So Many Floor Functions!

A better solution from Kushagra Sahni

Let $f(x) = x\lfloor x \lfloor x \lfloor x \rfloor \rfloor \rfloor$ , then when $x=3$ , $f(3) = 3^4 = 81$ . Similarly when $x=4$ , $f(4)=4^4=256$ , therefore, the solution $x$ is $\in (3,4)$ . Now, let $x=3+\dfrac 1y$ , where $y$ is a natural number larger than 1. Then, we have:

$\begin{aligned} x\lfloor x \lfloor x \lfloor x \rfloor \rfloor \rfloor & = 88 \\ \lfloor x \lfloor x \lfloor x \rfloor \rfloor \rfloor & = \color{#3D99F6}\frac {88}x \end{aligned}$

We note that $\color{#3D99F6}\dfrac {88}x$ is an integer. Let the integer be $p$ , then:

$\begin{aligned} \frac {88}x & = p \\ px & = 88 \\ p \left(3+\frac 1y\right) & = 88 \\ {\color{#3D99F6}\frac py} (3y + 1) & = 88 & \small \color{#3D99F6} \frac py \text{ is an integer and let it be }q \\ {\color{#3D99F6}q} (3y + 1) & = 88 \end{aligned}$

We note that $q$ can only be 1, 2, 4, 8 and 11, when $y = 29, \dfrac {43}3, 7, \dfrac {10}3$ and $\dfrac 73$ respectively. There are only two cases when $y$ is an integer 29 and 7. We note that:

$\begin{aligned} f\left(3+\frac 1{29}\right) & = \left(3+\frac 1{29}\right) \left \lfloor 3+\frac 1{29} \left \lfloor 3+\frac 1{29} \left \lfloor 3+\frac 1{29} \right \rfloor \right \rfloor \right \rfloor \approx 81.931 & \small \color{#D61F06} \text{Not a solution} \end{aligned}$

$\begin{aligned} f\left(3+\frac 17 \right) & = \left(3+\frac 17 \right) \left \lfloor 3+\frac 17 \left \lfloor 3+\frac 17 \left \lfloor 3+\frac 17 \right \rfloor \right \rfloor \right \rfloor = 88 & \small \color{#3D99F6} \text{The solution} \end{aligned}$

Therefore, $x = 3 + \dfrac 17 = \dfrac {22}7$ $\implies m+n = 22+7 = \boxed{29}$

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My solution

Let $f(x) = x\lfloor x \lfloor x \lfloor x \rfloor \rfloor \rfloor$ , then when $x=3$ , $f(3) = 3^4 = 81$ . Similarly when $x=4$ , $f(4)=4^4=256$ , therefore, the solution $x$ is $\in (3,4)$ . Then, we have:

$\begin{aligned} f(x) & = x\lfloor x \lfloor x \lfloor x \rfloor \rfloor \rfloor \\ & = (3+\{ x \}) \lfloor (3+\{ x \}) \lfloor (3+\{ x \}) \lfloor (3+\{ x \}) \rfloor \rfloor \rfloor \\ & = (3+\{ x \}) \lfloor (3+\{ x \}) \lfloor 9+3\{ x \} \rfloor \rfloor & \small \color{#3D99F6} \text{Put }\{x\} = \frac 13\\ f \left( 3\frac 13 \right) & = \left(3+\frac 13 \right) \left \lfloor \left(3+\frac 13 \right) \lfloor 9+1 \rfloor \right \rfloor \\ & = \frac {10}3 \left \lfloor 30+\frac {10}3 \right \rfloor = 110 > 81 \\ \implies \{x\} & < \frac 13 \\ \implies f \left(3 + {\color{#3D99F6}\{x\}} \right) & = (3+{\color{#3D99F6}\{x\}}) \lfloor 27+9{\color{#3D99F6}\{x\}} \rfloor & \small \color{#3D99F6} \text{Let } \{x\} = \frac 29 \\ f \left(3 + \frac 29 \right) & = \left(3+\frac 19 \right) (27+2) \\ & = \frac {28 \times 29}9 \approx 90.222 > 88 \\ \implies \{x\} & < \frac 29 \\ \implies f \left(3 + \{x\} \right) & = (3+\{x\}) \lfloor 27+{\color{#D61F06}1} \rfloor \\ \implies 28(3+\{x\}) & = 88 \\ 3+\{x\} & = \frac {88}{28} \\ \implies x & = \frac {22}7 \end{aligned}$

$\implies mn = 22 \times 7 = \boxed{154}$