# Problem 10.

Set $A$ on the plane $O x y$ is defined by the equation $x^{2}+y^{2}=2 x+2 y+23$. Set $B$ on the same plane is defined by the equation $|x-1|+|y-1|=5$. Set $C$ is the intersection of sets $A$ and $B$. What is the maximum value that the product of the lengths of $n$ segments $X Y_{1} \cdot X Y_{2} \cdot X Y_{3} \cdot \ldots \cdot X Y_{n}$ can take, where point $X$ is an arbitrarily chosen point from set $A$, and points $Y_{1}, Y_{2}, Y_{3}, \ldots, Y_{n}$ are all elements of set $C$?
Answer: 1250

Solution. The set $A$ is $(x-1)^{2}+(y-1)^{2}=25$, which is a circle with center $(1 ; 1)$ and radius 5. The set $B$ is a square with vertices $(-4 ; 1),(1 ; 6),(6 ; 1),(1 ;-4)$. The center of the square is the point $(1 ; 1)$, the diagonals of the square are 10, and the sides are $5 \sqrt{2}$. The set $C$ consists of 4 points on the circle: $(-4 ; 1),(1 ; 6),(6 ; 1),(1 ;-4)$, which are also the vertices of the inscribed square. Let's denote them as $Y_{1}, Y_{2}, Y_{3}, Y_{4}$. If we now take a point $X$ on the arc $Y_{2} Y_{3}$ (for definiteness), denote the distance from $X$ to the line $Y_{1} Y_{2}$ as $a$, and note that $\angle Y_{2} X Y_{1}=\frac{\angle Y_{2} O Y_{1}}{2}=45^{\circ}$, then we get: $X Y_{1} \cdot X Y_{2}=\frac{X Y_{1} \cdot X Y_{2} \cdot 2 \sin 45^{\circ}}{2 \sin 45^{\circ}}=S_{X Y_{1} Y_{2}} \cdot \frac{2}{\sin 45^{\circ}}=\frac{Y_{1} Y_{2} \cdot a}{2} \cdot \frac{2 \cdot 2}{\sqrt{2}}=\frac{5 \sqrt{2} \cdot a}{2} \cdot \frac{4}{\sqrt{2}}=10 a$. Similarly, $X Y_{3} \cdot X Y_{4}=10(5 \sqrt{2}-a)$. Then $X Y_{1} \cdot X Y_{2} \cdot X Y_{3} \cdot X Y_{4}=100 a(5 \sqrt{2}-a)$. The maximum of this quadratic function is achieved at $a=\frac{5 \sqrt{2}}{2}$ and is equal to 1250. Note that this means that the point $X$ is at the midpoint of the arc $Y_{2} Y_{3}$ (or any of the other three arcs).

## B-2

The set $A$ on the plane $O x y$ is defined by the equation $x^{2}+y^{2}=2 x+2 y+14$. The set $B$ on the same plane is defined by the equation $|x-1|+|y-1|=4$. The set $C$ is the intersection of sets $A$ and $B$. What is the maximum value that the product of the lengths of $n$ segments $X Y_{1} \cdot X Y_{2} \cdot X Y_{3} \cdot \ldots \cdot X Y_{n}$ can take, where the point $X$ is an arbitrarily chosen point from set $A$, and the points $Y_{1}, Y_{2}, Y_{3}, \ldots, Y_{n}$ are all elements of set $C$.

Answer: 512

## B-3

The set $A$ on the plane $O x y$ is defined by the equation $x^{2}+y^{2}=2 x+2 y+7$. The set $B$ on the same plane is defined by the equation $|x-1|+|y-1|=3$. The set $C$ is the intersection of sets $A$ and $B$. What is the maximum value that the product of the lengths of $n$ segments $X Y_{1} \cdot X Y_{2} \cdot X Y_{3} \cdot \ldots \cdot X Y_{n}$ can take, where the point $X$ is an arbitrarily chosen point from set $A$, and the points $Y_{1}, Y_{2}, Y_{3}, \ldots, Y_{n}$ are all elements of set $C$.

Answer: 162

## B-4

The set $A$ on the plane $O x y$ is defined by the equation $x^{2}+y^{2}=2 x-2 y+34$. The set $B$ on the same plane is defined by the equation $|x-1|+|y+1|=6$. The set $C$ is the intersection of sets $A$ and $B$. What is the maximum value that the product of the lengths of $n$ segments $X Y_{1} \cdot X Y_{2} \cdot X Y_{3} \cdot \ldots \cdot X Y_{n}$ can take, where the point $X$ is an arbitrarily chosen point from set $A$, and the points $Y_{1}, Y_{2}, Y_{3}, \ldots, Y_{n}$ are all elements of set $C$.

Answer: 2592