MathDB
14th ibmo - cuba 1999/q3.

Source: Spanish Communities

April 16, 2006
linear algebramatrixinductionpigeonhole principlecombinatorics unsolvedcombinatorics

Problem Statement

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.
Back to ProblemsView on AoPS