MathDB
Set and graph

Source: VMO 2020, Day 2 - P7

December 28, 2019
algebraSets

Problem Statement

Given a positive integer n>1n>1. Denote TT a set that contains all ordered sets (x;y;z)(x;y;z) such that x,y,zx,y,z are all distinct positive integers and 1≤x,y,z≤2n1\leq x,y,z\leq 2n. Also, a set AA containing ordered sets (u;v)(u;v) is called "connected" with TT if for every (x;y;z)∈T(x;y;z)\in T then {(x;y),(x;z),(y;z)}∩A≠∅\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing. a) Find the number of elements of set TT. b) Prove that there exists a set "connected" with TT that has exactly 2n(n−1)2n(n-1) elements. c) Prove that every set "connected" with TT has at least 2n(n−1)2n(n-1) elements.
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