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Question Number 160160    Answers: 0   Comments: 0

Σ_(k=1) ^n (1/((2k)!(2n−k)!))=?

$$\:\:\:\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{\left(\mathrm{2}{k}\right)!\left(\mathrm{2}{n}−{k}\right)!}=? \\ $$

Question Number 160159    Answers: 0   Comments: 0

prove that .. ∫_0 ^( ∞) (( x)/(cosh^( 3) (x ))) dx = G − (1/2) ■ G: catalan constant

$$ \\ $$$$\:\:\:\:\:\:\:{prove}\:\:{that}\:.. \\ $$$$ \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{x}}{{cosh}^{\:\mathrm{3}} \:\left({x}\:\right)}\:{dx}\:=\:\mathrm{G}\:−\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\blacksquare \\ $$$$\:\:\:\:\:\mathrm{G}:\:\:{catalan}\:{constant} \\ $$$$ \\ $$

Question Number 160142    Answers: 2   Comments: 0

Question Number 160141    Answers: 1   Comments: 0

Find the sum of all triples (x, y, z ; (x<y<z)) such that x, y, z, z−y, y−x, z−x are all prime positive integers. (First sum all the triples individually, then summ all the sums.)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{triples} \\ $$$$\left({x},\:{y},\:{z}\:;\:\left({x}<{y}<{z}\right)\right)\:\mathrm{such}\:\mathrm{that}\:{x},\:{y},\:{z}, \\ $$$${z}−{y},\:{y}−{x},\:{z}−{x}\:\mathrm{are}\:\mathrm{all}\:\mathrm{prime}\:\mathrm{positive} \\ $$$$\mathrm{integers}.\:\left(\mathrm{First}\:\mathrm{sum}\:\mathrm{all}\:\mathrm{the}\:\mathrm{triples}\right. \\ $$$$\left.\mathrm{individually},\:\mathrm{then}\:\mathrm{summ}\:\mathrm{all}\:\mathrm{the}\:\mathrm{sums}.\right) \\ $$

Question Number 160138    Answers: 0   Comments: 2

The nucleus of a certain atom has mass of 3.8×10^(−25 ) and is at rest. The nucleus is radioactive and suddenly eject from its self, a particle of mass 6.6×10^(−27) kg and speed 1.5×10^3 m/s. Find the recoil speed of the nucleus as is left behind.

$$\mathrm{The}\:\mathrm{nucleus}\:\mathrm{of}\:\mathrm{a}\:\mathrm{certain}\:\mathrm{atom}\:\mathrm{has} \\ $$$$\mathrm{mass}\:\mathrm{of}\:\mathrm{3}.\mathrm{8}×\mathrm{10}^{−\mathrm{25}\:} \mathrm{and}\:\mathrm{is}\:\mathrm{at}\:\mathrm{rest}.\:\mathrm{The} \\ $$$$\mathrm{nucleus}\:\mathrm{is}\:\mathrm{radioactive}\:\mathrm{and}\:\mathrm{suddenly} \\ $$$$\mathrm{eject}\:\mathrm{from}\:\mathrm{its}\:\mathrm{self},\:\mathrm{a}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass} \\ $$$$\mathrm{6}.\mathrm{6}×\mathrm{10}^{−\mathrm{27}} \mathrm{kg}\:\mathrm{and}\:\mathrm{speed}\:\mathrm{1}.\mathrm{5}×\mathrm{10}^{\mathrm{3}} \mathrm{m}/\mathrm{s}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{recoil}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{nucleus} \\ $$$$\mathrm{as}\:\mathrm{is}\:\mathrm{left}\:\mathrm{behind}. \\ $$$$ \\ $$

Question Number 160136    Answers: 1   Comments: 0

1+4+((16)/2)+((64)/6)+...+(4^n /(n!))=?

$$\mathrm{1}+\mathrm{4}+\frac{\mathrm{16}}{\mathrm{2}}+\frac{\mathrm{64}}{\mathrm{6}}+...+\frac{\mathrm{4}^{{n}} }{{n}!}=? \\ $$

Question Number 160178    Answers: 1   Comments: 1

lim_(x→π) ((sin (x/2)−1)/(cos (sin x)−1)) =?

$$\:\:\:\:\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\frac{{x}}{\mathrm{2}}−\mathrm{1}}{\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)−\mathrm{1}}\:=? \\ $$

Question Number 160175    Answers: 0   Comments: 0

Question Number 160123    Answers: 2   Comments: 0

if we write all natural numbers together like this 1234567891011121314151617..., what is the 1000th digit?

$${if}\:{we}\:{write}\:{all}\:{natural}\:{numbers} \\ $$$${together}\:{like}\:{this} \\ $$$$\mathrm{1234567891011121314151617}..., \\ $$$${what}\:{is}\:{the}\:\mathrm{1000}{th}\:{digit}? \\ $$

Question Number 160119    Answers: 1   Comments: 0

Question Number 160113    Answers: 1   Comments: 0

prove Φ:= ∫_0 ^( ∞) (( sinh(x))/(cosh^2 (x))).(1/x) dx =^(???) ((4G)/π)

$$ \\ $$$$\:\:{prove}\:\: \\ $$$$\:\:\:\:\:\:\Phi:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sinh}\left({x}\right)}{{cosh}^{\mathrm{2}} \left({x}\right)}.\frac{\mathrm{1}}{{x}}\:{dx}\:\overset{???} {=}\:\frac{\mathrm{4G}}{\pi} \\ $$

Question Number 160109    Answers: 0   Comments: 3

How many numbers are there which contain 5 digits and the sum and product of the digits are both prime numbers?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{numbers}\:\mathrm{are}\:\mathrm{there}\:\mathrm{which} \\ $$$$\mathrm{contain}\:\mathrm{5}\:\mathrm{digits}\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{and}\: \\ $$$$\mathrm{product}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{are}\:\mathrm{both}\:\mathrm{prime} \\ $$$$\mathrm{numbers}? \\ $$

Question Number 160102    Answers: 1   Comments: 1

Question Number 160092    Answers: 1   Comments: 0

x and y are positive integers and x × x − 8y = 4x . If x is not a multiple of 8, then what is the minimum possible value for y?

$${x}\:\mathrm{and}\:{y}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{and}\: \\ $$$${x}\:×\:{x}\:−\:\mathrm{8}{y}\:=\:\mathrm{4}{x}\:\:\:. \\ $$$$\mathrm{If}\:{x}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{8},\:\mathrm{then}\:\mathrm{what}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{minimum}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{for}\:{y}? \\ $$

Question Number 160091    Answers: 3   Comments: 1

51b2cd is a six-digit perfect square that is divisible by both 5 and 11. What is the sum of all possible values of it?

$$\mathrm{51b2cd}\:\mathrm{is}\:\mathrm{a}\:\mathrm{six}-\mathrm{digit}\:\mathrm{perfect}\:\mathrm{square}\: \\ $$$$\mathrm{that}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{both}\:\mathrm{5}\:\mathrm{and}\:\mathrm{11}.\: \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{values}\: \\ $$$$\mathrm{of}\:\mathrm{it}? \\ $$

Question Number 160080    Answers: 1   Comments: 0

lim_(x→0) ((x−∫_0 ^x e^t^2 dt)/(x(1−cos x)))=?

$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}−\int_{\mathrm{0}} ^{\mathrm{x}} \mathrm{e}^{\mathrm{t}^{\mathrm{2}} } \mathrm{dt}}{\mathrm{x}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{x}\right)}=? \\ $$

Question Number 160077    Answers: 1   Comments: 4

lim_(x→1) (((654)/(1−x^(654) ))−((678)/(1−x^(678) )))=?

$$\underset{\mathrm{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left(\frac{\mathrm{654}}{\mathrm{1}−\mathrm{x}^{\mathrm{654}} }−\frac{\mathrm{678}}{\mathrm{1}−\mathrm{x}^{\mathrm{678}} }\right)=? \\ $$

Question Number 160065    Answers: 1   Comments: 8

The largest value of non-negative integer a for which lim_(x→1) {((−ax+sin(x−1)+a)/(x+sin(x−1)−1))}^((1−x)/( 1−(√x))) =(1/4) is ........?

$$\mathrm{The}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:\mathrm{non}-\mathrm{negative}\:\mathrm{integer}\:{a} \\ $$$$\mathrm{for}\:\mathrm{which}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left\{\frac{−{ax}+\mathrm{sin}\left({x}−\mathrm{1}\right)+{a}}{{x}+\mathrm{sin}\left({x}−\mathrm{1}\right)−\mathrm{1}}\right\}^{\frac{\mathrm{1}−{x}}{\:\mathrm{1}−\sqrt{{x}}}} =\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\mathrm{is}\:........? \\ $$

Question Number 160064    Answers: 0   Comments: 2

Find the least positive integer n for which 2^n + 5^n - n is a multiple of 1000

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{least}\:\mathrm{positive}\:\mathrm{integer}\:\:\boldsymbol{\mathrm{n}}\:\:\mathrm{for} \\ $$$$\mathrm{which}\:\:\mathrm{2}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{5}^{\boldsymbol{\mathrm{n}}} \:-\:\boldsymbol{\mathrm{n}}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{1000} \\ $$

Question Number 160063    Answers: 0   Comments: 0

Find: Ω =lim_(n→∞) (1/(n!)) ∫_( 0) ^( 1) ((1 - x)^n + cosnx)e^x dx

$$\mathrm{Find}: \\ $$$$\Omega\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{n}!}\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\left(\left(\mathrm{1}\:-\:\mathrm{x}\right)^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{cos}\boldsymbol{\mathrm{nx}}\right)\mathrm{e}^{\boldsymbol{\mathrm{x}}} \:\mathrm{dx} \\ $$

Question Number 160062    Answers: 0   Comments: 0

Find: Ω =Π_(n=1) ^∞ ((n^(1/(n+1)) /2)) = ?

$$\mathrm{Find}: \\ $$$$\Omega\:=\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\mathrm{n}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{n}}+\mathrm{1}}} }{\mathrm{2}}\right)\:=\:? \\ $$

Question Number 160061    Answers: 1   Comments: 0

Find out some pairs (a,b) such that for some n≥1 a^n +b^n ,a^(2n) +b^(2n) ,a^(4n) +b^(4n) ,a^(8n) +b^(8n) ∈P

$$ \\ $$$${Find}\:{out}\:{some}\:{pairs}\:\left({a},{b}\right)\:{such}\:{that} \\ $$$${for}\:{some}\:{n}\geqslant\mathrm{1} \\ $$$${a}^{{n}} +{b}^{{n}} ,{a}^{\mathrm{2}{n}} +{b}^{\mathrm{2}{n}} ,{a}^{\mathrm{4}{n}} +{b}^{\mathrm{4}{n}} ,{a}^{\mathrm{8}{n}} +{b}^{\mathrm{8}{n}} \in\mathbb{P} \\ $$$$ \\ $$

Question Number 160056    Answers: 0   Comments: 0

Question Number 160058    Answers: 1   Comments: 2

Find n so that ((a^(n+1) +b^(n+1) )/(a^n +b^n )) may be the arithmetic mean between a and b.

$${Find}\:{n}\:{so}\:{that}\:\frac{{a}^{{n}+\mathrm{1}} +{b}^{{n}+\mathrm{1}} }{{a}^{{n}} +{b}^{{n}} }\:{may}\:{be} \\ $$$${the}\:{arithmetic}\:{mean}\:{between}\:{a} \\ $$$${and}\:{b}. \\ $$

Question Number 160052    Answers: 1   Comments: 0

find Φ(k)=Σ_(n=1) ^∞ (n^k /(n!)) with k≥1.

$${find}\:\Phi\left({k}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}^{{k}} }{{n}!}\:{with}\:{k}\geqslant\mathrm{1}. \\ $$

Question Number 160050    Answers: 1   Comments: 0

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