This one is tricky, but super cool!

Everyone should agree that there is at least one place, call it $A$, on the Equator (and certainly lots of places) where the temperature at noon is high while the point diametrically opposed, call it $B$, has a lower temperature (as it is in the dark while the first point is in the noon sun) and then 12 hours later, the temperature at point $A$ is lower than the temperature at point $B$. (This may not happen at every pair of diametrically opposed points, but that's okay.)

Now, to keep time straight, let's assume that $t=0$ corresponds to noon at location $A$ and roughly midnight (allowing for weird time zones) at location $B$. As with the "monk" problem and the "height-weight" problem, let's come up with some names for some continuous functions. Let $T_A(t)$ be temperature (in Celsius) at time $t$ (in hours), and similarly, let $T_B(t)$ be temperature (also in Celsius) at time $t$ (in hours). Both of these functions are continuous. Let $f(t)=T_A(t)-T_B(t)$, which is continuous since the difference of two continuous functions is continuous. What can you say about $f(0)$? What can you say about $f(12)$? What does the Intermediate Value Theorem tell us?