Assuming that temperature varies continuously, prove that there are always two opposite points on the Earth's surface that have the same temperature.
Aritro Pathak: consider any great circle.. T(x) is the temp at the point x .. let f(x)=T(x)-T(x+pi), then f(0)=T(0)-T(pi)..f(pi)=T(pi)-T(2pi)=T(pi)-T(0) then f(0) and f(pi) have different signs, so using mean value theorem, you have that f is 0 at some point.
FunFact: There are uncountable number of such pairs.
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An alternate version of this puzzle is the Mountain Man