4. In the Cartesian coordinate plane, the number of integer points that satisfy the system of inequalities $\left\{\begin{array}{l}y \leqslant 3 x \\ y \geqslant \frac{x}{3} \\ x+y \leqslant 100\end{array}\right.$ is
2551
4. 【Analysis and Solution】The number of integer points (including the boundary) in the region enclosed by the two coordinate axes and the line $x+y=100$ is $1+2+$ $3+\cdots+101=\frac{102 \times 101}{2}=5151$ (points).
The number of integer points in the region enclosed by $y=\frac{1}{3} x, x+y=100$ and the x-axis (excluding the boundary $y=\frac{1}{3} x$) is
$$
\begin{array}{l}
(1+1+1+1)+(2+2+2+2)+\cdots+(25+25+25+25) \\
=4 \times(1+2+\cdots+25)=1300 \text { (points). }
\end{array}
$$

The number of integer points in the region enclosed by $y=3 x, x+y=100$ and the y-axis (excluding the boundary $y=3 x$) is also 1300.
Therefore, the number of integer points that satisfy the given conditions is $5151-2 \times 1300=2551$ (points).