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What are the possible values of $m$?

Source: Moldova TST 1995

August 8, 2023

vector

Problem Statement

Let

S=\{a_1,a_2,\ldots,a_n\}

of nenul vectors in a plane. Show that

S{}

can be partitioned in nenul subsets

B_1, B_2,\ldots, B_m

with the properties: 1) each vector from

S{}

is part of only on subset; 2) if

a_i\in B_j

then the angle between vectors

a_i

and

c_j

, which is the sum of all vectors from

B_j

is not greater than

\frac{\pi}{2}

; 3) if

i\neq j

then the angle between vectors

c_i

and

c_j

, which is the sum of all vectors from

B_i

and

B_j

, respectively, is greater than

\frac{\pi}{2}

. What are the possible values of

m

?

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