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?

Consider the difference between the temperature at a point and at its symmetric point along the Equator. If the difference is zero, you're done. If not, walk along the Equator from the original point to the symmetric: when you arrive, the function will have changed sign, and if it's continuous, it means it must have been zero somewhere. QED.

Posted by e.g. on 2005-03-28 00:27:13