A generalization of the Borsuk Ulam theorem Is there a  compact  $n$-dimensional manifold $M$ or, more generaly,  a compact $n$-dimensional topological space  $M$  with the  following  property?
"For every  continuous  map $f:M \to \mathbb{R}^{n}$ there  are  points  $a,b,c \in M $  with $f(a)=f(b)=f(c)$."
This  is  motivated  by the following  obvious consequence of the   Borsuk-Ulam theorem:
"For every  continuous  map  $f:S^n \to \mathbb{R}^n$ there  are points $a, b \in S^n$  with $f(a)=f(b)$." 
 A: If M is allowed to be a simplicial complex, five $n$-simplices with a common $(n-1)$-dimensional face should do the job. If $M$ should be a manifold, take any closed non-orientable one.
The multiplicity of maps between manifolds can be studied with the help of characteristic classes, see this preprint of Roman Karasev and this preprint of Roman Karasev and Pavle Blagojevic. For example, any continuous map $\mathbb{R}\mathrm{P}^4 \to \mathbb{R}^4$ sends some four points in $\mathbb{R}\mathrm{P}^4$ to the same point in $\mathbb{R}^4$.
Another generalization of the Borsuk-Ulam theorem (more close to it in the spirit) deals with spaces with a group action (like $\mathbb{Z}_2$-action on the sphere) such that for any map to $\mathbb{R}^n$ some orbit is sent to a point. This is discussed in detail in Section 6 of
Matoušek, Jiří, Using the Borsuk-Ulam theorem. Lectures on topological methods in combinatorics and geometry. Written in cooperation with Anders Björner and Günter M. Ziegler, Universitext. Berlin: Springer (ISBN 978-3-540-00362-5/pbk; 978-3-540-76649-0/ebook). xii, 214~p. (2008). ZBL1234.05002.
