Problem 5. Answer: 1.

## Variant 0

Circles $K_{1}$ and $K_{2}$ have a common point $A$. Through point $A$, three lines are drawn: two pass through the centers of the circles and intersect them at points $B$ and $C$, the third is parallel to $BC$ and intersects the circles at points $M$ and $N$. Find the length of segment $MN$, if the length of segment $BC$ is $a$.
Answer: $a$

Solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_52f6ed243dad792d9207g-2.jpg?height=523&width=603&top_left_y=1709&top_left_x=815)

Notations: $\alpha$ - angle $B A M, \beta-$ angle $C A N$

Angle $A B C=\alpha$, angle $A C B=\beta$ (vertical angles).

Angle $A M B=90^{\circ}$, angle $A N C=90^{\circ}$ (subtend the diameter).

Angle $A B M=90^{\circ}-\alpha$, angle $A C N=90^{\circ}-\beta$ (acute angles in right triangles)

Conclusion: angles $M B C$ and $N C B$ are right angles, quadrilateral $M B C N$ is a rectangle and $M N=B C=a$

## Variant 2