MathDB

4

Part of 2023 Tuymaada Olympiad

Problems(2)

Game involving stones on piles

Source: 2023 Tuymaada Junior P4

7/12/2023
Two players play a game. They have n>2n > 2 piles containing n10+1n^{10}+1 stones each. A move consists of removing all the piles but one and dividing the remaining pile into nn nonempty piles. The player that cannot move loses. Who has a winning strategy, the player that moves first or his adversary?
combinatoricsGame Theorycombinatorial game theoryTuymaada
Circles (OCD) touching a fixed circle

Source: Tuymaada 2023 Senior P4

7/7/2023
Two points AA and BB and line ℓ\ell are fixed in the plane so that ℓ\ell is not perpendicular to ABAB and does not intersect the segment ABAB. We consider all circles with a centre OO not lying on ℓ\ell, passing through AA and BB and meeting ℓ\ell at some points CC and DD. Prove that all the circumcircles of triangles OCDOCD touch a fixed circle.
geometry