第41届环球城市数学竞赛秋季赛O水平.pdfVIP

41st INTERNATIONAL MATHEMATICAL TOURNAMENT OF TOWNS Junior O-Level Paper, Fall 2019 Grades 8 – 9 (ages 13-15) (The result is computed from the three problems with the highest scores.) points problems 1. An illusionist performs the following trick: he lays a full deck of 52 cards in a row. On each step a spectator selects some integer number, and the card lying on the place with this number in the row is thrown away. However, each time after the number is selected the illusionist 4 chooses from which side of the row it is counted. The illusionist announced beforehand that in the end only a definite card, namely the Three of Clubs, should remain. For which initial positions of the Three of Clubs the illusionist can guarantee the success of the trick? Alexey Voropaev 2. Let ω be a circle with the center O and A and C be two different points on ω . For any third point P of the circle let X and Y be the midpoints of the segments AP and CP . Finally, let H be the 4 orthocenter (the point of intersection of the altitudes) of the triangle OXY . Prove that the position of the point H does not depend on the choice of P . Artemiy Sokolov 3. There is a row of 100 cells each containing a token. For 1 dollar it is allowed to interchange two neighbouring tokens. Also it is allowed 4 to interchange with no charge any two tokens such that there are exactly 3 tokens between them.