8.4 On a grid sheet of paper sized $60 \times 70$ cells (horizontally and vertically respectively), Lena drew a coordinate system (the origin is at the center of the sheet, the $x$-axis is horizontal, the $y$-axis is vertical, and the axes are drawn to the edges of the sheet) and plotted the graph $y=0.83 x$. Then Lena shaded all the cells that the graph passes through, i.e., cells that contain points of the graph. How many shaded cells are there in total?
Answer: 108. Solution: The graph passes through the first and third quadrants. Let's count the number of shaded cells in the first quadrant (in the third quadrant, there will be the same number, since the graph is a straight line passing through the origin, and therefore centrally symmetric). The graph intersects 29 vertical grid lines: $x=1, x=2, \ldots, x=29$ (inside the quadrant) and 24 horizontal grid lines: $y=1, y=2, \ldots$, $y=24$ (since at $x=30$ the value of $y=24.9$). We also note that the points of intersection with these lines are not grid nodes, since $0.83 x$ is not an integer for $x=1,2, \ldots, 29$, because 83 is a prime number. Thus, there are a total of $29+24=53$ points of intersection with the grid lines inside the quadrant, and the number of segments in the first quadrant into which the graph is divided by these points is 54. Each such segment corresponds to a shaded cell (to which it belongs). Therefore, in both quadrants, there will be 108 shaded cells.

## 1st round. 11.11 .2018

## 8th grade