Let x,y>0 and xy=1.Find the minimum of the expression S=\(\left(x+y+1\right)\left(x^2+y^2\right)+\dfrac{4}{x+y}\)

Let x²+2xy+7(x+y)+2y²+10=0.Find the maximum,minimum of P=x+y+3

Let A=\(\dfrac{2x^3+x^2-2x-1}{x^4+x^2-2}\)

a,Compact A

b,Find the maximum,minimum of A

Solve the equation :\(m\left(m+1\right)\left(m+2\right)\left(m+3\right)=n^3\left(n+2\right)\) with m,n are integer number

Solve the equation 2004x⁴+2001x³+2008x²+2004x+2004=0

Find the value of n\(\in\)N for A=n²⁰¹²+n²⁰¹¹+1 is a prime number

Find the value of n\(\in\)N for A=n²⁰¹²+n²⁰¹¹+1 is a integer

Solve the equation:\(\left\{{}\begin{matrix}32x^3-48x^2+30x+\left(4y-7\right)\sqrt{1-y}=7\\3x+y-3=0\end{matrix}\right.\)

The three heights of a triangle are proportional to 3,4 and 5.Is the triangle an acute,right ,or obtuse triangle?

Given a triangle ABC,the point E on AB and the point F on AC such that EF//BC.Prove that:S_CEF\(\le\dfrac{1}{4}\)S_ABC

Solve the equation:2^x+2^y+2^z=1024 given x,y,z\(\in Z\)

Find n\(\in N\) for \(n^4+4^n\) is a prime

Give \(\left(a+1\right)^2+\left(b+2\right)^2+\left(c+3\right)^2\le2010\).Find the min of A=ab+b(c-1)+c(a-2)

Find the value of x,y in the equation:\(x^3+y^3=2013\)

Prove that:\(5^n\left(5^n+1\right)-6^n\left(3^n+2^n\right)⋮91\) with all n

Compact:A=\(\dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{4}}+........+\dfrac{1}{2006\sqrt{2005}+2005\sqrt{2006}}\)

Calculated:S=\(\dfrac{3}{1^2.3}+\dfrac{5}{\left(1^2+2^2\right).4}+\dfrac{7}{\left(1^2+2^2+3^2\right).5}+........+\dfrac{99}{\left(1^2+2^2+........+49^2\right).51}\)

Compact:\(\sqrt{6+\sqrt{6+\sqrt{6}.....}}\)

Solve the equation:

a,4xy-x-y=z²

b,x²-y³=7

Solve the equation:

a,\(x^4+x^3+x^2+x+1=\dfrac{31}{x-1}\)

b,1+x²+x⁴+.............+x²ⁿ=\(\dfrac{1}{1-x^2}\) with n\(\in N\)