Generalization of APMO 2015, problem 4
Posted: Thu Aug 20, 2015 1:03 pm
Can this problem from APMO 2015 be generalized for any $k\leq2n$? Which means:
Prove or disprove: In a plane, there are $2n$ distinct lines where $n$ is a positive integer. Among them, $n$ lines are colored blue and $n$ lines are colored red and no two lines are parallel. Let $\mathcal{B}$($\mathcal{R}$) be the set of all points that lie on at least one blue(red) line. Prove that, there exists a circle that intersects $\mathcal{B}$ and $\mathcal{R}$ in exactly $k$ points where $k\leq2n$.
Prove or disprove: In a plane, there are $2n$ distinct lines where $n$ is a positive integer. Among them, $n$ lines are colored blue and $n$ lines are colored red and no two lines are parallel. Let $\mathcal{B}$($\mathcal{R}$) be the set of all points that lie on at least one blue(red) line. Prove that, there exists a circle that intersects $\mathcal{B}$ and $\mathcal{R}$ in exactly $k$ points where $k\leq2n$.