Helly's Theorem states the following: Suppose that $X_1,X_2,...,X_n$ are convex sets in $\mathbb{R}^d$, such that for any $|I|\leq d+1$, $\cap_{i\in I}X_i \neq \emptyset$. Then $\cap_{i=1}^{n}X_i \neq \emptyset$.
I'm looking for interesting problems, the solutions of which use this theorem.
Here is one example: Let $K_1,K_2,...,K_n$ be closed intervals parallel to the $y$ axis. Assume that for any $|I|\leq d+2$ there exists a polynomial of degree at most $d$, the graph of which intersects all $K_i$ where $i\in I$. Show that there exists a polynomial of degree at most $d$, the graph of which intersects all the intervals $K_1,K_2,...,K_n$.