In a mayor/council/manager form of government, who is responsible for the chief administrative functions?
There is no such thing as a mayor/council/manager system of city government. As you can see in this link, there are mayor-council governments and council-manager governments, but there is no such thing as a government that has both an actual mayor (as opposed to a merely titular mayor who is just the presiding member of the council) and a city manager. The reason for this is that mayors and city managers both hold the administrative and executive powers in their respective forms of government.
In a mayor-council government, the mayor is the head of the executive branch. The mayor is elected separately from the council and generally has significant powers in terms of administering and executing the laws. In a council-manager government, there is a person called the mayor. However, this person has no significant executive powers. The mayor in such a system is a member of the city council and not an executive. In this system, the executive and administrative functions are in the hands of the city manager. This person is hired by the council and is supposed to administer the city on a professional basis rather than acting as a politician.
So, there is no form of government known as a mayor/council/manager system. Real mayors and city managers each hold the administrative and executive powers in their own forms of government.