2023 SSMO Accuracy Round Problems/Problem 9

Problem

Consider a $2023 \times 2023$ grid called $A = P_{2023}$ . We take one of the four smaller $2022 \times 2022$ grids located in $P_{2023}$ as $P_{2022}$ . We repeat the process of taking smaller grids until we eventually converge at the unit square $P_1.$

$[asy] size(7cm); filldraw((0,0)--(0,10)--(10,10)--(10,0)--cycle, opacity(0.2)+lightblue, blue); fill((0,0)--(0,9)--(9,9)--(9,0)--cycle, opacity(0.1)+lightblue); draw((0,9)--(9,9)--(9,0), blue); fill((1,0)--(1,8)--(9,8)--(9,0)--cycle, opacity(0.1)+lightblue); draw((1,0)--(1,8)--(9,8), blue); label("$A = P_{2023}$", (8.3, 9.52)); label("$P_{2022}$", (6.8, 8.52)); label("$\dots$", (4.78, 7.52)); [/asy]$

Of the $4^{2022}$ distinct tuples of shrinking grids $(P_{2023}, P_{2022}, \dots P_1)$ , let $T$ be the number of these tuples such that their last element is the center square of the original grid $A$ . Find the largest integer $a$ such $2^a \mid T.$

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2023 SSMO Accuracy Round Problems/Problem 9

Problem

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