3. Let $D$ be the midpoint of the arc $\widehat{A B}$ of the circumcircle of triangle $A B C$, which does not contain point $C$. Express the length of the segment $A D$ in terms of $a=$ $|B C|, b=|A C|$, and $c=|A B|$.
III/3. The angle $\angle A C D$ in $\angle D C B$ is subtended by equal chords, so they are congruent and $C D$ is the angle bisector of $\angle A C B=\gamma$. Therefore, $\angle D C B=\frac{\gamma}{2}$ and thus $\angle D A B=\frac{\gamma}{2}$. Let $E$ be the midpoint of side $A B$. Then $|A E|=\frac{c}{2}$, and therefore

$$
\begin{aligned}
|A D| & =\frac{c}{2 \cos \frac{\gamma}{2}}=\frac{c}{2 \sqrt{\frac{1+\cos \gamma}{2}}}= \\
& =\frac{c}{\sqrt{2\left(1+\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right)}}=c \sqrt{\frac{a b}{(a+b+c)(a+b-c)}}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_06_07_e3e00bf556075403c4d7g-09.jpg?height=494&width=466&top_left_y=1363&top_left_x=1383)

Conclusion $|A D|=\frac{c}{2 \cos \frac{\gamma}{2}}$ (or an equivalent expression involving $a, b, c$ and one angle of triangle $A B C$): 4 points. Simplification of the above expression and writing $|A D|$ (in any form) only in terms of $a, b, c$: 3 points.

If the contestant expresses $|A D|$ in terms of $a, b, c$ and two angles or some other dependent quantity, but does not simplify the expression: up to 2 points.