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Question Number 221352    Answers: 0   Comments: 0

Given real numbers a,b,c > 0 , such that a + b + c = a^3 + b^3 + c^3 , Prove ; (a^3 /(a^4 + b + c)) + (b^3 /(b^4 + c + a)) + (c^3 /(c^4 + a + b)) ≤ 1

$$ \\ $$$$\:\:\:\:\mathrm{Given}\:\mathrm{real}\:\mathrm{numbers}\:{a},{b},{c}\:>\:\mathrm{0}\:, \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:{a}\:+\:{b}\:+\:{c}\:=\:{a}^{\mathrm{3}} \:+\:{b}^{\mathrm{3}} \:+\:{c}^{\mathrm{3}} \:, \\ $$$$\:\mathrm{Prove}\:;\:\frac{{a}^{\mathrm{3}} }{{a}^{\mathrm{4}} \:+\:{b}\:+\:{c}}\:+\:\frac{{b}^{\mathrm{3}} }{{b}^{\mathrm{4}} \:+\:{c}\:+\:{a}}\:+\:\frac{{c}^{\mathrm{3}} }{{c}^{\mathrm{4}} \:+\:\:{a}\:+\:{b}}\:\leqslant\:\mathrm{1} \\ $$$$\: \\ $$

Question Number 221350    Answers: 1   Comments: 1

∫_0 ^∞ ((cos πx)/(Γ(2+x)Γ(2−x)))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{cos}\:\pi{x}}{\Gamma\left(\mathrm{2}+{x}\right)\Gamma\left(\mathrm{2}−{x}\right)}{dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 221348    Answers: 2   Comments: 0

lim_(x→2) ((4−2^x )/(x−2))

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{4}−\mathrm{2}^{{x}} }{{x}−\mathrm{2}} \\ $$

Question Number 221347    Answers: 1   Comments: 0

lim_(x→2) ((4−x^2 )/(x−2))

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{4}−{x}^{\mathrm{2}} }{{x}−\mathrm{2}} \\ $$

Question Number 221342    Answers: 0   Comments: 0

Question Number 221332    Answers: 1   Comments: 0

Question Number 221315    Answers: 0   Comments: 1

if function z is analytic within and on a simple closed curve C,−and z_0 is a point within C using cauchy′s integral formula ∮((sin𝛑z^2 +cos𝛑z^2 )/((x−1)(x−2)))dz

$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{function}}\:\boldsymbol{\mathrm{z}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{analytic}}\:\boldsymbol{\mathrm{within}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{simple}} \\ $$$$\boldsymbol{\mathrm{closed}}\:\boldsymbol{\mathrm{curve}}\:\boldsymbol{\mathrm{C}},−\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{z}}_{\mathrm{0}} \:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{point}}\:\boldsymbol{\mathrm{within}}\:\boldsymbol{\mathrm{C}} \\ $$$$\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{cauchy}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{integral}}\:\boldsymbol{\mathrm{formula}} \\ $$$$\oint\frac{\boldsymbol{\mathrm{sin}\pi\mathrm{z}}^{\mathrm{2}} +\boldsymbol{\mathrm{cos}\pi\mathrm{z}}^{\mathrm{2}} }{\left(\boldsymbol{\mathrm{x}}−\mathrm{1}\right)\left(\boldsymbol{\mathrm{x}}−\mathrm{2}\right)}\boldsymbol{\mathrm{dz}} \\ $$

Question Number 221322    Answers: 3   Comments: 0

Solve for x x^(1/a) +(√x^((1/a)+(1/b)) )=x^(1/b)

$$\mathrm{Solve}\:\mathrm{for}\:{x} \\ $$$${x}^{\frac{\mathrm{1}}{{a}}} +\sqrt{{x}^{\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}} }={x}^{\frac{\mathrm{1}}{{b}}} \\ $$

Question Number 221321    Answers: 0   Comments: 0

if 0<x<y<e^2 then y^(√x) +x^2 +6xy+18y^2 +(8/x)+((16)/(9x^2 y^2 ))>2+x^(√y)

$${if}\:\mathrm{0}<{x}<{y}<{e}^{\mathrm{2}} \:{then} \\ $$$${y}^{\sqrt{{x}}} +{x}^{\mathrm{2}} +\mathrm{6}{xy}+\mathrm{18}{y}^{\mathrm{2}} +\frac{\mathrm{8}}{{x}}+\frac{\mathrm{16}}{\mathrm{9}{x}^{\mathrm{2}} {y}^{\mathrm{2}} }>\mathrm{2}+{x}^{\sqrt{{y}}} \\ $$

Question Number 221306    Answers: 2   Comments: 0

Σ_(n=1) ^∞ ((csch^2 (πn))/n^2 )

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{csch}^{\mathrm{2}} \left(\pi{n}\right)}{{n}^{\mathrm{2}} } \\ $$

Question Number 221298    Answers: 5   Comments: 0

Find the area of △ABC. sides are (√(20)), (√(26)). and (√(34)) .

$${Find}\:{the}\:{area}\:{of}\:\bigtriangleup{ABC}. \\ $$$${sides}\:{are}\:\sqrt{\mathrm{20}},\:\sqrt{\mathrm{26}}.\:{and}\:\sqrt{\mathrm{34}}\:. \\ $$

Question Number 221288    Answers: 1   Comments: 0

Question Number 221271    Answers: 1   Comments: 0

Find the remainder when x^(100) is divided by (x^2 +x+1)

$$\:\:{Find}\:{the}\:{remainder}\:{when}\:{x}^{\mathrm{100}} \: \\ $$$$\:\:{is}\:{divided}\:{by}\:\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right) \\ $$

Question Number 221270    Answers: 1   Comments: 0

p,q∈P Use prime number p,q to find all prime number represented by p^q +q^p

$${p},{q}\in\mathbb{P}\: \\ $$$$\: \\ $$$$\mathrm{Use}\:\mathrm{prime}\:\mathrm{number}\:{p},{q}\:\mathrm{to}\:\mathrm{find}\:\mathrm{all}\:\mathrm{prime}\:\mathrm{number}\: \\ $$$$\mathrm{represented}\:\mathrm{by}\:{p}^{{q}} +{q}^{{p}} \\ $$

Question Number 221268    Answers: 1   Comments: 1

Question Number 221264    Answers: 0   Comments: 9

A wooden block and a solid lead ball are placed in a container fully filled with water Now if the lead ball is placed on top of the floating wooden block,what is the change in water level in the container??

$${A}\:{wooden}\:{block}\:{and}\:{a}\:{solid}\:{lead}\:{ball}\:{are}\: \\ $$$${placed}\:{in}\:{a}\:{container}\:{fully}\:{filled}\:{with}\:{water} \\ $$$${Now}\:{if}\:{the}\:{lead}\:{ball}\:{is}\:{placed}\:{on}\:{top}\:{of}\:{the} \\ $$$${floating}\:{wooden}\:{block},{what}\:{is}\:{the}\:{change} \\ $$$${in}\:{water}\:{level}\:{in}\:{the}\:{container}?? \\ $$

Question Number 221262    Answers: 2   Comments: 1

if a, b, c, > 0 , show that; (a^5 /b^2 ) + (b/c) + (c^3 /a^2 ) > 2a

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{if}\:{a},\:{b},\:{c},\:>\:\mathrm{0}\:\:,\:\mathrm{show}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{5}} }{{b}^{\mathrm{2}} }\:+\:\frac{{b}}{{c}}\:+\:\frac{{c}^{\mathrm{3}} }{{a}^{\mathrm{2}} }\:>\:\mathrm{2}{a} \\ $$$$ \\ $$

Question Number 221260    Answers: 1   Comments: 0

Question Number 221256    Answers: 0   Comments: 0

∫∫∫_( [0,1]) ((ln[(1+x)(1+y)(1+z])/(1 + xyz)) dxdydz

$$ \\ $$$$\:\:\int\int\int_{\:\left[\mathrm{0},\mathrm{1}\right]} \:\frac{\mathrm{ln}\left[\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+{y}\right)\left(\mathrm{1}+{z}\right]\right.}{\mathrm{1}\:+\:{xyz}}\:{dxdydz}\:\:\:\: \\ $$$$ \\ $$

Question Number 221255    Answers: 1   Comments: 3

Let X be a point inside a square ABCD, such that XA = 10 cm, XB = 6 cm and XC = 14 cm. Find the area of the square.

$$\mathrm{Let}\:\mathrm{X}\:\mathrm{be}\:\mathrm{a}\:\mathrm{point}\:\mathrm{inside}\:\mathrm{a}\:\mathrm{square}\:\mathrm{ABCD},\: \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{XA}\:=\:\mathrm{10}\:\mathrm{cm},\:\mathrm{XB}\:=\:\mathrm{6}\:\mathrm{cm}\:\mathrm{and} \\ $$$$\mathrm{XC}\:=\:\mathrm{14}\:\mathrm{cm}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}. \\ $$

Question Number 221254    Answers: 0   Comments: 0

∫∫∫_([0,1]^3 ) (1/((1−xyz)(1−xy)(1−yz)(1−zx))) dxdydz

$$\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{3}} } \:\frac{\mathrm{1}}{\left(\mathrm{1}−{xyz}\right)\left(\mathrm{1}−{xy}\right)\left(\mathrm{1}−{yz}\right)\left(\mathrm{1}−{zx}\right)}\:{dxdydz}\:\:\:\: \\ $$$$ \\ $$

Question Number 221247    Answers: 1   Comments: 1

Question Number 221245    Answers: 1   Comments: 0

Which is greater ((√7)+1) or ((√3)+(√5))?????

$${Which}\:{is}\:{greater}\:\:\:\:\:\:\:\left(\sqrt{\mathrm{7}}+\mathrm{1}\right)\:\:{or}\:\:\left(\sqrt{\mathrm{3}}+\sqrt{\mathrm{5}}\right)????? \\ $$

Question Number 221239    Answers: 1   Comments: 0

2^x =4^y =8^z and ((1/(2x))+1(1/(4y))+(1/(6z)))=((24)/7) z=??

$$\mathrm{2}^{{x}} =\mathrm{4}^{{y}} =\mathrm{8}^{{z}} \:\:{and}\:\left(\frac{\mathrm{1}}{\mathrm{2}{x}}+\mathrm{1}\frac{\mathrm{1}}{\mathrm{4}{y}}+\frac{\mathrm{1}}{\mathrm{6}{z}}\right)=\frac{\mathrm{24}}{\mathrm{7}}\:\:\: \\ $$$${z}=?? \\ $$

Question Number 221238    Answers: 3   Comments: 0

a=5+2(√6) then {(√a)−(1/( (√a)))}=??

$${a}=\mathrm{5}+\mathrm{2}\sqrt{\mathrm{6}}\:{then}\:\left\{\sqrt{{a}}−\frac{\mathrm{1}}{\:\sqrt{{a}}}\right\}=?? \\ $$

Question Number 221234    Answers: 2   Comments: 1

let 0 ≤ a,b,c, ≤ 2 , and a + b + c = 3 Prove that; (3/2) ≤ (1/(a + 1)) + (1/(b + 1)) + (1/(c +1 )) ≤ ((11)/6)

$$ \\ $$$$\:\:\mathrm{let}\:\mathrm{0}\:\leqslant\:{a},{b},{c},\:\leqslant\:\mathrm{2}\:,\:\mathrm{and}\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\frac{\mathrm{3}}{\mathrm{2}}\:\leqslant\:\frac{\mathrm{1}}{{a}\:+\:\mathrm{1}}\:+\:\frac{\mathrm{1}}{{b}\:+\:\mathrm{1}}\:+\:\frac{\mathrm{1}}{{c}\:+\mathrm{1}\:}\:\leqslant\:\frac{\mathrm{11}}{\mathrm{6}} \\ $$$$ \\ $$

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