MathDB

4

Part of 2023 All-Russian Olympiad

Problems(3)

Geometric inequality with excenters

Source: All-Russian MO Final stage 2023 9.4

4/23/2023
Given is a triangle ABCABC and a point XX inside its circumcircle. If IB,ICI_B, I_C denote the B,CB, C excenters, then prove that XB⋅XC<XIB⋅XICXB \cdot XC <XI_B \cdot XI_C.
geometry
Table tennis mini-tournament

Source: All-Russian MO 2023 Final stage 10.4

4/23/2023
There is a queue of nn{} girls on one side of a tennis table, and a queue of nn{} boys on the other side. Both the girls and the boys are numbered from 11{} to nn{} in the order they stand. The first game is played by the girl and the boy with the number 11{} and then, after each game, the loser goes to the end of their queue, and the winner remains at the table. After a while, it turned out that each girl played exactly one game with each boy. Prove that if nn{} is odd, then a girl and a boy with odd numbers played in the last game.
Proposed by A. Gribalko
combinatoricsAll Russian Olympiadgraph theorygameilostthegame
Beautiful geo finale of day 1

Source: All-Russian MO 2023 Final stage 11.4

4/23/2023
Let ω\omega be the circumcircle of triangle ABCABC with AB<ACAB<AC. Let II be its incenter and let MM be the midpoint of BCBC. The foot of the perpendicular from MM to AIAI is HH. The lines MH,BI,ABMH, BI, AB form a triangle TbT_b and the lines MH,CI,ACMH, CI, AC form a triangle TcT_c. The circumcircle of TbT_b meets ω\omega at B′B' and the circumcircle of TcT_c meets ω\omega at C′C'. Prove that B′,H,C′B', H, C' are collinear.
geometry