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Romania TST 2016 Day 2 P4

Source: Romania TST 2016 Day 2 P4

November 1, 2017
combinatorics

Problem Statement

Given any positive integer nn, prove that: (a) Every nn points in the closed unit square [0,1]×[0,1][0,1]\times [0,1] can be joined by a path of length less than 2n+42\sqrt{n}+4; and (b) There exist nn points in the closed unit square [0,1]×[0,1][0,1]\times [0,1] that cannot be joined by a path of length less than n−1\sqrt{n}-1.
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