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Question Number 151921 Answers: 0 Comments: 0

Question Number 152980 Answers: 2 Comments: 10

Question Number 151911 Answers: 0 Comments: 10

Determine the digit a and prime numbers x;y;z such that x<y, z<1000 and x + y^(2a) = z

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{digit}\:\boldsymbol{\mathrm{a}}\:\mathrm{and}\:\mathrm{prime} \\ $$$$\mathrm{numbers}\:\boldsymbol{\mathrm{x}};\boldsymbol{\mathrm{y}};\boldsymbol{\mathrm{z}}\:\mathrm{such}\:\mathrm{that}\:\boldsymbol{\mathrm{x}}<\boldsymbol{\mathrm{y}},\:\boldsymbol{\mathrm{z}}<\mathrm{1000} \\ $$$$\mathrm{and}\:\:\mathrm{x}\:+\:\mathrm{y}^{\mathrm{2}\boldsymbol{\mathrm{a}}} \:=\:\mathrm{z} \\ $$

Question Number 151907 Answers: 1 Comments: 2

If the area of a convex quadrilateral is 2k^2 and the sum of its diagonals is 4k^2 , then show that this quadrilateral is an orthodiagonal one.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{is}\:\mathrm{2}\boldsymbol{\mathrm{k}}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{diagonals} \\ $$$$\mathrm{is}\:\mathrm{4}\boldsymbol{\mathrm{k}}^{\mathrm{2}} ,\:\mathrm{then}\:\mathrm{show}\:\mathrm{that}\:\mathrm{this}\:\mathrm{quadrilateral} \\ $$$$\mathrm{is}\:\mathrm{an}\:\mathrm{orthodiagonal}\:\mathrm{one}. \\ $$

Question Number 151905 Answers: 1 Comments: 0

Question Number 151899 Answers: 1 Comments: 0

Question Number 151897 Answers: 1 Comments: 0

Question Number 151895 Answers: 0 Comments: 0

Question Number 151894 Answers: 1 Comments: 1

∫_0 ^a x (√((a^2 −x^2 )/(a^2 +x^2 )))

$$\int_{\mathrm{0}} ^{\mathrm{a}} \:\:\:\mathrm{x}\:\sqrt{\frac{\mathrm{a}^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} }{\mathrm{a}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} }} \\ $$

Question Number 151893 Answers: 1 Comments: 0

∫x(x^3 +a^3 )dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\int\mathrm{x}\left(\mathrm{x}^{\mathrm{3}} +\mathrm{a}^{\mathrm{3}} \right)\mathrm{dx} \\ $$

Question Number 151890 Answers: 1 Comments: 0

Question Number 151889 Answers: 1 Comments: 0

Compare: (((2020!)^3 ))^(1/(2020)) and 505∙2021^2

$$\mathrm{Compare}: \\ $$$$\sqrt[{\mathrm{2020}}]{\left(\mathrm{2020}!\right)^{\mathrm{3}} }\:\:\:\mathrm{and}\:\:\:\mathrm{505}\centerdot\mathrm{2021}^{\mathrm{2}} \\ $$

Question Number 151884 Answers: 2 Comments: 0

f(x)=(x-1)(x-2)...(x-2021) f^′ (2021) = ?

$$\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{x}-\mathrm{1}\right)\left(\mathrm{x}-\mathrm{2}\right)...\left(\mathrm{x}-\mathrm{2021}\right) \\ $$$$\mathrm{f}\:^{'} \left(\mathrm{2021}\right)\:=\:? \\ $$

Question Number 151876 Answers: 1 Comments: 0

Question Number 151874 Answers: 0 Comments: 0

Question Number 151863 Answers: 2 Comments: 0

lim_(x−0) ((1−Π_(k=1) ^n cos(kx))/x^2 )=????

$$\: \\ $$$$\boldsymbol{{li}}\underset{\boldsymbol{{x}}−\mathrm{0}} {\boldsymbol{{m}}}\frac{\mathrm{1}−\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\prod}}\boldsymbol{{cos}}\left(\boldsymbol{{kx}}\right)}{\boldsymbol{{x}}^{\mathrm{2}} }=???? \\ $$$$ \\ $$

Question Number 151851 Answers: 1 Comments: 0

Question Number 151849 Answers: 0 Comments: 2

Question Number 151841 Answers: 0 Comments: 0

The volue of the limit: lim_(n→∞) (2^(−n^2 ) /(Σ_(k=n+1) ^∞ 2^(−k^2 ) )) ; (a)0 (b)some c∈(0;1) (c)1

$$\mathrm{The}\:\mathrm{volue}\:\mathrm{of}\:\mathrm{the}\:\mathrm{limit}:\: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{2}^{−\boldsymbol{\mathrm{n}}^{\mathrm{2}} } }{\underset{\boldsymbol{\mathrm{k}}=\boldsymbol{\mathrm{n}}+\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{2}^{−\boldsymbol{\mathrm{k}}^{\mathrm{2}} } }\:\:\:;\:\:\:\left(\mathrm{a}\right)\mathrm{0}\:\:\left(\mathrm{b}\right)\mathrm{some}\:\mathrm{c}\in\left(\mathrm{0};\mathrm{1}\right)\:\:\left(\mathrm{c}\right)\mathrm{1} \\ $$

Question Number 151838 Answers: 0 Comments: 0

∫_0 ^( ∞) (((x^(log(⌊(⌊x⌋!)^((log(⌊x−1⌋!))^(−1) ) ⌋)+1) +1)^x )/(⌊x^(log(x^x )+1) ⌋!+1)) dx

$$\: \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\frac{\left({x}^{\mathrm{log}\left(\lfloor\left(\lfloor{x}\rfloor!\right)^{\left(\mathrm{log}\left(\lfloor{x}−\mathrm{1}\rfloor!\right)\right)^{−\mathrm{1}} } \rfloor\right)+\mathrm{1}} +\mathrm{1}\right)^{{x}} }{\lfloor{x}^{\mathrm{log}\left({x}^{{x}} \right)+\mathrm{1}} \rfloor!+\mathrm{1}}\:{dx} \\ $$$$\: \\ $$

Question Number 151832 Answers: 3 Comments: 0

Question Number 151859 Answers: 0 Comments: 2

Question Number 151860 Answers: 1 Comments: 3

∫_1 ^5 ∣2−∣3−x∣∣dx

$$\underset{\mathrm{1}} {\overset{\mathrm{5}} {\int}}\mid\mathrm{2}−\mid\mathrm{3}−\boldsymbol{\mathrm{x}}\mid\mid\mathrm{dx} \\ $$

Question Number 151828 Answers: 2 Comments: 0

Question Number 151826 Answers: 0 Comments: 0

if x;y;z>0 prove that: (x^2 +2)(y^2 +2)(z^2 +2) ≥ 9(xy+yz+zx)

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2}\right)\left(\mathrm{y}^{\mathrm{2}} +\mathrm{2}\right)\left(\mathrm{z}^{\mathrm{2}} +\mathrm{2}\right)\:\geqslant\:\mathrm{9}\left(\mathrm{xy}+\mathrm{yz}+\mathrm{zx}\right) \\ $$

Question Number 151823 Answers: 2 Comments: 0

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