2

Part of 2021 Francophone Mathematical Olympiad

Problems(2)

Sophie colors every triangle side in red, green or blue in 12 triangles fig.

Source: 2021 Francophone MO Juniors p2

Evariste has drawn twelve triangles as follows, so that two consecutive triangles share exactly one edge. https://cdn.artofproblemsolving.com/attachments/6/2/50377e7ad5fb1c40e36725e43c7eeb1e3c2849.png Sophie colors every triangle side in red, green or blue. Among the

3^{24}

possible colorings, how many have the property that every triangle has one edge of each color?

Coloringcombinatoricsgraph theorycountingFrancophone

Slbert and Beatrice play a game with 2021 on a table

Source: 2021 Francophone MO Seniors p2

Albert and Beatrice play a game.

2021

stones lie on a table. Starting with Albert, they alternatively remove stones from the table, while obeying the following rule. At the

n

-th turn, the active player (Albert if

n

is odd, Beatrice if

n

is even) can remove from

1

n

stones. Thus, Albert first removes

1

stone; then, Beatrice can remove

1

2

stones, as she wishes; then, Albert can remove from

1

3

stones, and so on. The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player's moves?

gamecombinatoricsgame strategywinning strategyFrancophone