A kangaroo from a point $(x, y)$ in the first quadrant where $x \geq 0, y \geq 0$ can jump to either the point $(x+1, y-1)$ or the point $(x-5, y+7)$, provided these points are also in the first quadrant. Which are the points in the quadrant from which the kangaroo cannot reach at least 1000 units away from the origin?
A kangaroo's jump from point $(x, y)$ to point $(x+1, y-1)$ is called a small jump, and the other is called a large jump.

For the quadrant $x \geq 5, y \geq 0$ (henceforth $S$), from any point in this quadrant, the kangaroo can reach any distance from the origin, for example, by making a large jump followed by six small jumps. The result of these jumps is indeed a jump from point $(x, y)$ to point $(x, y+2)$, and during the jumps, it does not leave the first quadrant, so this sequence can be repeated indefinitely.

![](https://cdn.mathpix.com/cropped/2024_05_02_d6f54424b0a837ad1a6bg-1.jpg?height=738&width=835&top_left_y=379&top_left_x=623)

Let's examine which points of the first quadrant are those from which the kangaroo cannot reach the $S$ quadrant. Since the kangaroo cannot make a large jump from points in the first quadrant outside of $S$, it can only reach $S$ using small jumps. Let's reverse the kangaroo's path! By translating the $S$ quadrant by the vectors $(-1,1), 2(-1,1), \ldots, 5(-1,1)$, we obtain the geometric locus of all points from which the kangaroo can reach $S$.

We claim that the missing stepped shape in the first quadrant gives the solution to the problem. We have just shown that from any point not in this "step," the kangaroo can reach any distance from the origin. On the other hand, from the points of the "step," the kangaroo cannot make a large jump (as it would leave the first quadrant), and if it makes a small jump, it will still land on a point of the step. This answers the question of the problem. It is easy to see that the points of the "step" are determined by the system of inequalities $x \geq 0$, $y \geq 0, [x]+[y] \leq 4$.