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2024 Pan-African
3
3
Part of
2024 Pan-African
Problems
(1)
Combinatorics
Source: Pan African Mathematics Olympiad P3
8/15/2024
Given an integer
n
≥
1
n \geq 1
n
≥
1
, Jo-Ané alternately writes crosses (
X
\mathcal{X}
X
) and circles (
O
\mathcal{O}
O
) in the cells of a square grid with
2
n
+
1
2n + 1
2
n
+
1
rows and
2
n
+
1
2n + 1
2
n
+
1
columns: she first writes a cross in a cell, then a circle in a second cell, then a cross in a third cell, and so on. When the table is completely filled, her score is calculated as the sum
X
+
O
\mathcal{X}+ \mathcal{O}
X
+
O
, where
X
\mathcal{X}
X
is the number of rows containing more crosses than circles and
O
\mathcal{O}
O
is the number of columns containing more circles than crosses.Determine, in terms of
n
n
n
, the highest possible score that Jo-Ané can obtain..
PAMO 2024