Consider the set of all triangles $ OPQ$ where $ O$ is the origin and $ P$ and $ Q$ are distinct points in the plane with nonnegative integer coordinates $ (x,y)$ such that $ 41x\plus{}y \equal{} 2009$. Find the number of such distinct triangles whose area is a positive integer.
1. **Identify the points on the line:**
   We are given the equation \(41x + y = 2009\). We need to find all points \((x, y)\) with nonnegative integer coordinates that satisfy this equation. 

   Since \(y = 2009 - 41x\), \(x\) must be an integer such that \(0 \leq 41x \leq 2009\). This implies:
   \[
   0 \leq x \leq \left\lfloor \frac{2009}{41} \right\rfloor = 49
   \]
   Therefore, \(x\) can take any integer value from 0 to 49, inclusive.

2. **Calculate the distance from the origin to the line:**
   The distance \(D\) from the origin \((0,0)\) to the line \(41x + y = 2009\) is given by the formula:
   \[
   D = \frac{|41(0) + 1(0) - 2009|}{\sqrt{41^2 + 1^2}} = \frac{2009}{\sqrt{1682}} = \frac{2009}{41\sqrt{2}}
   \]

3. **Determine the coordinates of points \(P\) and \(Q\):**
   Let \(P = (x_1, 2009 - 41x_1)\) and \(Q = (x_2, 2009 - 41x_2)\) be two distinct points on the line. 

4. **Calculate the distance between points \(P\) and \(Q\):**
   The distance \(L\) between \(P\) and \(Q\) is:
   \[
   L = \sqrt{(x_2 - x_1)^2 + (41x_1 - 41x_2)^2} = \sqrt{(x_2 - x_1)^2 + 41^2(x_2 - x_1)^2} = \sqrt{(1 + 41^2)(x_2 - x_1)^2} = 41\sqrt{2}|x_2 - x_1|
   \]

5. **Calculate the area of triangle \(OPQ\):**
   The area of triangle \(OPQ\) is given by:
   \[
   [\triangle OPQ] = \frac{1}{2} \times D \times L = \frac{1}{2} \times \frac{2009}{41\sqrt{2}} \times 41\sqrt{2}|x_2 - x_1| = \frac{2009}{2} |x_2 - x_1|
   \]
   For the area to be a positive integer, \(\frac{2009}{2} |x_2 - x_1|\) must be an integer. This implies that \(|x_2 - x_1|\) must be even.

6. **Determine the valid pairs \((x_1, x_2)\):**
   Since \(|x_2 - x_1|\) must be even, \(x_1\) and \(x_2\) must have the same parity. We can split the possible values of \(x\) into two sets:
   \[
   S_1 = \{0, 2, 4, \ldots, 48\} \quad \text{(even numbers)}
   \]
   \[
   S_2 = \{1, 3, 5, \ldots, 49\} \quad \text{(odd numbers)}
   \]
   Each set contains 25 elements. The number of ways to choose 2 distinct elements from a set of 25 is:
   \[
   \binom{25}{2} = \frac{25 \times 24}{2} = 300
   \]
   Since we have two sets, the total number of valid pairs \((x_1, x_2)\) is:
   \[
   2 \times 300 = 600
   \]

Conclusion:
\[
\boxed{600}
\]