## Task A-1.6.

In triangle $A B C$, the measure of angle $\varangle A B C$ is $120^{\circ}$, and $|A B|=6$ and $|B C|=9$. Let points $P$ and $Q$ be on side $\overline{A C}$ such that triangle $B P Q$ is equilateral. Determine $|P Q|$.
Scoring: The student's solution should be scored exclusively according to the single scoring scheme that yields the higher score.

In the first solution, 1 point is awarded for recognizing the similarity of triangles $ABC$ and one of the triangles $BCQ$ and $APB$ based on equal angles, and another 2 points for highlighting the similarity of all three triangles, i.e., the similarity of triangles $BCQ$ and $APB$. Another 2 points are awarded for writing the necessary ratios for each of these two pairs. Then, 1 point is awarded for expressing the lengths $y$ and $z$ in terms of $x$, 1 point for setting up an equation with a single unknown, and 1 point for the correct solution.

In the second solution, 1 point is awarded for drawing point $D$, and 1 point each for $|DB|$ and $|AD|$ (it is not necessary to use the Pythagorean theorem for the height of an equilateral triangle). Additionally, 1 point is awarded for calculating $|AC|$. Introducing point $N$ and connecting the height $|BN|$ to the side $|PQ|$ earns 1 point. The idea that the length of the height $|BN|$ can be calculated using the area earns 2 points, and the calculation itself earns 1 point. The final solution earns 1 point.

The final solution is recognized in any equivalent form, e.g., $\frac{54}{\sqrt{171}}$ or $\frac{18 \sqrt{19}}{19}$.