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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))

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

Question Number 33322 Answers: 0 Comments: 4

a) px^2 + 3x + q=0 has roots [((α+β)/(αβ))]× αβ find p and q if x^2 − 7x + 4 = 0 with real and distinct roots has same roots.. b)if α and β are roots of x^2 + kx +2k+8=0. a) find k if one root is twice the other.

$$\left.{a}\right)\:{px}^{\mathrm{2}} +\:\mathrm{3}{x}\:+\:{q}=\mathrm{0}\:{has}\:{roots}\: \\ $$$$\:\:\left[\frac{\alpha+\beta}{\alpha\beta}\right]×\:\alpha\beta \\ $$$${find}\:{p}\:{and}\:{q}\:{if}\: \\ $$$$\:\:{x}^{\mathrm{2}} −\:\mathrm{7}{x}\:+\:\mathrm{4}\:=\:\mathrm{0}\:{with}\:{real}\:{and}\: \\ $$$${distinct}\:{roots}\:{has}\:{same}\:{roots}.. \\ $$$$\left.{b}\right){if}\:\alpha\:{and}\:\beta\:{are}\:{roots}\:{of}\: \\ $$$$\:{x}^{\mathrm{2}} +\:{kx}\:+\mathrm{2}{k}+\mathrm{8}=\mathrm{0}. \\ $$$$\left.{a}\right)\:{find}\:{k}\:{if}\:{one}\:{root}\:{is}\:{twice}\:{the}\:{other}. \\ $$

Question Number 33317 Answers: 1 Comments: 0

find the radius of a circle wich inscribes an equalateral triangle with perimeter of 24cm

$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{radius}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{circle}} \\ $$$$\boldsymbol{\mathrm{wich}}\:\boldsymbol{\mathrm{inscribes}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{equalateral}}\:\boldsymbol{\mathrm{triangle}} \\ $$$$\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{perimeter}}\:\boldsymbol{\mathrm{of}}\:\mathrm{24}\boldsymbol{\mathrm{cm}} \\ $$

Question Number 33316 Answers: 1 Comments: 0

If ((f(2x+2y))/(f(2x−2y))) = ((sin (x+y))/(sin (x−y))) . Then find f(x) ?

$${If}\:\:\frac{{f}\left(\mathrm{2}{x}+\mathrm{2}{y}\right)}{{f}\left(\mathrm{2}{x}−\mathrm{2}{y}\right)}\:=\:\frac{\mathrm{sin}\:\left({x}+{y}\right)}{\mathrm{sin}\:\left({x}−{y}\right)}\:. \\ $$$${Then}\:{find}\:{f}\left({x}\right)\:? \\ $$

Question Number 33313 Answers: 0 Comments: 1

find lim_(x→0) ((ln(1+sinx) −sin(ln(1+x)))/x^2 )

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{{ln}\left(\mathrm{1}+{sinx}\right)\:−{sin}\left({ln}\left(\mathrm{1}+{x}\right)\right)}{{x}^{\mathrm{2}} } \\ $$

Question Number 33312 Answers: 1 Comments: 0

calculate lim_(x→0) ((e^(−3x^2 ) −1)/x^2 ) .