Gauss-Lucas Theorem
The Gauss-Lucas Theorem is an interesting theorem relating the roots of a polynomial P to the roots of its derivative P’.
Specifically, it states that the roots of P’ all lie within the convex hull of the roots of P.
The Borsuk-Ulam Theorem and Stolen Necklaces
This 3b1b video overviews the Borsuk-Ulam Theorem, and its applications for the Necklace Splitting problem.
It’s a super nice connection between topology and combinatorics.