Folklore

Find the maximum value of the expression $x^{2}+y^{2}$, if $|x-y| \leq 2$ and $|3 x+y| \leq 6$.
The set of points satisfying the system of inequalities $|x-y|<2$ and $|3x+y|<6$ forms the intersection of two strips, and thus represents the boundary and interior of a parallelogram $ABCD$, where $A(1,3), B(2, 0), C(-1,-3), D(-2,0)$ (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_dc8ed1d6cdac2bf77445g-39.jpg?height=563&width=575&top_left_y=1193&top_left_x=753)

We need to find the maximum square of the distance from a point in this parallelogram to the origin $O$. Clearly, this is $OA^2 = 1^2 + 3^2 = 10$.

## Answer

10.

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