OME 1985 Problem 2

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 $-(a,b) = (-a,-b)$ .