A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$. The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$, where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$. If the number is rational, let $r=1$. 

[i](Ahaan Rungta, 2 points)[/i]
1. The speed of a wave on a string is given by the formula:
   \[
   v = \sqrt{\frac{T}{\mu}}
   \]
   where \( T \) is the tension in the rope and \( \mu \) is the linear mass density of the rope.

2. Initially, the tension in the rope is \( T \) and the velocity of the wave is \( v_1 \):
   \[
   v_1 = \sqrt{\frac{T}{\mu}}
   \]

3. When the tension is halved, the new tension is \( \frac{T}{2} \). The new velocity \( v_2 \) is:
   \[
   v_2 = \sqrt{\frac{\frac{T}{2}}{\mu}} = \sqrt{\frac{T}{2\mu}} = \frac{1}{\sqrt{2}} \sqrt{\frac{T}{\mu}} = \frac{v_1}{\sqrt{2}}
   \]

4. When the tension is halved again, the new tension is \( \frac{T}{4} \). The new velocity \( v_3 \) is:
   \[
   v_3 = \sqrt{\frac{\frac{T}{4}}{\mu}} = \sqrt{\frac{T}{4\mu}} = \frac{1}{2} \sqrt{\frac{T}{\mu}} = \frac{v_1}{2}
   \]

5. We need to find the value of \( \frac{v_1}{v_3} + \frac{v_3}{v_1} \):
   \[
   \frac{v_1}{v_3} = \frac{v_1}{\frac{v_1}{2}} = 2
   \]
   \[
   \frac{v_3}{v_1} = \frac{\frac{v_1}{2}}{v_1} = \frac{1}{2}
   \]
   \[
   \frac{v_1}{v_3} + \frac{v_3}{v_1} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}
   \]

6. The expression \( \frac{5}{2} \) can be written as \( \frac{m\sqrt{r}}{n} \) where \( m = 5 \), \( r = 1 \), and \( n = 2 \). Since \( r = 1 \), it is rational.

7. Therefore, \( m + n + r = 5 + 2 + 1 = 8 \).

The final answer is \( \boxed{8} \).