270. A kangaroo jumps in the corner $x \geqslant 0, y \geqslant 0$ of the coordinate plane $O x y$ as follows: from the point ( $x ; y$ ) the kangaroo can jump to the point ( $x-5$; $y+7)$ or to the point $(x+1 ; y-1)$, and jumping to points with a negative coordinate is not allowed. From which initial points $(x ; y)$ can the kangaroo not reach a point that is more than 1000 units away from the origin? Draw the set of all such points ( $x ; y$ ) and find its area.
270. Answer: the set of points from which one cannot escape *to infinity* has an area of 15; this is the stepped figure $T$, shown in Fig. 112.

From any point outside $T$, one can reach the region $x \geq 5$ in several steps $(1; -1)$, and then make steps $(-5; 7) + 5(1; -1) = (0; 2)$

$\nabla$ Even more interesting shapes of figures are formed in this problem if the number of allowed jumps is 3 or more.

![](https://cdn.mathpix.com/cropped/2024_05_21_7b8a6b2625543c3dae75g-236.jpg?height=528&width=378&top_left_y=416&top_left_x=119)

Fig. 111

![](https://cdn.mathpix.com/cropped/2024_05_21_7b8a6b2625543c3dae75g-236.jpg?height=516&width=476&top_left_y=417&top_left_x=580)

Fig. 112