Undergrad algebraic topology proof
Prove the following:
(a) For any continuous map $f:S\rightarrow \mathbb{R}$, there exists a pair of antipodal points which take the same value under $f$.
(b) If $U$ and $V$ are bounded, connected, open subsets of $\mathbb{R^2}$, then there exists a straight line that divides each of $U$ and $V$ in half by area. (You may declare continuity without proof for this problem.)
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Hi Phillip, thank you for answering my question! The response was clear and thorough, so I was wondering if you'd be able to help complete the rest of the questions that I've posted here? Thank you!

I am busy at the moment, but I may have time to look into them later. You should definitely extend your deadline and allow more time. Not many users can answer those abstract questions in short notice.
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