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length of the vector c_1a_1+c_2a_2+c_3a_3 is at least 2

Source: 1983 Polish MO Finals p5

February 25, 2020
Vectorsinequalitiesalgebracombinatorial geometry

Problem Statement

On the plane are given unit vectors a1→,a2→,a3→\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3}. Show that one can choose numbers c1,c2,c3∈{−1,1}c_1,c_2,c_3 \in \{-1,1\} such that the length of the vector c1a1→+c2a2→+c3a3→c_1\overrightarrow{a_1}+c_2\overrightarrow{a_2}+c_3\overrightarrow{a_3} is at least 22.
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