I did find this problem a little interesting too and the solution isnt too difficult, although i did wonder why you would want to sum the voltages around that loop.
One test is worth a thousand expert opinions, but one expert
Originally I was asked to determine the Thevenin equivalent of the circuit shown in 1.pdf below.
Then I have my own question: is there really no "current" flowing through C1 when terminals A, B is short-circuited? I doubt it.
This is how I think: referring to fig.1 in 1.pdf, there is an alternating voltage at node N1, electrons in wires S1 and S2 must be forced to move back and forth between the upper plate of C1 and S1, and between the lower plate and S2, hence the charge on the plates is also changing, resulting in voltage across C1. It might be difficult to measure, small enough to be neglected, but it exists I believe.
So I put it into simulation as shown in fig2.
V1= sin(2*pi*f*t +90 degrees)
f=50Hz
R1=200 ohms
R2=100 ohms
C1=1mF, assume initially uncharged
switch U3 is closed at t=0
switch U1 is open at arbitrarily t=20ms
do a transient analysis
If we focus on the current through C1 AFTER switch U1 is open, we see "current" flowing back and forth even it is extremely small, as shown fig3.
Maybe I should have asked my question like this: if I apply KVL around the left loop of the citcuit shown in fig1 in 1.pdf, the voltage across C1 can be neglected, but really there is an extremely small alternating voltage across that capacitor C1, do you agree?