LC circuit with series diode (finding the linear average)
We know the impedance of L and C with f given.  

You could compute the peak current on the 1st cycle from Vin/Z(ω), but you don't have to.  

The question was what is the linear average current?    

First you know that the series capacitor blocks DC in the steady state.
Second the diode only allows one polarity of current to flow, so that capacitor eventually will charge from one polarity up to the peak voltage of  Vin(t) then the diode will be blocking all current.

So the average current in the steady state is zero.