Olympiad problems

1999 All-Russian Olympiad Regional Round, 9.5

All cells of the checkered plane are painted in $5$ colors so that in any figure of the species [img]https://cdn.artofproblemsolving.com/attachments/f/f/49b8d6db20a7e9cca7420e4b51112656e37e81.png[/img] all colors are different. Prove that in any figure of the species $ \begin{tabular}{ | l | c| c | c | r| } \hline & & & &\\ \hline \end{tabular}$, all colors are different..

1997 Denmark MO - Mohr Contest, 5

Tags: combinatorics , combinatorial geometry , tiles , tiling

A $7\times 7$ square is cut into pieces following types: [img]https://cdn.artofproblemsolving.com/attachments/e/d/458b252c719946062b655340cbe8415d1bdaf9.png[/img] Show that exactly one of the pieces is of type (b). [img]https://cdn.artofproblemsolving.com/attachments/4/9/f3dd0e13fed9838969335c82f5fe866edc83e8.png[/img]

2018 Regional Olympiad of Mexico Northeast, 5

Tags: domino , coloring , tiles , combinatorics

A $300\times 300$ board is arbitrarily filled with $2\times 1$ dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.) Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met: $\bullet$ Each token is painted in one and only one of the colors. $\bullet$ The same number of tiles are painted in each color. $\bullet$ No piece is a neighbor of more than two pieces of the same color. Note: Two dominoes are [i]neighbors [/i]if they share an edge.

2011 Peru MO (ONEM), 4

Tags: combinatorics , tiling , tiles , domino

A domino is a $1 \times 2$ (or 2 $\times 1$) rectangular piece; namely, made up of two squares. There is an $8 \times 8$ board such that each domino can be cover exactly two of its squares. John places $n$ dominoes on the board, so that each one covers exactly two squares of the board and it is no longer possible to place a piece more without overlapping with any of those already placed. Determine the smallest value of $n$ for which the described situation is possible.

2012 Peru MO (ONEM), 3

Tags: combinatorics , tiles , tiling , domino

A domino is a $1\times2$ or $2\times 1$ rectangle. Diego wants to completely cover a $6\times 6$ board using $18$ dominoes. Determine the smallest positive integer $k$ for which Diego can place $k$ dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remaining dominoes.