Olympiad problems

2003 Estonia National Olympiad, 5

Is it possible to cover an $n \times n$ chessboard which has its center square cut out with tiles shown in the picture (each tile covers exactly $4$ squares, tiles can be rotated and turned around) if a) $n = 5$, b) $n = 2003$? [img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]

chessboard tiling combinatorics

2004 Switzerland - Final Round, 10

Let $n > 1$ be an odd natural number. The squares of an $n \times n$ chessboard are alternately colored white and black so that the four corner squares are black. An $L$-triomino is an $L$-shaped piece that covers exactly three squares of the board. For which values of $n$ is it possible to cover all black squares with $L$-triominoes, so that no two $L$-triominos overlap? For these values of $n$ determine the smallest possible number of $L$-triominoes that are necessary for this.

tiles tiling combinatorics combinatorial games

2024 Czech and Slovak Olympiad III A, 3

Find the largest natural number $n$ such that any set of $n$ tetraminoes, each of which is one of the four shapes in the picture, can be placed without overlapping in a $20 \times 20$ table (no tetramino extends beyond the borders of the table), such that each tetramino covers exactly 4 cells of the 20x20 table. An individual tetramino is allowed to turn and flip at will. [img]https://cdn.artofproblemsolving.com/attachments/b/9/0dddb25c2aa07536b711ded8363679e47972d6.png[/img]

tiles tiling combinatorics