2

Part of 2024 Francophone Mathematical Olympiad

Problems(2)

Alice and Bob move tile around a n-gon, who will win?

Source: Francophone 2024, Junior P2

n \ge 2

points on a circle, Alice and Bob play the following game. Initially, a tile is placed on one of the points and no segment is drawn. The players alternate in turns, with Alice to start. In a turn, a player moves the tile from its current position

P

n-1

Q

and draws the segment

PQ

. This move is not allowed if the segment

PQ

is already drawn. If a player cannot make a move, the game is over and the opponent wins. Determine, for each

n

, which of the two players has a winning strategy.

combinatoricscombinatorics proposedgamewinning strategy

3D combinatorial geometry!

Source: Francophone 2024, Senior P2

Given a positive integer

n \ge 2

\mathcal{P}

\mathcal{Q}

be two sets, each consisting of

n

points in three-dimensional space. Suppose that these

2n

points are distinct. Show that it is possible to label the points of

\mathcal{P}

P_1,P_2,\dots,P_n

and the points of

\mathcal{Q}

Q_1,Q_2,\dots,Q_n

such that for any indices

i

j

, the balls of diameters

P_iQ_i

P_jQ_j

have at least one common point.

combinatoricscombinatorial geometry3D geometrygeometry