When the cup of water is placed in the boiling water, the cup is cooler than the surrounding water and heat will flow into it. Eventually, the water in the cup will increase to 100 degree C if any more energy goes into the cup, then the cup will begin boiling.
But at this point, the boiling water in the pot and the 100 degree C water in the cup are at the same temperature. Since there is no temperature difference, there will be no more heat flow into the cup, so it will never boil.
Background
Consider the familiar setting of statistical mechanics, an assembly of $N=4$ distinguishable gas particles. Let, the total energy be $4E_0$.
The possible ways energy of sharing energy between the particles gives us $5$ possible distributions $D(i)$:
$$\begin{vmatrix}
& 0 &E_0 & 2E_0 
