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Found problems: 1

MIPT Undergraduate Contest 2019, 2.3
Tags: geometry , rectangle , topology , mappings , functions
Let $A$ and $B$ be rectangles in the plane and $f : A \rightarrow B$ be a mapping which is uniform on the interior of $A$, maps the boundary of $A$ homeomorphically to the boundary of $B$ by mapping the sides of $A$ to corresponding sides in $B$. Prove that $f$ is an affine transformation.