Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_i$ for every $i=\overline{0,4}$?

[hide=thanks ]Thanks to the user Vlados021 for translating the problem.[/hide]
1. **Assume the radii of the circles**: Let the radii of the circles be \( r_0, r_1, r_2, r_3, r_4 \) such that \( r_i = q^i \) for \( i = 0, 1, 2, 3, 4 \). Therefore, the radii are \( 1, q, q^2, q^3, q^4 \).

2. **Consider the broken line \( A_0A_1A_2A_3A_4 \)**: Each segment \( A_iA_{i+1} \) must be equal in length. Let this common length be \( d \).

3. **Distance constraints**: 
   - The distance \( A_0A_1 \) is at most \( 1 + q \) because \( A_0 \) is on the circle of radius 1 and \( A_1 \) is on the circle of radius \( q \).
   - The distance \( A_3A_4 \) is at least \( q^4 - q^3 \) because \( A_3 \) is on the circle of radius \( q^3 \) and \( A_4 \) is on the circle of radius \( q^4 \).

4. **Inequality setup**: For the broken line to be possible, the maximum distance between any two consecutive points must be at least the minimum distance between the farthest points:
   \[
   1 + q \ge q^4 - q^3
   \]

5. **Simplify the inequality**:
   \[
   1 + q \ge q^4 - q^3
   \]
   \[
   q^4 - q^3 - q - 1 \le 0
   \]

6. **Solve the inequality**: To find the maximum value of \( q \), we need to solve the polynomial inequality \( q^4 - q^3 - q - 1 \le 0 \). This can be done by finding the roots of the polynomial and determining the intervals where the inequality holds.

7. **Find the roots**: The polynomial \( q^4 - q^3 - q - 1 \) has a root at \( q = \frac{1 + \sqrt{5}}{2} \), which is the golden ratio \( \phi \approx 1.618 \).

8. **Verify the root**: Substitute \( q = \frac{1 + \sqrt{5}}{2} \) into the polynomial:
   \[
   \left( \frac{1 + \sqrt{5}}{2} \right)^4 - \left( \frac{1 + \sqrt{5}}{2} \right)^3 - \left( \frac{1 + \sqrt{5}}{2} \right) - 1 = 0
   \]
   This confirms that \( q = \frac{1 + \sqrt{5}}{2} \) is indeed a root.

9. **Conclusion**: The maximal value of \( q \) for which it is possible to draw the broken line \( A_0A_1A_2A_3A_4 \) with four equal segments is \( q = \frac{1 + \sqrt{5}}{2} \).

The final answer is \( \boxed{\frac{1 + \sqrt{5}}{2}} \).