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?
Solve As shown in Figure 7-5, the region enclosed by the lines $y=3 x, y=\frac{1}{3} x$ and $x+y=100$ is a triangular region, with the three vertices of this triangle being $O(0,0), A(75,25)$, and $B(25,75)$.

To calculate the number of integer points $N$ on the boundary and inside $\triangle O A C$, for any lattice point $M(m, n)$ on the boundary or inside $\triangle O A C$: when $0 \leqslant m<75$, $0 \leqslant n \leqslant \frac{1}{3} m$. When $75 \leqslant m \leqslant 100$, $0 \leqslant n \leqslant 100-m$, then
$$
\begin{aligned}
N= & \sum_{m=0}^{74}\left(\left[\frac{1}{3} m\right]+1\right)+\sum_{m=75}^{100}(100-m+1) \\
& =\sum_{k=0}^{24}\left(\left[\frac{1}{3} \cdot 3 k\right]+1+\left[\frac{1}{3} \cdot(3 k+1)\right]+1+\left[\frac{1}{3} \cdot(3 k+2)\right]+1\right)+\sum_{m=75}^{100}(100-m+1) \\
& =\sum_{k=0}^{24}(3 k+3)+\sum_{m=75}^{100}(100-m+1) \\
& =975+351=1326 .
\end{aligned}
$$

Similarly, the number of integer points on the boundary and inside $\triangle O B D$ also equals $N=1326$.
The number of integer points on the line segment $O A$ equals $25+1=26$.
Similarly, there are 26 integer points on the line segment $O B$.
Let $L$ be the number of integer points inside $\triangle O A B$, then $L+2 N-26 \times 2$ represents the number of integer points on the boundary and inside $\triangle O C D$, 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 lattice points $L=2551$.