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Question Number 120206 Answers: 0 Comments: 0

Find the remainder when 1×3×5×7×…×2017×2019 is divided by 1000.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\: \\ $$$$\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}×\ldots×\mathrm{2017}×\mathrm{2019}\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\mathrm{1000}. \\ $$

Question Number 120204 Answers: 0 Comments: 2

Call a 7−digit telephone number d_1 d_2 d_3 −d_4 d_5 d_6 d_7 memorable if the prefix sequence d_1 d_2 d_3 is exactly the same as either of the sequences d_4 d_5 d_6 or d_5 d_6 d_7 (posibly both). Assuming that each d_i can be any of the ten decimal digits 0,1,2,...,9 . Find the number of different memorable telephone numbers

$${Call}\:{a}\:\mathrm{7}−{digit}\:{telephone}\:{number}\:{d}_{\mathrm{1}} {d}_{\mathrm{2}} {d}_{\mathrm{3}} −{d}_{\mathrm{4}} {d}_{\mathrm{5}} {d}_{\mathrm{6}} {d}_{\mathrm{7}} \\ $$$${memorable}\:{if}\:{the}\:{prefix}\:{sequence}\:{d}_{\mathrm{1}} {d}_{\mathrm{2}} {d}_{\mathrm{3}} \\ $$$${is}\:{exactly}\:{the}\:{same}\:{as}\:{either}\:{of}\:{the}\:{sequences} \\ $$$${d}_{\mathrm{4}} {d}_{\mathrm{5}} {d}_{\mathrm{6}} \:{or}\:{d}_{\mathrm{5}} {d}_{\mathrm{6}} {d}_{\mathrm{7}} \:\left({posibly}\:{both}\right). \\ $$$${Assuming}\:{that}\:{each}\:{d}_{{i}} \:{can}\:{be}\:{any}\:{of}\:{the}\:{ten} \\ $$$${decimal}\:{digits}\:\mathrm{0},\mathrm{1},\mathrm{2},...,\mathrm{9}\:.\:{Find}\:{the}\:{number} \\ $$$${of}\:{different}\:{memorable}\:{telephone} \\ $$$${numbers} \\ $$

Question Number 120202 Answers: 0 Comments: 0

Show for the equation a^n = b^2 −1 where n>1 and a>2 are any natural numbers , there are no positive integer solutions for a and b ?

$${Show}\:{for}\:{the}\:{equation}\:{a}^{{n}} \:=\:{b}^{\mathrm{2}} −\mathrm{1}\: \\ $$$${where}\:{n}>\mathrm{1}\:{and}\:{a}>\mathrm{2}\:{are}\:{any}\:{natural} \\ $$$${numbers}\:,\:{there}\:{are}\:{no}\:{positive}\:{integer} \\ $$$${solutions}\:{for}\:{a}\:{and}\:{b}\:? \\ $$

Question Number 120201 Answers: 2 Comments: 0

lim_(x→∞) ((ln (x+e^x +e^(2x) ))/x)=?

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left({x}+{e}^{{x}} +{e}^{\mathrm{2}{x}} \right)}{{x}}=? \\ $$

Question Number 120196 Answers: 1 Comments: 0

The ratio of male and female participants in a competition is 3:2. Given that 15% of participants are prize winner, and among the prize winner, the ratio of male to female is 2:1. Find the ratio of male and female among the participants that are not prize winner.

$$\mathrm{The}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{male}\:\mathrm{and}\:\mathrm{female}\:\mathrm{participants} \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{competition}\:\mathrm{is}\:\mathrm{3}:\mathrm{2}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{15\%}\: \\ $$$$\mathrm{of}\:\mathrm{participants}\:\mathrm{are}\:\mathrm{prize}\:\mathrm{winner},\:\mathrm{and}\:\mathrm{among} \\ $$$$\mathrm{the}\:\mathrm{prize}\:\mathrm{winner},\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{male}\:\mathrm{to}\:\mathrm{female} \\ $$$$\mathrm{is}\:\mathrm{2}:\mathrm{1}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{male}\:\mathrm{and}\:\mathrm{female} \\ $$$$\mathrm{among}\:\mathrm{the}\:\mathrm{participants}\:\mathrm{that}\:\mathrm{are}\:\mathrm{not}\:\mathrm{prize} \\ $$$$\mathrm{winner}. \\ $$

Question Number 120195 Answers: 3 Comments: 0

C .lim_(z→0) ((Re(z).Im(z))/(Re(z)+Im(z))) =?

$$\mathbb{C}\:.\underset{\mathrm{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{Re}\left(\mathrm{z}\right).\mathrm{Im}\left(\mathrm{z}\right)}{\mathrm{Re}\left(\mathrm{z}\right)+\mathrm{Im}\left(\mathrm{z}\right)}\:=? \\ $$

Question Number 120194 Answers: 1 Comments: 0

Question Number 120188 Answers: 0 Comments: 0

Let n be a positive integer . Prove that Σ_(k=0) ^n 2^k ((n),(k) ) ((( n−k)),((⌊((n−k)/2)⌋)) ) = (((2n+1)),(( n)) )

$${Let}\:{n}\:{be}\:{a}\:{positive}\:{integer}\:. \\ $$$${Prove}\:{that}\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\mathrm{2}^{{k}} \:\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}\:\begin{pmatrix}{\:\:{n}−{k}}\\{\lfloor\frac{{n}−{k}}{\mathrm{2}}\rfloor}\end{pmatrix}\:=\:\begin{pmatrix}{\mathrm{2}{n}+\mathrm{1}}\\{\:\:\:\:\:{n}}\end{pmatrix} \\ $$

Question Number 120186 Answers: 1 Comments: 0

Question Number 120185 Answers: 1 Comments: 0

solve for x, y∈Z (√x)+(√y)=(√(2020))

$${solve}\:{for}\:{x},\:{y}\in{Z} \\ $$$$\sqrt{{x}}+\sqrt{{y}}=\sqrt{\mathrm{2020}} \\ $$

Question Number 120183 Answers: 0 Comments: 0

Question Number 123960 Answers: 0 Comments: 0

Question Number 120279 Answers: 0 Comments: 2

Question Number 120277 Answers: 2 Comments: 1

lim_(x→∞) x^3 {(√(x^2 +(√(x^4 +1)))) − x(√2) } ?

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{3}} \:\left\{\sqrt{{x}^{\mathrm{2}} +\sqrt{{x}^{\mathrm{4}} +\mathrm{1}}}\:−\:{x}\sqrt{\mathrm{2}}\:\right\}\:? \\ $$

Question Number 120166 Answers: 1 Comments: 1

Question Number 120160 Answers: 1 Comments: 1

Question Number 120156 Answers: 1 Comments: 0

...♠ nice calculus♠... if cos^(−1) (x)+cos^(−1) (y)+cos^(−1) (z)=π✓ show that :: x^2 +y^2 +z^2 +2xyz=1 ✓ ... ♣M.N.july.1970♣...

$$\:\:\:\:\:\:\:\:...\spadesuit\:{nice}\:\:{calculus}\spadesuit... \\ $$$$\:\:{if}\:\:{cos}^{−\mathrm{1}} \left({x}\right)+{cos}^{−\mathrm{1}} \left({y}\right)+{cos}^{−\mathrm{1}} \left({z}\right)=\pi\checkmark \\ $$$$ \\ $$$$\:\:\:\:\:\:{show}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +\mathrm{2}{xyz}=\mathrm{1}\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:...\:\clubsuit\mathscr{M}.\mathscr{N}.{july}.\mathrm{1970}\clubsuit... \\ $$

Question Number 120154 Answers: 1 Comments: 0

solve using LambertW function ((8/7))^x +17=25x

$${solve}\:{using}\:{LambertW}\:{function} \\ $$$$\left(\frac{\mathrm{8}}{\mathrm{7}}\right)^{{x}} +\mathrm{17}=\mathrm{25}{x} \\ $$

Question Number 120152 Answers: 0 Comments: 0

Determinate and construct the set of points M which have as affix z in each case: 1) arg(i−z)=0[π] 2) arg(z+1−i)=(π/6)[2π]

$$\mathrm{Determinate}\:\mathrm{and}\:\mathrm{construct}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:\mathrm{M} \\ $$$$\mathrm{which}\:\mathrm{have}\:\mathrm{as}\:\mathrm{affix}\:\mathrm{z}\:\:\mathrm{in}\:\mathrm{each}\:\mathrm{case}: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{arg}\left(\mathrm{i}−\mathrm{z}\right)=\mathrm{0}\left[\pi\right] \\ $$$$\left.\mathrm{2}\right)\:\mathrm{arg}\left(\mathrm{z}+\mathrm{1}−\mathrm{i}\right)=\frac{\pi}{\mathrm{6}}\left[\mathrm{2}\pi\right] \\ $$$$ \\ $$

Question Number 120151 Answers: 0 Comments: 0

Represent in complex plane the set of points M which have as affix z such that ∣z∣=2 and arg(z+1)=(π/4)[π]

$$\mathrm{Represent}\:\mathrm{in}\:\mathrm{complex}\:\mathrm{plane}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$$\mathrm{M}\:\mathrm{which}\:\mathrm{have}\:\mathrm{as}\:\mathrm{affix}\:\mathrm{z}\:\mathrm{such}\:\mathrm{that}\:\mid\mathrm{z}\mid=\mathrm{2}\:\mathrm{and} \\ $$$$\mathrm{arg}\left(\mathrm{z}+\mathrm{1}\right)=\frac{\pi}{\mathrm{4}}\left[\pi\right] \\ $$

Question Number 120147 Answers: 1 Comments: 1

Question Number 120133 Answers: 0 Comments: 5

Question Number 120132 Answers: 0 Comments: 0

Suppose you are in a imagnary train which travels at the half of the speed of light. Suppose You have a brother who is 7 year smaller than you. He stands on the platform which you had left. After 1 hour of travelling on the train you come back on the platform. Then you observe something strange. You can see your brother looks older . So what is his age?(He was ten years old)

$${Suppose}\:{you}\:{are}\:{in}\:{a}\:{imagnary}\:{train}\:{which}\:{travels}\:{at}\:{the}\:{half}\: \\ $$$${of}\:{the}\:{speed}\:{of}\:{light}.\:{Suppose}\:{You}\:{have}\:{a}\:{brother}\:{who}\:{is}\:\mathrm{7}\:{year} \\ $$$${smaller}\:{than}\:{you}.\:{He}\:{stands}\:{on}\:{the}\:{platform}\:{which}\:{you}\:{had}\:{left}. \\ $$$${After}\:\mathrm{1}\:{hour}\:{of}\:{travelling}\:{on}\:{the}\:{train}\:{you}\:{come}\:{back}\:{on}\:{the} \\ $$$${platform}.\:{Then}\:{you}\:{observe}\:{something}\:{strange}.\:{You}\:{can}\:{see} \\ $$$${your}\:{brother}\:{looks}\:{older}\:.\:{So}\:{what}\:{is}\:{his}\:{age}?\left({He}\:{was}\:{ten}\:{years}\right. \\ $$$$\left.{old}\right) \\ $$

Question Number 120131 Answers: 1 Comments: 1

Question Number 120127 Answers: 2 Comments: 0

Question Number 120122 Answers: 0 Comments: 3

{ ((s_(12) =30)),((s_8 =4)) :} a_3 =?

$$\begin{cases}{{s}_{\mathrm{12}} =\mathrm{30}}\\{{s}_{\mathrm{8}} =\mathrm{4}}\end{cases}\:\:\:{a}_{\mathrm{3}} =? \\ $$

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