MathDB

2

Part of 2021 Dutch IMO TST

Problems(2)

2-player game on a m x n board

Source: 2021 Dutch IMO TST 2.2

12/28/2021
Stekel and Prick play a game on an m×n m \times n board, where mm and nn are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs (m,n)(m, n) who has a winning strategy.
combinatoricsgame strategygamewinning strategy
PQ passes through centroid, 3 centroids, <ACB =<CEP=<CFQ=90^o, CP=AP, CQ=BQ

Source: 2021 Dutch IMO TST 3.2

12/29/2021
Let ABCABC be a right triangle with ∠C=90o\angle C = 90^o and let DD be the foot of the altitude from CC. Let EE be the centroid of triangle ACDACD and let FF be the centroid of triangle BCDBCD. The point PP satisfies ∠CEP=90o\angle CEP = 90^o and ∣CP∣=∣AP∣|CP| = |AP|, while point QQ satisfies ∠CFQ=90o\angle CFQ = 90^o and ∣CQ∣=∣BQ∣|CQ| = |BQ|. Prove that PQPQ passes through the centroid of triangle ABCABC.
Centroidright triangleright anglesgeometry