Similarly for others: [ S = \fracy^2x^2+xy+y^2 + \fracz^2y^2+yz+z^2 + \fracx^2z^2+zx+x^2. ]
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The most difficult tier.
“A finite set of points in the plane has the property that the perpendicular bisector of any segment joining two points contains at least one other point from the set. Prove that all points are collinear.” Similarly for others: [ S = \fracy^2x^2+xy+y^2 +