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exists a subset E of Z \to Z so that ... ?

Source: Spanish Mathematical Olympiad 1985 P2

August 2, 2018

SubsetIntegersalgebra

Problem Statement

Determine if there exists a subset

E

of

Z \times Z

with the properties: (i)

E

is closed under addition, (ii)

E

contains

(0,0),

(iii) For every

(a,b) \ne (0,0), E

contains exactly one of

(a,b)

and

-(a,b)

.
Remark: We define

(a,b)+(a',b') = (a+a',b+b')

and

-(a,b) = (-a,-b)

.

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