Answer: 867. Solution: Shift the entire picture to the left by 1. We get a set of lines passing through the point $(9,9)$ and intersecting the line $y=100-x$. The picture will become symmetric with respect to the line $y=x$, so the sum of the abscissas on it is equal to the sum of the ordinates. Through the point $(9,9)$, 18 lines are drawn, and the line $y=100-x$ intersects 17 of them. For each point on the line $y=100-x$, the sum of the coordinates is 100, so the total sum of the abscissas and ordinates is 1700, and the sum of the abscissas is half of that, which is 850. However, on the symmetric picture, each abscissa is 1 less than the original, so the desired sum is $850+17=867$.

Criterion: No less than 1 point for the correct answer without a correct justification. 1 point for the idea of shifting the picture so that it becomes symmetric with respect to the axes. 5 points if all ideas are found, but the answer is incorrect due to an incorrect count of the number of intersection points (for example, the student counts 18 or 36 points).
10.4. Through the point with coordinates $(10,9)$, lines (including those parallel to the coordinate axes) have been drawn, dividing the plane into angles of $10^{\circ}$. Find the sum of the x-coordinates of the points of intersection of these lines with the line $y=101-x$.