Assuming that the earth is spherical, that its temperature is continuous, and that some other more or less realistic conditions hold, we might think that the Earth's core temperature should be about the average of its surface temperatures.
This is not the case, as the core is hotter than all but a few spots on the surface. Can someone explain where the assumptions break down? Is it that the temperature is not static?
In the limit of very, very long time you can expect that situation to obtain, but
In the time since I first wrote this, David Hammen provided a much more complete discussion of the geothermal heat budget. What you see here leaves off the latent heat of fusions at the inner core boundary and the conversion of gravitational potential to heat as the Earth settles out into density segregated layers.
Your assumption would be accurate in the extreme far future. The surface of a sphere after a long time has a roughly uniform temperature where the heat it absorbs balances with the heat it loses and the temperature would be uniform throughout.
With the Earth though, a huge amount of heat was generated when the planetary disk coalesced. Because the only (essentially only anyways) way the Earth can lose heat energy is through black-body radiation it takes a very long time to shed the excess heat from formation. The Earth is a very large sphere so it has a small ratio of surface area to volume ratio which makes the surface somewhat of a heat loss bottleneck.
Also, there are a few factors that dramatically slow the heat loss. The Earth's atmosphere traps a lot of heat captured by the Sun's radiation. The Sun is adding heat to the Earth so in order for the Earth to cool it must shed all of the heat it's absorbing as well as the extra heat from the core as it slowly makes its way to the surface.
Also, there are many radioactive isotopes in the Earth's core generating large amounts of heat through fission.
You should check out the Wikipedia article on the geothermal gradient.
