4. In the Cartesian coordinate plane, the number of integer points (i.e., points with both coordinates as integers) that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \leqslant 3 x \\
y \geqslant \frac{1}{3} x \\
x+y \leqslant 100
\end{array}\right.
$$

is $\qquad$ . (as shown in the figure)
4. The number of integer points in the region (including the boundaries) enclosed by the two coordinate axes and the line $x+y=100$ is
$$
1+2+3+\cdots+101=\frac{102 \times 101}{2}=5151 \text { points. }
$$

The number of integer points in the region (excluding the boundary $y=\frac{1}{3} x$) enclosed by $y=\frac{1}{3} x$, $x+y=100$, and the $x$-axis 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 (excluding the boundary $y=3 x$) enclosed by $y=3 x$, $x+y=100$, and the $y$-axis is also 1300 points.
$\therefore$ The number of integer points that satisfy the given conditions is $5151-2 \times 1300=2551$ points.