9.2
Problems(3)
equilateral 2-player game in 72-gon (I Soros Olympiad 1994-95 R1 9.2)
Source:
7/30/2021
Given a regular -gon. Lenya and Kostya play the game "Make an equilateral triangle." They take turns marking with a pencil on one still unmarked angle of the -gon: Lenya uses red. Kostya uses blue. Lenya starts the game, and the one who marks first wins if its color is three vertices that are the vertices of some equilateral triangle, if all the vertices are marked and no such a triangle exists, the game ends in a draw. Prove that Kostya can play like this so as not to lose.
combinatoricscombinatorial geometryRegularEquilateralgameStrategy
hour + minute, angle (I Soros Olympiad 1994-95 R2 9.2)
Source:
5/25/2024
What can be the angle between the hour and minute hands of a clock if it is known that its value has not changed after minutes?
geometryalgebra
2 circumcircles of similar triangles and 2 lines concurrent
Source: : I Soros Olympiad 1994-95 Ukraine R2 9.2 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics
6/6/2024
Triangles and are similar to each other and have the same orientation. Prove that the circles circumcribed around these triangles and the straight lines , have a common point.
geometrycircumcirclesimilar triangles