`1/(x+2y)+1/(2y+3z)+1/(x+3z)=(12x)/(2y+3z)+(12.2y)/(x+3z)+(12.3z)/(x+2y)=2016`
`⇒(12x)/(2y+3z)+12+(12.2y)/(x+3z)+12+(12.3z)/(x+2y)+12=2052`
`⇔(12(x+2y+3z))/(x+2y)+(12(x+2y+3z))/(2y+3z)+(12(x+2y+3z))/(x+3z)=2052`
`⇔12(x+2y+3z)(1/(x+2y)+1/(2y+3z)+1/(x+3z))=2052`
`⇔(x+2y+3z).12.2016=2052`
`⇔S=(2052)/(2016.12)`
`⇔S=(343)/(4044)`