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Question Number 33323 by Rasheed.Sindhi last updated on 14/Apr/18

Question related to Q#33217 If A_1 ,A_2 ,...A_n are n points with integer coordinates of a plane such that every triangle whose vertices are any three of the above points has its centroid with at least one non-integer coordinate. Find the maximum possible n. Recall that if P(x_1 ,y_1 ),Q(x_2 ,y_2 ),R(x_3 ,y_3 ) are three vertices then centroid G is (((x_1 +x_2 +x_3 )/3) , ((y_1 +y_2 +y_3 )/3))

$$\mathrm{Question}\:\mathrm{related}\:\mathrm{to}\:\mathrm{Q}#\mathrm{33217} \\ $$$$\mathrm{If}\:\mathrm{A}_{\mathrm{1}} ,\mathrm{A}_{\mathrm{2}} ,...\mathrm{A}_{\mathrm{n}} \:\mathrm{are}\:\mathrm{n}\:\mathrm{points}\:\mathrm{with}\:\mathrm{integer} \\ $$$$\mathrm{coordinates}\:\mathrm{of}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{such}\:\mathrm{that}\:\mathrm{every}\:\mathrm{triangle} \\ $$$$\mathrm{whose}\:\mathrm{vertices}\:\mathrm{are}\:\mathrm{any}\:\mathrm{three}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$$$\mathrm{points}\:\mathrm{has}\:\mathrm{its}\:\mathrm{centroid}\:\mathrm{with}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one} \\ $$$$\mathrm{non}-\mathrm{integer}\:\mathrm{coordinate}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{possible}\:\mathrm{n}.\:\:\: \\ $$$$\mathrm{Recall}\:\mathrm{that}\:\mathrm{if}\:\mathrm{P}\left(\mathrm{x}_{\mathrm{1}} ,\mathrm{y}_{\mathrm{1}} \right),\mathrm{Q}\left(\mathrm{x}_{\mathrm{2}} ,\mathrm{y}_{\mathrm{2}} \right),\mathrm{R}\left(\mathrm{x}_{\mathrm{3}} ,\mathrm{y}_{\mathrm{3}} \right) \\ $$$$\mathrm{are}\:\mathrm{three}\:\mathrm{vertices}\:\mathrm{then}\:\mathrm{centroid}\:\mathrm{G}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} }{\mathrm{3}}\:,\:\frac{\mathrm{y}_{\mathrm{1}} +\mathrm{y}_{\mathrm{2}} +\mathrm{y}_{\mathrm{3}} }{\mathrm{3}}\right) \\ $$

Commented by Rasheed.Sindhi last updated on 17/Apr/18

n=972 There sbould be at most 972 points in order to meet the condition.

$$\mathrm{n}=\mathrm{972} \\ $$$$\mathrm{There}\:\mathrm{sbould}\:\mathrm{be}\:\mathrm{at}\:\mathrm{most}\:\mathrm{972}\:\mathrm{points}\: \\ $$$$\mathrm{in}\:\mathrm{order}\:\mathrm{to}\:\mathrm{meet}\:\mathrm{the}\:\mathrm{condition}. \\ $$

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