# Floor Functions With Unlucky 13

Algebra Level 5

\begin{aligned} \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{aligned}

Do the equations above have real solutions?

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.

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