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Determine all functions f:R→R such that the equality f([x] y)= f(x) [f(y) ] holds for all x,y ∈R . Here by [x] we denote the greatest integer not exceeding x.

$${Determine}\:{all}\:{functions}\:{f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${such}\:{that}\:{the}\:{equality}\:{f}\left(\left[{x}\right]\:{y}\right)=\:{f}\left({x}\right)\:\left[{f}\left({y}\right)\:\right] \\ $$$${holds}\:{for}\:{all}\:{x},{y}\:\in\mathbb{R}\:.\:{Here}\:\:{by}\:\left[{x}\right]\:{we}\: \\ $$$${denote}\:{the}\:{greatest}\:{integer}\:{not}\:{exceeding}\:{x}. \\ $$$$ \\ $$

Question Number 117844 Answers: 0 Comments: 1

Question Number 117841 Answers: 1 Comments: 0

∫ ln (1−e^(−2x) ) dx =?

$$\int\:\mathrm{ln}\:\left(\mathrm{1}−{e}^{−\mathrm{2}{x}} \right)\:{dx}\:=? \\ $$

Question Number 117838 Answers: 1 Comments: 0

Let be P the set of prime numbers and A=P∪{0,1} Prove that Π_(n∉A) (n/( (√(n^2 −1)))) =(2/π)(√3)

$$\:\:{Let}\:{be}\:{P}\:\:{the}\:{set}\:{of}\:{prime}\:{numbers}\:{and}\: \\ $$$${A}={P}\cup\left\{\mathrm{0},\mathrm{1}\right\} \\ $$$${Prove}\:{that}\:\:\:\underset{{n}\notin{A}} {\prod}\:\frac{{n}}{\:\sqrt{{n}^{\mathrm{2}} −\mathrm{1}}}\:=\frac{\mathrm{2}}{\pi}\sqrt{\mathrm{3}}\: \\ $$

Question Number 117832 Answers: 1 Comments: 0

Let a,b>0 and x∈]0;(π/2)[ Prove ((a/(sinx))+1)((b/(cosx))+1)≥(1+(√(2ab)))^2

$$\left.{Let}\:{a},{b}>\mathrm{0}\:\:{and}\:{x}\in\right]\mathrm{0};\frac{\pi}{\mathrm{2}}\left[\:\right. \\ $$$$\:\:{Prove}\:\:\:\left(\frac{{a}}{{sinx}}+\mathrm{1}\right)\left(\frac{{b}}{{cosx}}+\mathrm{1}\right)\geqslant\left(\mathrm{1}+\sqrt{\mathrm{2}{ab}}\right)^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 117828 Answers: 1 Comments: 0

1)((√3)−1)((√3)+1)=(√3)×(√3)−(√3)−1 =3−(√3)−1 =2−(√3) 2)(2x+(√3))(2x−(√3))=(2x)^2 −2x(√3)+2x(√3)−3 =4x^2 −3

$$\left.\mathrm{1}\right)\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)\left(\sqrt{\mathrm{3}}+\mathrm{1}\right)=\sqrt{\mathrm{3}}×\sqrt{\mathrm{3}}−\sqrt{\mathrm{3}}−\mathrm{1} \\ $$$$=\mathrm{3}−\sqrt{\mathrm{3}}−\mathrm{1} \\ $$$$=\mathrm{2}−\sqrt{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\left(\mathrm{2}{x}+\sqrt{\mathrm{3}}\right)\left(\mathrm{2}{x}−\sqrt{\mathrm{3}}\right)=\left(\mathrm{2}{x}\right)^{\mathrm{2}} −\mathrm{2}{x}\sqrt{\mathrm{3}}+\mathrm{2}{x}\sqrt{\mathrm{3}}−\mathrm{3} \\ $$$$=\mathrm{4}{x}^{\mathrm{2}} −\mathrm{3} \\ $$

Question Number 117827 Answers: 0 Comments: 5

Question Number 117826 Answers: 1 Comments: 0

Prove that the Euler Constant is qlso equal to lim_(x→−1) Γ(x)−(1/(x(x+1)))

$$\:\:{Prove}\:{that}\:{the}\:{Euler}\:{Constant}\:{is}\:{qlso}\:{equal}\:{to} \\ $$$$\underset{{x}\rightarrow−\mathrm{1}} {\mathrm{lim}}\:\:\Gamma\left({x}\right)−\frac{\mathrm{1}}{{x}\left({x}+\mathrm{1}\right)} \\ $$

Question Number 117825 Answers: 1 Comments: 0

Let ABC be a triangle such as 2cosA+3sinB=4 and 3cosB+2sinA=3 Prove that the angle C is right.

$${Let}\:{ABC}\:{be}\:{a}\:{triangle}\:{such}\:{as}\: \\ $$$$\:\mathrm{2}{cosA}+\mathrm{3}{sinB}=\mathrm{4}\:{and}\:\:\mathrm{3}{cosB}+\mathrm{2}{sinA}=\mathrm{3} \\ $$$${Prove}\:{that}\:{the}\:{angle}\:{C}\:{is}\:{right}. \\ $$$$\: \\ $$

Question Number 117820 Answers: 3 Comments: 0

lim_(x→+∞) (x.sin (1/x))^x^2

$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\left({x}.\mathrm{sin}\:\frac{\mathrm{1}}{{x}}\right)^{{x}^{\mathrm{2}} } \\ $$

Question Number 117818 Answers: 0 Comments: 2

let ABC be a triangle AB=c AC=b BC=a Show that ABC is right ⇔ tan((B/2))=((a+c)/b)

$$\:{let}\:{ABC}\:\:{be}\:{a}\:{triangle}\:{AB}={c}\:\:{AC}={b}\:\:{BC}={a} \\ $$$${Show}\:{that}\:{ABC}\:{is}\:{right}\:\Leftrightarrow\:\:{tan}\left(\frac{{B}}{\mathrm{2}}\right)=\frac{{a}+{c}}{{b}}\: \\ $$

Question Number 117817 Answers: 2 Comments: 1

Question Number 117816 Answers: 1 Comments: 0

find out for n≥1 Π_(k=0) ^(n−1) Γ(1+(k/n))

$$\:\:{find}\:{out}\:\:\:{for}\:{n}\geqslant\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\Gamma\left(\mathrm{1}+\frac{{k}}{{n}}\right) \\ $$$$ \\ $$

Question Number 117811 Answers: 1 Comments: 0

∫_( 0) ^( 𝛑) ln∣sinh(x)∣dx

$$\int_{\:\mathrm{0}} ^{\:\boldsymbol{\pi}} \boldsymbol{\mathrm{ln}}\mid\boldsymbol{\mathrm{sinh}}\left(\boldsymbol{\mathrm{x}}\right)\mid\boldsymbol{\mathrm{dx}} \\ $$

Question Number 117806 Answers: 4 Comments: 2

... nice calculus... i :: 1 +(4/9)+(9/(36))+((16)/(100))+...= ?? ii:: ∫_0 ^( (π/2)) x^2 cot(x) dx=?? m.n.1970

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:{i}\:::\:\:\:\mathrm{1}\:+\frac{\mathrm{4}}{\mathrm{9}}+\frac{\mathrm{9}}{\mathrm{36}}+\frac{\mathrm{16}}{\mathrm{100}}+...=\:?? \\ $$$$\:\:\:\:\:{ii}::\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {x}^{\mathrm{2}} {cot}\left({x}\right)\:{dx}=?? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$ \\ $$

Question Number 117800 Answers: 1 Comments: 0

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