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Tiling a cube

Source: Serbia MO 2023 P2

April 29, 2023

Problem Statement

Given is a cube of side length 20212021. In how many different ways is it possible to add somewhere on the boundary of this cube a 1×1×11\times 1\times 1 cube in such a way that the new shape can be filled in with 1×1×k1\times 1\times k shapes, for some natural number kk, k≥2k\geq 2?
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