfloor 8x \rfloor = 13 \end{eqnarray} \]

Do the equations above have real solutions?

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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