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Omni-MATH / Let P₁,P₂,…,P_(n) be n distinct points over a line in the plane (n≥2). Consider all the circumferences with diameters P_(i)P_(j) (1≤i,j≤n) and they are painted with k given colors.…

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problem

Let P1,P2,…,PnP_1,P_2,\dots,P_n be nn distinct points over a line in the plane (n≥2n\geq2). Consider all the circumferences with diameters PiPjP_iP_j (1≤i,j≤n1\leq{i,j}\leq{n}) and they are painted with kk given colors. Lets call this configuration a (n,kn,k)-cloud. For each positive integer kk, find all the positive integers nn such that every possible (n,kn,k)-cloud has two mutually exterior tangent circumferences of the same color.
Plain-text mathematical notation (without MathML)
Let P₁,P₂,…,P_(n) be n distinct points over a line in the plane (n≥2). Consider all the circumferences with diameters P_(i)P_(j) (1≤i,j≤n) and they are painted with k given colors. Lets call this configuration a (n,k)-cloud.

For each positive integer k, find all the positive integers n such that every possible (n,k)-cloud has two mutually exterior tangent circumferences of the same color.
Original LaTeX notation
Let $P_1,P_2,\dots,P_n$ be $n$ distinct points over a line in the plane ($n\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\leq{i,j}\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud.

For each positive integer $k$, find all the positive integers $n$ such that every possible ($n,k$)-cloud has two mutually exterior tangent circumferences of the same color.

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Source and history

task_id
e90976e7-626e-5d60-90d3-fb9b040befc6
task_key
test--e90976e7-626e-5d60-90d3-fb9b040befc6
task_revision_id
1
config / split
/ test

Official source

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