Prove that at any time there are two opposite points along the Equator, which have exactly the same temperature. Assume the temperature function varies continuously as you move along the Equator.

Counterargument: This is patently impossible. If there are such points on the Equator, there must also be similar points on any circle around the Earth, such as a meridian. But in that case, we'd have one point in the north hemisphere, in winter, and the other in the south, in summer; that doesn't make sense!

What's wrong with this reasoning?

(In reply to Solution by David)

Even if there were a perfect symmetry (or anti-symmetry if you like) between winter and summer, that would merely mean that the points of equal temperature on a meridian would be located where that meridian meets the equator.  Deviations from this perfect asymmetry would just shift these points away from the equator.  The major, regular, asymmetry of course is that of day/night--it being coolest just before dawn and warmest in the late afternoon, so due to the irregularities, it makes quite a bit of sense that one point would be north of the equator and another south.  And lacking any irregularities, there's still left the possibility that the two points on the meridian do lie directly on the equator.  In fact, as 24 hours goes by, the points probably shift as to which one is north of the equator and which one south, and twice a day probably pass through the equator.

Posted by Charlie on 2005-03-28 13:35:38