4. How many integer solutions $(x, y)$ does the inequality $|x|+|y| \leq n$ have?
# Solution

Graphical interpretation of the inequality - a square with sides on the lines $x+y=n$, $-x+y=n, x-y=n,-x-y=n$. The side lying on the line $x+y=n$ corresponds to $\mathrm{n}+1$ points with integer coordinates, the side lying on the line $-x+y=n$ corresponds to $\mathrm{n}+1$ points with integer coordinates, one of which belongs to the side lying on the line $x+y=n$, so we count $\mathrm{n}$ points. The next line adds another $\mathrm{n}$ points, the fourth line adds $\mathrm{n}-1$ points. Thus, there are $4 \mathrm{n}$ points on the sides of the square. On the sides of the nearest inner square - 4(n-1) points. The origin - 1 point. The total number of points is $1+4 * 1+4 * 2+\ldots .+4 * n=1+2(n+1) n$.

## Evaluation Criteria

Only the correct answer - 0 points.

The solution is started, the counting of solutions (analytically or graphically) is carried out, but the solution is not completed - 1-3 points depending on the progress.

The answer is obtained, but there is one or more errors in the calculation of the number of points - 4-6 points.

Complete and correctly justified solution of the problem - 7 points.