Consider an $n \times n$ grid consisting of $n^2$ until cells, where $n \geq 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places $k$ frogs on the cells so that each of the $n^2$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of $k$ for which this is always possible regardless of the colouring chosen by Dionysus. Proposed by Tommy Walker Mackay, United Kingdom
Problem
Source: Balkan MO 2022 P4
Tags: combinatorics, Balkan Mathematics Olympiad
huricane
06.05.2022 21:03
Uhm... https://artofproblemsolving.com/community/c6h1546234p9380562. Unacceptable for the BMO jury to have not observed this! EDIT: Besides China 2018, this was also a problem given at a Romanian contest in 2018. Wonder how the Romanians fared on this problem...
Achilleas
07.05.2022 09:32
huricane wrote: Uhm... https://artofproblemsolving.com/community/c6h1546234p9380562. Unacceptable for the BMO jury to have not observed this! EDIT: Besides China 2018, this was also a problem given at a Romanian contest in 2018. Wonder how the Romanians fared on this problem... The Romanian contest is the 2018 Danube Mathematical Competition. The book, "Romanian Mathematical Competitions, RMC 2019" published by the Romanian Mathematical Society, p. 71, refers to a 2017 Chinese Mathematical Olympiad. As it is linked above, it is the 5th question of the 2018 China Mathematical Olympiad.
NTstrucker
07.05.2022 13:35
Achilleas wrote: The Romanian contest is the 2018 Danube Mathematical Competition. The book, "Romanian Mathematical Competitions, RMC 2019" published by the Romanian Mathematical Society, p. 71, refers to a 2017 Chinese Mathematical Olympiad. As it is linked above, it is the 5th question of the 2018 China Mathematical Olympiad. 2018 China Mathematical Olympiad held in November of 2017, you'll definitely see that if you click the link you attached. AoPS has a different chronology with common sense.