MathDB
Some sets of a plane

Source: RMO 2006, 10th grade

April 17, 2006
analytic geometrycombinatorics proposedcombinatorics

Problem Statement

Let n∈N\displaystyle n \in \mathbb N, n≥2\displaystyle n \geq 2. Determine n\displaystyle n sets Ai\displaystyle A_i, 1≤i≤n\displaystyle 1 \leq i \leq n, from the plane, pairwise disjoint, such that: (a) for every circle C\displaystyle \mathcal C from the plane and for every i∈{1,2,…,n}\displaystyle i \in \left\{ 1,2,\ldots,n \right\} we have Ai∩Int(C)≠ϕ\displaystyle A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi; (b) for all lines d\displaystyle d from the plane and every i∈{1,2,…,n}\displaystyle i \in \left\{ 1,2,\ldots,n \right\}, the projection of Ai\displaystyle A_i on d\displaystyle d is not d\displaystyle d.
Back to ProblemsView on AoPS