A Pre-RMO question! -20

$127^{x_{124}}=128$ $(126^{x_{123}})^{x_{124}}=126^{x_{123}x_{124}}=128$ $(125^{x_{122}})^{x_{123}x_{124}}=128$ Doing this many times you get $4^{x_1x_2x_3x_4...x_{123}x_{124}}=128$ $(2^2)^{x_1x_2x_3x_4...x_{123}x_{124}}=2^{2(x_1x_2x_3x_4...x_{123}x_{124})}=2^7$ $\Rightarrow 2{x_1x_2x_3x_4...x_{123}x_{124}}=7$ ${x_1x_2x_3x_4...x_{123}x_{124}}=\frac{7}{2}$ $\gcd(7,2)=1;2\in Z ; 7\in Z$ $7+2=\boxed{9}$

$4^{x_1} = 5 \implies x_1 =\log_{4}5, 5^{x_2} = 6 \implies x_2 =\log_{5}6 \cdots 127^{x_124} = 128 \implies x_{124} =\log_{127}128$

$\text{Product} = x_1 \times x_2 \times \cdots \times x_{124}$

$\text{Product} = \log_{4}5 \times \log_{5}6 \times \cdots \times \log_{127}128$

$\text{Product} = \dfrac{\log(5)}{\log(4)} \times \dfrac{\log(6)}{\log(5)} \times \cdots \times \dfrac{\log(128)}{\log(127)}$

$\text{Product} = \dfrac{\cancel{\log(5)}}{\log(4)} \times \dfrac{\cancel{\log(6)}}{\cancel{\log(5)}} \times \cdots \times \dfrac{\log(128)}{\cancel{\log(127)}}$

$\text{Product} = \dfrac{\log(128)}{\log(4)} =\log_{4}128$

$\text{Product} = \dfrac{7}{2} \implies m=7 , n=2$

$\implies m+n=7+2=\boxed{9}$