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.

The answer is accepted.