Solution

At the initial moment, when the iceboat is stationary, the air moves relative to it with a velocity $\vec{v}$.

As the iceboat starts moving and acquires its own velocity $\vec{v}_i$, the relative velocity of the wind changes. According to the velocity addition formula, the relative velocity of the wind is:
$$\vec{v}_{rel} = \vec{v} - \vec{v}_i$$

Since the absolute wind velocity $\vec{v}$ is perpendicular to the iceboat's velocity $\vec{v}_i$, these vectors form a right-angled triangle.

The wind will continue to exert pressure on the sail and accelerate the iceboat as long as the relative wind velocity $\vec{v}_{rel}$ pushes against it. The iceboat reaches its maximum speed when the wind stops exerting force on the sail. This condition is met when the relative wind velocity vector $\vec{v}_{rel}$ becomes parallel to the sail.

According to the problem, the sail is set at an angle of $30^\circ$ to the direction of the iceboat's motion. Therefore, in the limiting case, the angle between the vectors $\vec{v}_i$ and $\vec{v}_{rel}$ in our right-angled triangle is exactly $30^\circ$.

From this velocity vector triangle, we can express the relationship:
$$\tan 30^\circ = \frac{v}{v_i}$$

Solving for the maximum speed of the iceboat $v_i$:
$$v_i = \frac{v}{\tan 30^\circ} = v \cot 30^\circ$$

Since $\cot 30^\circ = \sqrt{3}$, we obtain:
$$v_i = v\sqrt{3}$$