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(A) Let the vertices of the regular $10$-gon be $z_k = e^{i \frac{2k\pi}{10}}$ for $k = 0, 1, \ldots, 9$. Fix the vertex at $z_0 = 1$. The lengths of

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$10$

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(A) Let the vertices of the regular $10$-gon be $z_k = e^{i \frac{2k\pi}{10}}$ for $k = 0, 1, \ldots, 9$. Fix the vertex at $z_0 = 1$. The lengths of the chords are $l_k = |1 - z_k| = |1 - e^{i \frac{2k\pi}{10}}| = 2 \sin \frac{k\pi}{10}$ for $k = 1, 2, \ldots, 9$.We need to calculate the product $P = \prod_{k=1}^{9} 2 \sin \frac{k\pi}{10}$.Using the identity $\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} = \frac{n}{2^{n-1}}$,we have:$P = 2^9 \prod_{k=1}^{9} \sin \frac{k\pi}{10} = 2^9 	imes \frac{10}{2^{10-1}} = 2^9 	imes \frac{10}{2^9} = 10$.

Consider a regular $10$-gon with its vertices on the unit circle. With one vertex fixed,draw straight lines to the other $9$ vertices. Call them $L_1, L_2, \ldots, L_9$ and denote their lengths by $l_1, l_2, \ldots, l_9$ respectively. Then,the product $l_1 \times l_2 \times \ldots \times l_9$ is

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