Is it possible to heat an artificial lake using the temperatures from deep bore holes? If one were to drill a deep hole similar to the Kola bore hole, reaching temperatures around $200 \ ^\circ C$, could it be used to heat up the water in an artificial lake? The way I imagine it working, would be for the water from the lake to flow down into the hole, where it will get heated up by the temperature at that great depth and then hot water would flow back up due to convection.
 A: This is an interesting question and I think the answer is not necessarily obvious. There are people interesting in extracting geothermal energy and using earth as either a heat sink or a heat source. For Deep Boreholes the interest is as a heat source. One concept is shown below, and for these type of concepts instead of natural convection, the fluid is injected hotter fluid is brought to the surface. In Iceland an Hawaii, they also know where the pools of magma located near the surface are so they can try to take advantage of that.

However, and probably more so in the case where there is not magma nearby it seems that there is a limitation in how fast the heat can be transferred from the surrounding rocks to the rock in contact with the bore hole. Since the thermal conductivity of dry rock is pretty low, and the water in the bore hole can remove heat fairly quickly (especially if pumped) that you may reach some sort of equilibrium.
So I think the answer to your question depends a lot on the details. How deep the borehole is, diameter of the hole, the type rock, is magma nearby, if the bore hole is not that deep is groundwater moving past the bore hole etc.
But the general concept of drilling down and using the earth as heat sink or source is used all the time for geothermal heat pumps in lots of places where the rock temperature is warmer than the air temperature in the winter and cooler than the air temperature in the summer. But for your lake heating question it seems like it could be more complicated.
