5. How many integer solutions $(x, y)$ does the equation $|x|+|y|=n$ have?

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# Solution

Graphical interpretation of the equation - the boundaries of 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 $n$ points. The next line adds another $n$ points, and the fourth line adds $n-1$ points. Thus, there are 4$n$ points on the sides of the square.

## Grading Criteria

Only the correct answer - 0 points.

The solution is started, and the counting of solutions (analytically or graphically) is underway, 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 counting of the number of points - 4-6 points.

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