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8

Part of 2018 All-Russian Olympiad

Problems(3)

All-Russian MO 2018 Grade 10 P8

Source: All-Russian MO 2018 Grade 10 P8

5/12/2018
The board used for playing a game consists of the left and right parts. In each part there are several fields and there’re several segments connecting two fields from different parts (all the fields are connected.) Initially, there is a violet counter on a field in the left part, and a purple counter on a field in the right part. Lyosha and Pasha alternatively play their turn, starting from Pasha, by moving their chip (Lyosha-violet, and Pasha-purple) over a segment to other field that has no chip. It’s prohibited to repeat a position twice, i.e. can’t move to position that already been occupied by some earlier turns in the game. A player losses if he can’t make a move. Is there a board and an initial positions of counters that Pasha has a winning strategy?
combinatorics
All-Russian Olympiad Day 2 Problem 10.7.

Source: All-Russian Olympiad 2018

4/25/2018
ABCDABCD is a convex quadrilateral. Angles AA and CC are equal. Points MM and NN are on the sides ABAB and BCBC such that MN∣∣ADMN||AD and MN=2ADMN=2AD. Let KK be the midpoint of MNMN and HH be the orthocenter of △ABC\triangle ABC. Prove that HKHK is perpendicular to CDCD.
geometry
All-Russian MO 2018 Grade 11 P8

Source: All-Russian MO 2018 Grade 11 P8

4/28/2018
Initially, on the lower left and right corner of a 2018×20182018\times 2018 board, there're two horses, red and blue, respectively. AA and BB alternatively play their turn, AA start first. Each turn consist of moving their horse (AA-red, and BB-blue) by, simultaneously, 2020 cells respect to one coordinate, and 1717 cells respect to the other; while preserving the rule that the horse can't occupied the cell that ever occupied by any horses in the game. The player who can't make the move loss, who has the winning strategy?
combinatoricsanalytic geometrygrid