MathDB

154

Part of 1971 All Soviet Union Mathematical Olympiad

Problems(1)

ASU 154 All Soviet Union MO 1971 signs +,- on vertices of a dodecagon

Source:

7/3/2019
a) The vertex A1A_1 of the regular 1212-gon (dodecagon) A1A2...A12A_1A_2...A_{12} is marked with "−-" and all the rest −−-- with "++". You are allowed to change the sign simultaneously in the 66 vertices in succession. Prove that is impossible to obtain dodecagon with A2A_2 marked with "−-" and the rest of the vertices −−-- with "++".
b) Prove the same statement if it is allowed to change the signs not in six, but in four vertices in succession.
c) Prove the same statement if it is allowed to change the signs in three vertices in succession.
combinatoricsdodecagonImpossible