A hyperbola in the coordinate plane passing through the points $(2,5)$, $(7,3)$, $(1,1)$, and $(10,10)$ has an asymptote of slope $\frac{20}{17}$. The slope of its other asymptote can be expressed in the form $-\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.

[i]Proposed by Michael Ren[/i]
To solve this problem, we need to find the slope of the other asymptote of the hyperbola. Given that the hyperbola passes through the points \((2,5)\), \((7,3)\), \((1,1)\), and \((10,10)\), and has one asymptote with a slope of \(\frac{20}{17}\), we can use properties of hyperbolas and their asymptotes to find the slope of the other asymptote.

1. **Identify the slopes of the asymptotes:**
   The slopes of the asymptotes of a hyperbola are given by \(\pm \frac{b}{a}\) or \(\pm \frac{a}{b}\), depending on the orientation of the hyperbola. Since we are given one slope as \(\frac{20}{17}\), we can denote the slopes of the asymptotes as \(\frac{20}{17}\) and \(-\frac{m}{n}\).

2. **Use the property of hyperbolas:**
   For a hyperbola, the product of the slopes of the asymptotes is always \(-1\). This is because the asymptotes are perpendicular to each other in the coordinate plane.

   \[
   \left(\frac{20}{17}\right) \left(-\frac{m}{n}\right) = -1
   \]

3. **Solve for the unknown slope:**
   Substitute the given slope into the equation and solve for \(\frac{m}{n}\):

   \[
   \frac{20}{17} \cdot -\frac{m}{n} = -1
   \]

   \[
   \frac{20m}{17n} = 1
   \]

   \[
   20m = 17n
   \]

   \[
   \frac{m}{n} = \frac{17}{20}
   \]

   Therefore, the slope of the other asymptote is \(-\frac{17}{20}\).

4. **Express the slope in the form \(-\frac{m}{n}\):**
   Here, \(m = 17\) and \(n = 20\). These are relatively prime positive integers.

5. **Compute \(100m + n\):**
   Substitute \(m\) and \(n\) into the expression \(100m + n\):

   \[
   100m + n = 100 \cdot 17 + 20 = 1700 + 20 = 1720
   \]

The final answer is \(\boxed{1720}\).