10. Given a regular 2017-gon inscribed in a unit circle, let $S$ be the set of distances between any two distinct vertices of the 2017-gon, and $Q=\prod_{s \in S} s$. Then $Q^{2}=$ $\qquad$
10. 2017.

Let \( n = 2017 = 2m + 1 \). Suppose the regular \( n \)-sided polygon inscribed in the unit circle is \( P_0 P_1 \cdots P_{2m} \). Place the regular polygon in the complex plane with the center at the origin, and the vertices correspond to the complex numbers \( P_0 = 1, P_k = \omega^k (k = 1, 2, \cdots, 2m) \), where
\[
\omega = \cos \frac{2\pi}{2m+1} + i \sin \frac{2\pi}{2m+1}.
\]
Then,
\[
Q = \prod_{1 \leq i \leq m} \left| \omega^i - 1 \right|
= \prod_{1 \leq i \leq m} \left| \bar{\omega}^i - 1 \right| = \prod_{1 \leq i \leq m} \left| \omega^{2m+1-i} - 1 \right|,
\]
\[
Q^2 = \left( \prod_{1 \leq i \leq m} \left| \omega^i - 1 \right| \right) \left( \prod_{1 \leq i \leq m} \left| \omega^{2m+1-i} - 1 \right| \right)
= \prod_{1 \leq j \leq 2m} \left| \omega^j - 1 \right|.
\]

Since \( 1, \omega, \cdots, \omega^{2m} \) are the roots of the equation \( x^{2m+1} - 1 = 0 \), and \( \omega, \cdots, \omega^{2m} \) are the roots of the equation \( x^{2m} + x^{2m-1} + \cdots + x + 1 = 0 \), then
\[
x^{2m} + x^{2m-1} + \cdots + x + 1
= (x - \omega)(x - \omega^2) \cdots (x - \omega^{2m}).
\]
Let \( x = 1 \), we get
\[
2m + 1 = (1 - \omega)(1 - \omega^2) \cdots (1 - \omega^{2m})
= |1 - \omega| |1 - \omega^2| \cdots |1 - \omega^{2m}| = Q^2.
\]

Thus, \( Q^2 = 2017 \).