In the Cartesian plane, all points with both integer coordinates are painted blue. Blue colon they are said to be mutually visible if the line segment connecting them has no other blue dots. Prove that There is a set of $ 2019$ blue dots that are mutually visible two by two.
Problem
Source: Brazil National Olympiad 2019 - level 2 - #6
Tags: Brazilian Math Olympiad
RodSalgDomPort
07.12.2019 00:06
Even for $n=5$ the problem does not work.
RodSalgDomPort
07.12.2019 00:08
See the parity of the coordinates: For 5 points, there will be 2 of them such that their x and their y have the same parity. There will be a point in the segment between them
matinyousefi
06.04.2020 12:56
Can anyone post the right problem?
parmenides51
06.04.2020 13:47
official wording Quote: No plano cartesiano, todos os pontos com ambas coordenadas inteiras são pintados de azul. Dois pontos azuis são ditos mutuamente visíveis se o segmento de reta que os conecta não possui outros pontos azuis. Prove que existe um conjunto de 2019 pontos azuis que são mutuamente visíveis dois a dois. google translate wrote: On the Cartesian plane, all points with both full coordinates are painted blue. Two blue dots they are said to be mutually visible if the line segment that connects them has no other blue dots. Prove that there is a set of 2019 blue dots that are mutually visible two by two.
numero_abestado
19.04.2020 00:59
matinyousefi wrote: Can anyone post the right problem? Actually, the original problem was wrong
parmenides51
15.09.2022 22:40
numero_abestado wrote: matinyousefi wrote: Can anyone post the right problem? Actually, the original problem was wrong can you justify why the problem was wrong? I reposted it here.