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Filling the blank

Source: 2022 China Second Round A1

December 22, 2022
combinatorics

Problem Statement

Given r∈Rr\in\mathbb{R}. Alice and Bob plays the following game: An equation with blank is written on the blackboard as below: S=∣□−□∣+∣□−□∣+∣□−□∣S=|\Box-\Box|+|\Box-\Box|+|\Box-\Box| Every round, Alice choose a real number from [0,1][0,1] (not necessary to be different from the numbers chosen before) and Bob fill it in an empty box. After 6 rounds, every blank is filled and SS is determined at the same time. If S≥rS\ge r then Alice wins, otherwise Bob wins. Find all rr such that Alice can guarantee her victory.
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