$10 \cdot 98$ Find the maximum positive integer $n$, such that there exist $n$ integer points in the coordinate plane, where any 3 points form a triangle, and the centroid of the triangle is not an integer point.
[Solution] Let the points $\left(x_{1}, y_{1}\right) ;\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)$ be 3 integer points, then the coordinates of the centroid of the triangle formed by them are
$$
\left(\frac{1}{3}\left(x_{1}+x_{2}+x_{3}\right), \frac{1}{3}\left(y_{1}+y_{2}+y_{3}\right)\right) .
$$

It is easy to verify that the following 8 integer points
$$
(0,0),(0,3),(3,1),(3,4),(1,0),(1,3),(4,1),(4,4) \text { form a triangle with any 3 }
$$
points and the centroid of the triangle is not an integer point. Therefore, the maximum positive integer $n \geqslant 8$.

On the other hand, we will prove that any 9 integer points do not satisfy the condition in the problem. If not, suppose there are 9 integer points that satisfy the condition. For an integer point $(x, y)$, let the remainders of $x, y$ when divided by 3 be $r_{1}, r_{2}$, where $r_{1}$ and $r_{2}$ are non-negative integers, then the integer point $(x, y)$ is called of type $\left(r_{1}, r_{2}\right)$. Thus, we divide all integer points on the plane into 9 categories. Since the centroid of a triangle formed by 3 points of the same type is also an integer point, any 3 points among the 9 points cannot belong to the same category. Therefore, these 9 points must belong to at least 5 of the 9 categories. Select 1 point from each of the 5 categories, we get 5 points that belong to different categories, and these 5 points have only 3 different values for their x-coordinates: $0,1,2$. If 3 points among the 5 points have the same x-coordinate, then because they belong to different categories, their y-coordinates must be $0,1,2$, thus the centroid of the triangle formed by these 3 points is an integer point, which is impossible. Therefore, the x-coordinates of the 5 points must take 3 different values, and the number of points for each value must be $2,2,1$. Since whether the centroid is an integer point is invariant under translation, we can assume that the point corresponding to the number 1 is $(0,0)$, then the x-coordinates of the other 4 points are two 1s and two 2s. The 6 categories of points with x-coordinates 1 and 2 can be divided into 3 groups
$$
\{(1,1),(2,2)\},\{(1,0),(2,0)\},\{(1,2),(2,1)\} .
$$

By the pigeonhole principle, among the other 4 points, there must be two points belonging to the same group. These two points, together with the point $(0,0)$, form a triangle whose centroid is an integer point, which is a contradiction.
In conclusion, the maximum positive integer $n=8$.