Borsuk-Ulam theorem

Theorem. Every continuous function from an n-sphere to n-dimensional Euclidean space maps some pair of antipodal points to the same point: $$f(x)=f(-x).$$

It "follows" that at any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures (or equal barometric pressures etc).

The Borsuk–Ulam theorem implies the Brouwer fixed point theorem.