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2player game with vectors

Source: Polish MO Recond Round 1980 p1

September 9, 2024
combinatoricsvector

Problem Statement

Students A A and B B play according to the following rules: student A A selects a vector a1→ \overrightarrow{a_1} of length 1 in the plane, then student B B gives the number s1 s_1 , equal to 1 1 or − - 1; then the student A A chooses a vector a1→ \overrightarrow{a_1} of length 1 1 , and in turn the student B B gives a number s2 s_2 equal to 1 1 or −1 -1 etc. B B wins if for a certain n n vector ∑j=1nεjaj→ \sum_{j=1}^n \varepsilon_j \overrightarrow{a_j} has a length greater than the number R R determined before the start of the game. Prove that student BB can achieve a win in no more than R2+1R^2 + 1 steps regardless of partner AA's actions.
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