3-4. From thirty points $A_{1}, A_{2}, \ldots, A_{30}$, located on a straight line $M N$ at equal distances from each other, thirty straight roads extend. These roads are located on one side of the line $M N$ and form the following angles with $M N$:

|  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $\alpha$ | $60^{\circ}$ | $30^{\circ}$ | $15^{\circ}$ | $20^{\circ}$ | $155^{\circ}$ | $45^{\circ}$ | $10^{\circ}$ | $35^{\circ}$ | $140^{\circ}$ | $50^{\circ}$ | $125^{\circ}$ | $65^{\circ}$ | $85^{\circ}$ | $86^{\circ}$ | $80^{\circ}$ |
|  | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| $\alpha$ | $75^{\circ}$ | $78^{\circ}$ | $115^{\circ}$ | $95^{\circ}$ | $25^{\circ}$ | $28^{\circ}$ | $158^{\circ}$ | $30^{\circ}$ | $25^{\circ}$ | $5^{\circ}$ | $15^{\circ}$ | $160^{\circ}$ | $170^{\circ}$ | $20^{\circ}$ | $158^{\circ}$ |

Thirty cars start simultaneously from all thirty points, driving along these roads without turning, at the same speed. A barrier is installed at each intersection. As soon as the first car in time passes an intersection, the barrier closes and blocks the path of all subsequent cars that reach this intersection. Which cars will pass all intersections on their path, and which will get stuck?
Solve 4. Answer: cars with numbers 14, 23, and 24 will not be detained anywhere.

Let $a_{n}$ be the road leading out of point $A_{n}, \alpha_{n}$ be the angle that road $a_{n}$ forms with the line $M N$, and $P_{m n}$ be the intersection of roads $a_{n}$ and $a_{m}$.

(1) If car $a_{m}$ is detained at intersection $P_{n m}$, then angle $\alpha_{n}$ is closer to $90^{\circ}$ than angle $\alpha_{m}$ (road $a_{n}$ is steeper than road $a_{m}$).

(2) If $mq > m$, it is considered similarly.

(5) If car $a_{n}$ passes through all intersections, then all roads intersecting $a_{n}$ are less steep than road $a_{n}$; this follows from (4) and (1).

Thus, from (3) and (5), it follows that car $a_{n}$ passes through all intersections if and only if all roads intersecting $a_{n}$ are less steep than road $a_{n}$. Now it is easy to obtain the required result.