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Question Number 21294    Answers: 1   Comments: 0

Let z_1 , z_2 , z_3 be complex numbers, not all real, such that ∣z_1 ∣ = ∣z_2 ∣ = ∣z_3 ∣ = 1 and 2(z_1 + z_2 + z_3 ) − 3z_1 z_2 z_3 ∈ R. Prove that max(arg z_1 , arg z_2 , arg z_3 ) ≥ (π/6) . Where 0 < arg(z_1 ), arg(z_2 ), arg(z_3 ) < 2π.

$$\mathrm{Let}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{complex}\:\mathrm{numbers},\:\mathrm{not} \\ $$$$\mathrm{all}\:\mathrm{real},\:\mathrm{such}\:\mathrm{that}\:\mid{z}_{\mathrm{1}} \mid\:=\:\mid{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{3}} \mid\:=\:\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{2}\left({z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} \right)\:−\:\mathrm{3}{z}_{\mathrm{1}} {z}_{\mathrm{2}} {z}_{\mathrm{3}} \:\in\:{R}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{max}\left(\mathrm{arg}\:{z}_{\mathrm{1}} ,\:\mathrm{arg}\:{z}_{\mathrm{2}} ,\:\mathrm{arg}\:{z}_{\mathrm{3}} \right)\:\geqslant \\ $$$$\frac{\pi}{\mathrm{6}}\:.\:\mathrm{Where}\:\mathrm{0}\:<\:\mathrm{arg}\left({z}_{\mathrm{1}} \right),\:\mathrm{arg}\left({z}_{\mathrm{2}} \right),\:\mathrm{arg}\left({z}_{\mathrm{3}} \right) \\ $$$$<\:\mathrm{2}\pi. \\ $$

Question Number 21293    Answers: 1   Comments: 0

Let n be an even positive integer such that (n/2) is odd and let α_0 , α_1 , ...., α_(n−1) be the complex roots of unity of order n. Prove that Π_(k=0) ^(n−1) (a + bα_k ^2 ) = (a^(n/2) + b^(n/2) )^2 for any complex numbers a and b.

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{an}\:\mathrm{even}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such} \\ $$$$\mathrm{that}\:\frac{{n}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{and}\:\mathrm{let}\:\alpha_{\mathrm{0}} ,\:\alpha_{\mathrm{1}} ,\:....,\:\alpha_{{n}−\mathrm{1}} \:\mathrm{be} \\ $$$$\mathrm{the}\:\mathrm{complex}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{of}\:\mathrm{order}\:{n}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({a}\:+\:{b}\alpha_{{k}} ^{\mathrm{2}} \right)\:=\:\left({a}^{\frac{{n}}{\mathrm{2}}} \:+\:{b}^{\frac{{n}}{\mathrm{2}}} \right)^{\mathrm{2}} \\ $$$$\mathrm{for}\:\mathrm{any}\:\mathrm{complex}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}. \\ $$

Question Number 21292    Answers: 0   Comments: 0

Let a,b∈Z 0<a<b How would you find the maximum/ largest prime gap in (a, b)? Note: Prime gaps are the distance between consecutive primes. e.g. 7 and 11 has a prime gap 4 p_k ∈P ∴∀p_x ∀p_(x+1) ∈(a,b):p_(x+1) >p_x p_(x+1) and p_x are consecutive primes Lets denote δ_x =p_(x+1) −p_x as prime gap for (1, 20), the primes are 2,3,5,7,11,13,17 The prime gaps are: 1,2,2,4,2,4 Therefore the largest δ = 4 Is there a more general method?

$$\mathrm{Let}\:\:{a},{b}\in\mathbb{Z} \\ $$$$\mathrm{0}<{a}<{b} \\ $$$$\: \\ $$$$\mathrm{How}\:\mathrm{would}\:\mathrm{you}\:\mathrm{find}\:\mathrm{the}\:\mathrm{maximum}/ \\ $$$$\mathrm{largest}\:\mathrm{prime}\:\mathrm{gap}\:\mathrm{in}\:\left({a},\:{b}\right)? \\ $$$$ \\ $$$$\mathrm{Note}: \\ $$$$\mathrm{Prime}\:\mathrm{gaps}\:\mathrm{are}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between} \\ $$$$\mathrm{consecutive}\:\mathrm{primes}. \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\:\mathrm{7}\:\mathrm{and}\:\mathrm{11}\:\mathrm{has}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{gap}\:\mathrm{4} \\ $$$$\: \\ $$$${p}_{{k}} \in\mathbb{P} \\ $$$$\therefore\forall{p}_{{x}} \forall{p}_{{x}+\mathrm{1}} \in\left({a},{b}\right):{p}_{{x}+\mathrm{1}} >{p}_{{x}} \\ $$$${p}_{{x}+\mathrm{1}} \:\mathrm{and}\:{p}_{{x}} \:\mathrm{are}\:\mathrm{consecutive}\:\mathrm{primes} \\ $$$$\mathrm{Lets}\:\mathrm{denote}\:\delta_{{x}} ={p}_{{x}+\mathrm{1}} −{p}_{{x}} \:\mathrm{as}\:\mathrm{prime}\:\mathrm{gap} \\ $$$$\: \\ $$$$\mathrm{for}\:\left(\mathrm{1},\:\mathrm{20}\right),\:\mathrm{the}\:\mathrm{primes}\:\mathrm{are}\:\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},\mathrm{17} \\ $$$$\mathrm{The}\:\mathrm{prime}\:\mathrm{gaps}\:\mathrm{are}: \\ $$$$\mathrm{1},\mathrm{2},\mathrm{2},\mathrm{4},\mathrm{2},\mathrm{4} \\ $$$$\mathrm{Therefore}\:\mathrm{the}\:\mathrm{largest}\:\delta\:=\:\mathrm{4} \\ $$$$\: \\ $$$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{more}\:\mathrm{general}\:\mathrm{method}? \\ $$

Question Number 21297    Answers: 1   Comments: 0

Question Number 21282    Answers: 0   Comments: 0

soit (u_n )_(n∈N^∗ ) une suite a termes positifs telle que: ∀n∈N^∗ ,Σ_(k=1) ^n u_k ^3 =(Σ_(k=1) ^n u_k )^2 montrer que ∀n∈N^∗ , u_n =n

$${soit}\:\left({u}_{{n}} \right)_{{n}\in\mathbb{N}^{\ast} } {une}\:{suite}\:{a}\:{termes}\:{positifs}\:{telle}\:{que}: \\ $$$$\forall{n}\in\mathbb{N}^{\ast} ,\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{u}_{{k}} ^{\mathrm{3}} =\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{u}_{{k}} \right)^{\mathrm{2}} \\ $$$${montrer}\:{que}\:\forall{n}\in\mathbb{N}^{\ast} ,\:{u}_{{n}} ={n} \\ $$

Question Number 21280    Answers: 1   Comments: 0

cosec^2 67^0 −tan^2 23^0 =

$${cosec}^{\mathrm{2}} \mathrm{67}^{\mathrm{0}} −{tan}^{\mathrm{2}} \mathrm{23}^{\mathrm{0}} = \\ $$

Question Number 21279    Answers: 1   Comments: 0

1+sinθ/cosθ=

$$\mathrm{1}+{sin}\theta/{cos}\theta= \\ $$

Question Number 21296    Answers: 1   Comments: 3

Question Number 21276    Answers: 1   Comments: 0

Question Number 21273    Answers: 0   Comments: 0

cos 70+4cos 70

$$\mathrm{cos}\:\mathrm{70}+\mathrm{4cos}\:\mathrm{70} \\ $$

Question Number 21272    Answers: 1   Comments: 0

Let α, β ∈ (−π, π) be such that cos(α − β) = 1 and cos(α + β) = (1/e). The number of pairs of α, β satisfying the above system of equation is

$$\mathrm{Let}\:\alpha,\:\beta\:\in\:\left(−\pi,\:\pi\right)\:\mathrm{be}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{cos}\left(\alpha\:−\:\beta\right)\:=\:\mathrm{1}\:\mathrm{and}\:\mathrm{cos}\left(\alpha\:+\:\beta\right)\:=\:\frac{\mathrm{1}}{{e}}. \\ $$$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{pairs}\:\mathrm{of}\:\alpha,\:\beta\:\mathrm{satisfying} \\ $$$$\mathrm{the}\:\mathrm{above}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equation}\:\mathrm{is} \\ $$

Question Number 21270    Answers: 0   Comments: 2

why any infinitely differentiable function is a power series? mathematically, if f(x) is A infinitely differentiable function then why f(x)=ax^0 +bx^1 +cx^2 +dx^3 +ex^4 +..... for example sin(x)=x−(x^3 /(3!))+(x^5 /(5!))−(x^7 /(7!))+.....up to∞

$$\mathrm{why}\:\mathrm{any}\:\mathrm{infinitely}\:\mathrm{differentiable}\: \\ $$$$\mathrm{function}\:\mathrm{is}\:\mathrm{a}\:\mathrm{power}\:\mathrm{series}? \\ $$$${mathematically}, \\ $$$${if}\:{f}\left({x}\right)\:{is}\:{A}\:{infinitely}\:{differentiable} \\ $$$${function}\:{then}\:{why} \\ $$$${f}\left({x}\right)={ax}^{\mathrm{0}} +{bx}^{\mathrm{1}} +{cx}^{\mathrm{2}} +{dx}^{\mathrm{3}} +{ex}^{\mathrm{4}} +..... \\ $$$${for}\:{example} \\ $$$${sin}\left({x}\right)={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}−\frac{{x}^{\mathrm{7}} }{\mathrm{7}!}+.....{up}\:{to}\infty \\ $$$$ \\ $$

Question Number 21268    Answers: 1   Comments: 0

if A+B+C=π so proof sin A+sin B+sin C=4cos (A/2)cos (B/2)cos (C/2)

$${if}\:{A}+{B}+{C}=\pi \\ $$$${so}\:{proof}\: \\ $$$$\mathrm{sin}\:{A}+\mathrm{sin}\:{B}+\mathrm{sin}\:{C}=\mathrm{4cos}\:\frac{{A}}{\mathrm{2}}\mathrm{cos}\:\frac{{B}}{\mathrm{2}}\mathrm{cos}\:\frac{{C}}{\mathrm{2}} \\ $$

Question Number 21267    Answers: 0   Comments: 0

cos A+cos B+cos C=1+(r/R)

$$\mathrm{cos}\:{A}+\mathrm{cos}\:{B}+\mathrm{cos}\:{C}=\mathrm{1}+\frac{{r}}{{R}} \\ $$

Question Number 21266    Answers: 1   Comments: 0

((b^2 −c^2 )/(tan A))+((c^2 −a^2 )/(tan B))+((a^2 −b^2 )/(tan C))=0

$$\frac{{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{tan}\:{A}}+\frac{{c}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{tan}\:{B}}+\frac{{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }{\mathrm{tan}\:{C}}=\mathrm{0} \\ $$

Question Number 21264    Answers: 1   Comments: 0

2x+3y+2z=2

$$\mathrm{2x}+\mathrm{3y}+\mathrm{2z}=\mathrm{2} \\ $$

Question Number 21262    Answers: 0   Comments: 0

Question Number 21260    Answers: 1   Comments: 0

if (θ−ϕ) subtle and sin θ+sin ϕ=(√(3(cos ϕ)) −cos θ) so proof sin 3θ+sin 3ϕ=0

$${if}\:\left(\theta−\varphi\right)\:{subtle}\:{and}\:\mathrm{sin}\:\theta+\mathrm{sin}\:\varphi=\sqrt{\mathrm{3}\left(\mathrm{cos}\:\varphi\right.} \\ $$$$\left.−\mathrm{cos}\:\theta\right) \\ $$$${so}\:{proof}\:\mathrm{sin}\:\mathrm{3}\theta+\mathrm{sin}\:\mathrm{3}\varphi=\mathrm{0} \\ $$

Question Number 21259    Answers: 1   Comments: 0

2cos (π/(13))cos ((9π)/(13))+cos ((3π)/(13))+cos ((5π)/(13))=0

$$\mathrm{2cos}\:\frac{\pi}{\mathrm{13}}\mathrm{cos}\:\frac{\mathrm{9}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{13}}=\mathrm{0} \\ $$

Question Number 21253    Answers: 1   Comments: 0

the hcf of two numbers is 75 and their lcm is 3375 if one of the numbers is 675 find the other number

$${the}\:{hcf}\:{of}\:{two}\:{numbers}\:{is}\:\mathrm{75}\:{and}\:{their}\:{lcm}\:{is}\:\mathrm{3375}\:{if}\:{one}\:{of}\:{the}\:{numbers}\:{is}\:\mathrm{675}\:{find}\:{the}\:{other}\:{number} \\ $$

Question Number 21252    Answers: 0   Comments: 0

prove,∀x_1 ,...,x_n y_1 ,...,y_n ∈R^+ (√(x_1 x_2 ...x_n ))+(√(y_1 y_2 ...y_n ))≤(√((x_1 +y_1 )(x_2 +y_2 )...(x_n +y_n )))

$${prove},\forall{x}_{\mathrm{1}} ,...,{x}_{{n}} {y}_{\mathrm{1}} ,...,{y}_{{n}} \in\mathbb{R}^{+} \\ $$$$\sqrt{{x}_{\mathrm{1}} {x}_{\mathrm{2}} ...{x}_{{n}} }+\sqrt{{y}_{\mathrm{1}} {y}_{\mathrm{2}} ...{y}_{{n}} }\leqslant\sqrt{\left({x}_{\mathrm{1}} +{y}_{\mathrm{1}} \right)\left({x}_{\mathrm{2}} +{y}_{\mathrm{2}} \right)...\left({x}_{{n}} +{y}_{{n}} \right)} \\ $$

Question Number 21251    Answers: 0   Comments: 0

resolve : ∀x∈R, (√(x+1))∣x−2∣=(√(x+2))∣x−1∣−2

$${resolve}\:: \\ $$$$\forall{x}\in\mathbb{R},\:\sqrt{{x}+\mathrm{1}}\mid{x}−\mathrm{2}\mid=\sqrt{{x}+\mathrm{2}}\mid{x}−\mathrm{1}\mid−\mathrm{2} \\ $$$$ \\ $$

Question Number 21249    Answers: 0   Comments: 1

A particle slides down a frictionless parabolic (y = x^2 ) track (A − B − C) starting from rest at point A. Point B is at the vertex of parabola and point C is at a height less than that of point A. After C, the particle moves freely in air as a projectile. If the particle reaches highest point at P, then (a) KE at P = KE at B (b) height at P = height at A (c) total energy at P = total energy at A (d) time of travel from A to B = time of travel from B to P.

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{slides}\:\mathrm{down}\:\mathrm{a}\:\mathrm{frictionless} \\ $$$$\mathrm{parabolic}\:\left({y}\:=\:{x}^{\mathrm{2}} \right)\:\mathrm{track}\:\left({A}\:−\:{B}\:−\:{C}\right) \\ $$$$\mathrm{starting}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{at}\:\mathrm{point}\:{A}.\:\mathrm{Point}\:{B} \\ $$$$\mathrm{is}\:\mathrm{at}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{of}\:\mathrm{parabola}\:\mathrm{and}\:\mathrm{point}\:{C} \\ $$$$\mathrm{is}\:\mathrm{at}\:\mathrm{a}\:\mathrm{height}\:\mathrm{less}\:\mathrm{than}\:\mathrm{that}\:\mathrm{of}\:\mathrm{point}\:{A}. \\ $$$$\mathrm{After}\:{C},\:\mathrm{the}\:\mathrm{particle}\:\mathrm{moves}\:\mathrm{freely}\:\mathrm{in}\:\mathrm{air} \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{projectile}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{reaches} \\ $$$$\mathrm{highest}\:\mathrm{point}\:\mathrm{at}\:{P},\:\mathrm{then} \\ $$$$\left({a}\right)\:\mathrm{KE}\:\mathrm{at}\:{P}\:=\:\mathrm{KE}\:\mathrm{at}\:{B} \\ $$$$\left({b}\right)\:\mathrm{height}\:\mathrm{at}\:{P}\:=\:\mathrm{height}\:\mathrm{at}\:{A} \\ $$$$\left({c}\right)\:\mathrm{total}\:\mathrm{energy}\:\mathrm{at}\:{P}\:=\:\mathrm{total}\:\mathrm{energy}\:\mathrm{at} \\ $$$${A} \\ $$$$\left({d}\right)\:\mathrm{time}\:\mathrm{of}\:\mathrm{travel}\:\mathrm{from}\:{A}\:\mathrm{to}\:{B}\:=\:\mathrm{time}\:\mathrm{of} \\ $$$$\mathrm{travel}\:\mathrm{from}\:{B}\:\mathrm{to}\:{P}. \\ $$

Question Number 21248    Answers: 0   Comments: 0

The locus of the centre of a circle which touches the given circles ∣z − z_1 ∣ = ∣3 + 4i∣ and ∣z − z_2 ∣ = ∣1 + i(√3)∣ is a hyperbola, then the length of its transverse axis is

$$\mathrm{The}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{which} \\ $$$$\mathrm{touches}\:\mathrm{the}\:\mathrm{given}\:\mathrm{circles}\:\mid{z}\:−\:{z}_{\mathrm{1}} \mid\:= \\ $$$$\mid\mathrm{3}\:+\:\mathrm{4}{i}\mid\:\mathrm{and}\:\mid{z}\:−\:{z}_{\mathrm{2}} \mid\:=\:\mid\mathrm{1}\:+\:{i}\sqrt{\mathrm{3}}\mid\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{hyperbola},\:\mathrm{then}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{its} \\ $$$$\mathrm{transverse}\:\mathrm{axis}\:\mathrm{is} \\ $$

Question Number 21247    Answers: 1   Comments: 0

If [ ] represents the greatest integer function and f(x) = x − [x] then number of real roots of the equation f(x) + f((1/x)) = 1 are infinite. True/False

$$\mathrm{If}\:\left[\:\right]\:\mathrm{represents}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer} \\ $$$$\mathrm{function}\:\mathrm{and}\:{f}\left({x}\right)\:=\:{x}\:−\:\left[{x}\right]\:\mathrm{then} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${f}\left({x}\right)\:+\:{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{1}\:\mathrm{are}\:\mathrm{infinite}. \\ $$$$\boldsymbol{\mathrm{True}}/\boldsymbol{\mathrm{False}} \\ $$

Question Number 21353    Answers: 1   Comments: 0

A censusman on duty visited a house which the lady inmates declined to reveal their individual ages, but said − “we do not mind giving you the sum of the ages of any two ladies you may choose”. Thereupon the censusman said − “In that case please give me the sum of the ages of every possible pair of you”. The gave the sums as follows : 30, 33, 41, 58, 66, 69. The censusman took these figures and happily went away. How did he calculate the individual ages of the ladies from these figures?

$$\mathrm{A}\:\mathrm{censusman}\:\mathrm{on}\:\mathrm{duty}\:\mathrm{visited}\:\mathrm{a}\:\mathrm{house} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{lady}\:\mathrm{inmates}\:\mathrm{declined}\:\mathrm{to} \\ $$$$\mathrm{reveal}\:\mathrm{their}\:\mathrm{individual}\:\mathrm{ages},\:\mathrm{but}\:\mathrm{said}\:− \\ $$$$``\mathrm{we}\:\mathrm{do}\:\mathrm{not}\:\mathrm{mind}\:\mathrm{giving}\:\mathrm{you}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{ages}\:\mathrm{of}\:\mathrm{any}\:\mathrm{two}\:\mathrm{ladies}\:\mathrm{you}\:\mathrm{may} \\ $$$$\mathrm{choose}''.\:\mathrm{Thereupon}\:\mathrm{the}\:\mathrm{censusman} \\ $$$$\mathrm{said}\:−\:``\mathrm{In}\:\mathrm{that}\:\mathrm{case}\:\mathrm{please}\:\mathrm{give}\:\mathrm{me}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ages}\:\mathrm{of}\:\mathrm{every}\:\mathrm{possible}\:\mathrm{pair}\:\mathrm{of} \\ $$$$\mathrm{you}''.\:\mathrm{The}\:\mathrm{gave}\:\mathrm{the}\:\mathrm{sums}\:\mathrm{as}\:\mathrm{follows}\:: \\ $$$$\mathrm{30},\:\mathrm{33},\:\mathrm{41},\:\mathrm{58},\:\mathrm{66},\:\mathrm{69}.\:\mathrm{The}\:\mathrm{censusman} \\ $$$$\mathrm{took}\:\mathrm{these}\:\mathrm{figures}\:\mathrm{and}\:\mathrm{happily}\:\mathrm{went} \\ $$$$\mathrm{away}.\:\mathrm{How}\:\mathrm{did}\:\mathrm{he}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{individual} \\ $$$$\mathrm{ages}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ladies}\:\mathrm{from}\:\mathrm{these}\:\mathrm{figures}? \\ $$

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