China Mathematical Competition

By symmetry, we only need to consider the part of the area above the $x$-axis. Suppose line $y=k$ intercepts the right branch of the hyperbola and line $x=100$ at points $A_k$ and $B_k$ ($k=1,2,\dots ,99$), respectively. Then the number of integral points within the segment $A_k{B}_k$ is $99-k$. Therefore, the number of integral points within the area above the $x$-axis is $$\sum _{k=1}^{99}(99-k)=99\times 49=4851.$$ Finally, we obtain the total number of integral points within the whole area as $2\times 4851+98=9800$. $\square$