Let's find such an $n$ and such numbers $a_{i}, b_{i}, c_{i},(i=1,2, \ldots, n)$, for which the points $(x, y)$ satisfying the condition

$$
\sum_{i=1}^{n}\left|a_{i} x+b_{i} y+c_{i}\right|=10
$$

when plotted on the coordinate system, form the vertices of a regular octagon!
I. solution. Let's choose the coordinate system such that its origin is the center of the octagon and its axes are the diagonals of the octagon. Let the points on the positive \(x\) and \(y\) axes be denoted by \(X\) and \(Y\), respectively, and the origin by \(O\), and let the length of the segment \(OX\) be the unit (1. figure). First, let's write the equation of the octagon side starting from \(X\) and extending into the first quadrant.

![](https://cdn.mathpix.com/cropped/2024_05_02_1ca908e4bf88544eda10g-1.jpg?height=661&width=788&top_left_y=285&top_left_x=663)

The coordinates of the vertex adjacent to \(X\) are \(\left(\cos 45^{\circ} ; \sin 45^{\circ}\right)\), so the equation of the line of this side is

\[
y=\frac{\sqrt{2}}{\sqrt{2}-2}(x-1)
\]

Rearranging this, we get the equation

\[
\sqrt{2} x + (x + y) = \sqrt{2} + 1
\]

The equation of the side starting from \(Y\) and extending into the first quadrant can be obtained by swapping \(x\) and \(y\), since these two lines are symmetric with respect to the line \(y = x\). Thus, the equation is

\[
\sqrt{2} y + (x + y) = \sqrt{2} + 1
\]

We can combine these two cases into one equation if we notice that we need (2) when \(x > y\) and (3) when \(y > x\). Thus, we get the equation

\[
\sqrt{2} \max (x, y) + (x + y) = \sqrt{2} + 1
\]

where \(\max (x, y)\) denotes the larger of the numbers \(x\) and \(y\). This is the equation of the union of the rays starting from the point \(\left(\cos 45^{\circ}, \sin 45^{\circ}\right)\) and passing through \(X\) and \(Y\). However, we only need the part of this that lies in the first quadrant, and in the other quadrants, we need the reflections of the corresponding axes. We can satisfy both requirements at once if we replace \(x\) and \(y\) with their absolute values:

\[
\sqrt{2} \max (|x|, |y|) + (|x| + |y|) = \sqrt{2} + 1
\]

This is already the equation of the octagon, but it is not yet in the form (1). Notice that

\[
2 \max (|x|, |y|) = |x + y| + |x - y|
\]

since for any numbers, the sum of their absolute values and the difference of their absolute values are such that one is always the sum of the absolute values and the other is the difference between the larger and smaller absolute values:

\[
|x + y| + |x - y| = (|x| + |y|) + \max (|x|, |y|) - \min (|x|, |y|)
\]

Thus, the equation (5) is equivalent to

\[
|x + y| + |x - y| + |\sqrt{2} x| + |\sqrt{2} y| = 2 + \sqrt{2}
\]

Multiplying this by \(10 / (2 + \sqrt{2})\) gives us the desired form (1). Therefore, \(n = 4\).

# II. solution.

![](https://cdn.mathpix.com/cropped/2024_05_02_1ca908e4bf88544eda10g-2.jpg?height=547&width=658&top_left_y=122&top_left_x=663)

2. figure

For any \(a, b, c\), the value of the function

\[
F(x, y) = a x + b y + c
\]

is 0 at the points of the line

\[
a x + b y + c = 0
\]

and \(d\) at the points of the line

\[
a x + b y + c = d
\]

The value of the function \(F(x, y)\) is constant on lines parallel to \(e\), meaning that the value of \(F(x, y)\) at any point \(P(x; y)\) depends only on which side of \(e\) the point \(P\) is on and the distance between \(P\) and \(e\). The equation of the line passing through the origin and perpendicular to \(e\) is

\[
b x - a y = 0
\]

At the point \(Q(a; b)\) on this line, \(F(a, b) = a^2 + b^2 + c\), so \(F(a, b) - F(0, 0) = a^2 + b^2\) (2. figure). Since \(Q\) is \(\sqrt{a^2 + b^2}\) units away from the origin, this means that the absolute value of \(F\) is \(\sqrt{a^2 + b^2}\) times the distance from \(P\) to \(e\). This proves that the distance from a point \(P(x, y)\) to the line \(a_i x + b_i y + c_i = 0\), denoted by \(d_i(P)\),

\[
d_i(P) = \frac{|a_i x + b_i y + c_i|}{\sqrt{a_i^2 + b_i^2}}
\]

In the following, we assume that \(\sqrt{a_i^2 + b_i^2}\) is constant and independent of \(i\), and we denote it by \(10 / d\). With this constraint, the problem can be rewritten as follows:

Find \(n\) lines and a distance \(d\) in the plane such that the geometric locus of points