To solve the problem, we start by denoting key variables and using the angle bisector length formula and trigonometric identities.

Step 1: Key Notations

Let (AB = c), (AC = b), (\angle A = 120^\circ), (\angle B = 2x), (\angle C = 2y) (so (x + y = 30^\circ)). We need to find (\frac{b}{c}).

Step 2: Angle Bisector Ratio

The ratio of angle bisectors of (\angle B) and (\angle C) is (\frac{d_b}{d_c} = \frac{1}{2}). Using the angle bisector length formula:
[d_b = \frac{2ac \cos x}{a + c}\quad \text{and}\quad d_c = \frac{2ab \cos y}{a + b}]
Simplifying the ratio (\frac{d_b}{d_c} = \frac{1}{2}) gives:
[ \frac{c(a + b)\cos x}{b(a + c)\cos y} = \frac{1}{2}]

Step 3: Law of Sines

By the Law of Sines: (\frac{b}{c} = \frac{\sin 2x}{\sin 2y}). Assuming (\frac{b}{c}=2) (integer ratio), (\sin 2x = 2\sin 2y). Since (y=30^\circ - x), (\sin 2x = 2\sin(60^\circ - 2x)), which holds for appropriate (x).

Conclusion

The ratio (\frac{AC}{AB} = \frac{b}{c} = 2).

Answer: (\boxed{2})

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