MathDB

Given right

n

-gon wit the point

O

-- its centre. All the vertices are marked either with

+1

-1

. We may change all the signs in the vertices of regular

k

-gon (

2 \le k \le n

) with the same centre

O

. (By

2

-gon we understand a segment, being halved by

O

.) Prove that in a), b) and c) cases there exists such a set of

(+1)

s and

(-1)

s, that we can never obtain a set of

(+1)

s only.
a)

n = 15

,
b)

n = 30

,
c)

n > 2

,
d) Let us denote

K(n)

the maximal number of

(+1)

and

(-1)

sets such, that it is impossible to obtain one set from another. Prove, for example, that

K(200) = 2^{80}

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