MathDB

P5

Part of 2021 Peru Cono Sur TST.

Problems(1)

2n concentric circles

Source: 2021 Peru Cono Sur TST P5

7/11/2023
Let n≥2n\ge 2 be an integer. They are given n+1n + 1 red points in the plane. Prove that there exist 2n2n circles C1,C2,…,Cn,D1,D2,…,DnC_1 , C_2 , \ldots , C_n , D_1 , D_2 , \ldots , D_n such that: ∙\bullet C1,C2,…,CnC_1 , C_2 ,\ldots , C_n are concentric. ∙\bullet D1,D2,…,DnD_1 , D_2 ,\ldots , D_n are concentric. ∙\bullet For k=1,2,3,…,nk = 1, 2, 3,\ldots , n the circles CkC_k and DkD_k are disjoint. ∙\bullet For k=1,2,3,…,nk = 1, 2, 3,\ldots , n it is true that CkC_k contains exactly kk red dots in its interior and DkD_k contains exactly n+1−kn + 1 - k red dots in its interior.
combinatorics