KVPY 2015 8

$\begin{aligned} 2^a + 4^b + 8^c & = 328 \\ \Rightarrow 2^a + 2^{2b} + 2^{3c} & = 328 \quad \quad \quad \quad \quad \quad \small \color{#3D99F6}{\text{We note that 328 is a sum of powers of 2.}} \\ & = 256 + 72 \\ & = 256 + 64 + 8 \\ & = 2^8 + 2^6 + 2^3 \end{aligned}$

Equating the exponents, we note that $a + 2b + 3c = 8+6+3 = 17$ .

For integers solutions, $\Rightarrow \begin{cases} a = 8 & b = 3 & c=1 & \Rightarrow abc = 24 \\ a = 6 & b = 4 & c = 1 & \Rightarrow abc = 24 \\ a = 3 & b = 4 & c = 2 & \Rightarrow abc = 24 \end{cases}$

$\Rightarrow \dfrac{a + 2b + 3c}{abc} = \boxed{\dfrac{17}{24}}$

I solved and got 17/32