IMO 2017 Shortlist C8
Let n be a given positive integer. In the Cartesian plane, ea h latti e point with nonnegative oordinates initially onta...
Category: Combinatorics
Problem
Let n be a given positive integer. In the Cartesian plane, ea h latti e point
with nonnegative oordinates initially ontains a butter y, and there are no other butter-
ies. The neighborhood of a latti e point c onsists of all latti e points within the axis-aligned
p2n1qˆp2n1q square entered at c, apart from c itself. We all a butter y lonely, rowded,
or omfortable, depending on whether the number of butter ies in its neighborhood N is re-
spe tively less than, greater than, or equal to half of the number of latti e points in N.
Every minute, all lonely butter ies y away simultaneously. This pro ess goes on for as
long as there are any lonely butter ies. Assuming that the pro ess eventually stops, determine
the number of omfortable butter ies at the nal state.
(Bulgaria) 8 IMO 2017, Rio de Janeiro