An algebra problem by Saurabh Patil

The strategy is to use the weighted AM-GM inequality . Assign coefficients to the variables that correspond to the exponents:

$\begin{array}{l} a \Rightarrow \frac{1}{3} \\ b \Rightarrow \frac{1}{2} \\ c \Rightarrow \frac{1}{4} \end{array}$

Now weighted AM-GM inequality:

$\begin{aligned} \frac{3\left(\frac{1}{3}a\right)+2\left(\frac{1}{2}b\right)+4\left(\frac{1}{4}c\right)}{3+2+4} &\ge \sqrt[3+2+4]{\left(\frac{1}{3}a\right)^3 \left(\frac{1}{2}b\right)^2 \left(\frac{1}{4}c\right)^4} \\ \\ \frac{a+b+c}{3^2} &\ge \sqrt[9]{\frac{a^3b^2c^4}{2^{10} \cdot 3^3}} \\ \\ \frac{5^9}{3^{18}} &\ge \frac{a^3b^2c^4}{2^{10} \cdot 3^3} \\ \\ 2^{10} \cdot 3^{-15} \cdot 5^9 &\ge a^3b^2c^4 \end{aligned}$

Thus, the maximum is $2^{10} \cdot 3^{-15} \cdot 5^9,$ and $x+y+z=\boxed{4}.$