Beware: Wet Floor!

Relevant Wiki: Hermite's Identity

Let $x= a + r$ where $a \in \mathbb{Z}$ and $0 \leq r < 1$

$\implies$ $\lfloor x \rfloor = a$

$\implies$ $\lfloor 2x \rfloor = \lfloor x \rfloor + \lfloor x + \dfrac{1}{2} \rfloor = 2a$

if $0 \leq r < \frac{1}{2}$

or

$\lfloor 2x \rfloor = \lfloor x \rfloor + \lfloor x + \dfrac{1}{2} \rfloor = 2a+1$

if $\frac{1}{2} \leq r < 1$

Case 1: if $0 \leq r < \frac{1}{2}$

The equation becomes,

$\dfrac{2}{a+r} = \dfrac{1}{a} + \dfrac{1}{2a}$

$\implies a=3r$

If $a \geq 2$ , then $r \geq \dfrac{2}{3} > \dfrac{1}{2}$ which is impossible

So $a=1$ and $r=\dfrac{1}{3}$ giving us $x=\dfrac{4}{3}$

Case 2: if $\frac{1}{2} \leq r < 1$

Then the equation becomes

$\dfrac{2}{a+r} = \dfrac{1}{a} + \dfrac{1}{2a+1}$

$\implies a^2 +a=3ar +r$

If $a \geq 3$ , then $r \geq \dfrac{6}{5} > 1$ which is impossible

So we are left with $a=1$ and $a=2$

If $a=1$ then $r=\dfrac{1}{2}$ and $x=\dfrac{3}{2}$

If $a=2$ then $r=\dfrac{6}{7}$ and $x=\dfrac{20}{7}$

$\therefore$ The largest value is $x=\dfrac{20}{7}$ and $p+q=\boxed{27}.$

$\begin{aligned} \frac 2x & = \frac 1{\lfloor x \rfloor} + \frac 1{\lfloor 2x \rfloor} \\ & = \frac {\lfloor 2x \rfloor + \lfloor x \rfloor}{\lfloor x \rfloor\lfloor 2x \rfloor} \\ 2 \lfloor x \rfloor \lfloor 2x \rfloor & = {\color{#3D99F6}x} \left(\lfloor 2x \rfloor + \lfloor x \rfloor\right) & \small \color{#3D99F6} \text{Using }x = \lfloor x \rfloor + \{x\} \\ & = {\color{#3D99F6}\left( \lfloor x \rfloor + \{x\}\right)} \left(\lfloor 2x \rfloor + \lfloor x \rfloor\right) & \small \color{#3D99F6} \text{where }\{x\} \text{ is the fractional part of }x. \\ \lfloor x \rfloor \lfloor 2x \rfloor - \lfloor x \rfloor ^2 & = \{x\} \left(\lfloor 2x \rfloor + \lfloor x \rfloor\right) \\ \implies \{x\} & = \frac {\lfloor x \rfloor \left(\lfloor 2x \rfloor - \lfloor x \rfloor\right)}{\lfloor 2x \rfloor + \lfloor x \rfloor} \end{aligned}$

Note that $0 \le \{x\} < 1$ . And the largest possible $x$ is when $\{x\} < 1$ :

$\begin{aligned} \frac {\lfloor x \rfloor \left(\lfloor 2x \rfloor - \lfloor x \rfloor\right)}{\lfloor 2x \rfloor + \lfloor x \rfloor} & < 1 \\ \lfloor x \rfloor \left(\lfloor 2x \rfloor - \lfloor x \rfloor\right) & < \lfloor 2x \rfloor + \lfloor x \rfloor \\ \implies x & < 3 \end{aligned}$

For $x$ slightly less than 3,

$\begin{aligned} \frac 2x & = \frac 12 + \frac 15 = \frac 7{10} \\ \implies x & = \frac {20}7 \end{aligned}$

Therefore, $p+q = 20+7 = \boxed{27}$ .