Example 3 In the Cartesian coordinate system, the number of integer points that satisfy $(1) y \geqslant 3 x$; (2) $y \geqslant \frac{1}{3} x ;(3) x+y \leqslant 100$ is how many?
As shown in Figure 7-5, the region enclosed by the lines $y=3x$, $y=\frac{1}{3}x$, and $x+y=100$ forms a triangular region. The three vertices of this triangle are $O(0,0)$, $A(75,25)$, and $B(25,75)$. $\square$

Next, we calculate the number of integer points $N$ on the boundary and inside $\triangle OAC$. For a grid point $M(m, n)$ on the boundary or inside $\triangle OAC$, when $0 \leqslant m < 75$, $0 \leqslant n \leqslant \frac{1}{3} m$. When $75 \leqslant m \leqslant 100$, $0 \leqslant n \leqslant 100-m$. Therefore,
$$\begin{aligned}
N= & \sum_{m=0}^{74}\left(\left[\frac{1}{3} m\right]+1\right)+\sum_{m=75}^{100}(100-m+1) \\
& =\sum_{k=0}^{24}\left(\left[\frac{1}{3} \cdot 3 k\right]+1+\left[\frac{1}{3} \cdot(3 k+1)\right]+1+\left[\frac{1}{3} \cdot(3 k+2)\right]+1\right)+\sum_{m=75}^{100}(100-m+1) \\
& =\sum_{k=0}^{24}(3 k+3)+\sum_{m=75}^{100}(100-m+1) \\
& =975+351=1326
\end{aligned}$$

Similarly, the number of integer points on the boundary and inside $\triangle OBD$ is also $N=1326$.
The number of integer points on the line segment $OA$ is $25+1=26$.
Similarly, there are 26 integer points on the line segment $OB$.
Let $L$ be the number of integer points inside $\triangle OAB$. Then $L+2N-26 \times 2$ represents the number of integer points on the boundary and inside $\triangle OCD$, which numerically equals $\sum_{x=1}^{100}([100-x]+1)=\sum_{x=0}^{100}(101-x)=5151$.

Therefore, $L+2 \times 1326-2 \times 26=5151$.
Thus, the number of grid points $L=2551$.