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Omni-MATH / We say that a finite set S of points in the plane is [i]balanced[/i] if, for any two different points A and B in S, there is a point C in S such that AC=BC. We say that S is [i]cen…

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We say that a finite set S\mathcal{S} of points in the plane is [i]balanced[/i] if, for any two different points AA and BB in S\mathcal{S}, there is a point CC in S\mathcal{S} such that AC=BCAC=BC. We say that S\mathcal{S} is [i]centre-free[/i] if for any three different points AA, BB and CC in S\mathcal{S}, there is no points PP in S\mathcal{S} such that PA=PB=PCPA=PB=PC. (a) Show that for all integers n≥3n\ge 3, there exists a balanced set consisting of nn points. (b) Determine all integers n≥3n\ge 3 for which there exists a balanced centre-free set consisting of nn points.
Plain-text mathematical notation (without MathML)
We say that a finite set S of points in the plane is [i]balanced[/i] if, for any two different points A and B in S, there is a point C in S such that AC=BC. We say that S is [i]centre-free[/i] if for any three different points A, B and C in S, there is no points P in S such that PA=PB=PC.

(a) Show that for all integers n≥3, there exists a balanced set consisting of n points.

(b) Determine all integers n≥3 for which there exists a balanced centre-free set consisting of n points.
Original LaTeX notation
We say that a finite set $\mathcal{S}$ of points in the plane is [i]balanced[/i] if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is [i]centre-free[/i] if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.

(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.

(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.

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task_id
61fea923-c617-5e6f-86d1-328d1202b7cf
task_key
test--61fea923-c617-5e6f-86d1-328d1202b7cf
task_revision_id
1
config / split
/ test

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