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Omni-MATH / Let k and s be positive integers such that s<(2k+1)². Initially, one cell out of an n×n grid is coloured green. On each turn, we pick some green cell c and colour green some s out …

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problem

Let kk and ss be positive integers such that s<(2k+1)2s<(2k + 1)^2. Initially, one cell out of an n×nn \times n grid is coloured green. On each turn, we pick some green cell cc and colour green some ss out of the (2k+1)2(2k + 1)^2 cells in the (2k+1)×(2k+1)(2k + 1) \times (2k + 1) square centred at cc. No cell may be coloured green twice. We say that ss is k−sparsek-sparse if there exists some positive number CC such that, for every positive integer nn, the total number of green cells after any number of turns is always going to be at most CnCn. Find, in terms of kk, the least kk-sparse integer ss. [I]
Plain-text mathematical notation (without MathML)
Let k and s be positive integers such that s<(2k+1)². Initially, one cell out of an n×n grid is coloured green. On each turn, we pick some green cell c and colour green some s out of the (2k+1)² cells in the (2k+1)×(2k+1) square centred at c. No cell may be coloured green twice. We say that s is k−sparse if there exists some positive number C such that, for every positive integer n, the total number of green cells after any number of turns is always going to be at most Cn. Find, in terms of k, the least k-sparse integer s.

[I]
Original LaTeX notation
Let $k$ and $s$ be positive integers such that $s<(2k + 1)^2$. Initially, one cell out of an $n \times n$ grid is coloured green. On each turn, we pick some green cell $c$ and colour green some $s$ out of the $(2k + 1)^2$ cells in the $(2k + 1) \times (2k + 1)$ square centred at $c$. No cell may be coloured green twice. We say that $s$ is $k-sparse$ if there exists some positive number $C$ such that, for every positive integer $n$, the total number of green cells after any number of turns is always going to be at most $Cn$. Find, in terms of $k$, the least $k$-sparse integer $s$.

[I]

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Source and history

task_id
47490048-d134-5a0f-8a14-be4985214c00
task_key
test--47490048-d134-5a0f-8a14-be4985214c00
task_revision_id
1
config / split
/ test

Official source

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