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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