Given that .
Calculate: 
=

 and  are positive integers such that , where  is a prime number.
The number of pairs  is ... ?

The sum of all possible numbers \(n\) such that \(n^2+n+1589\) is a perfect square is ... ?

\(\dfrac{1}{\left(x+29\right)^2}+\dfrac{1}{\left(x+30\right)^2}=\dfrac{5}{4}\)

What is the product of all real solutions to the equation above?

ABC is a triangle such that \(\widehat{A}=2\widehat{B}\). Given that AC = b and AB = c then BC = ?

\(\left[{}\begin{matrix}\sqrt{b^2+bc}\\\sqrt{c^2+bc}\\\sqrt{b^2+c^2}\\\sqrt{2bc}\end{matrix}\right.\)

How many positive numbers \(n\) satisfy that \(n^3-3n^2+2\) is a prime number?

Find the maximum and the minimum of \(N=\dfrac{4x-8}{x^2-4x+8}\).

Help me !

Given 2011 natural numbers \(x_1,x_2,x_3,...,x_{2011}\) satisfy the expression below :

\(\dfrac{1}{(x_1)^{11}}+\dfrac{1}{(x_2)^{11}}+\dfrac{1}{(x_3)^{11}}+...+\dfrac{1}{\left(x_{2011}\right)^{11}}=\dfrac{2011}{2048}\)

Calculate : \(M=\dfrac{1}{\left(x_1\right)^1}+\dfrac{1}{\left(x_2\right)^2}+...+\dfrac{1}{\left(x_{2011}\right)^{2011}}\)

Help me, I have to finish my homework today :

Solve the equations :

\(\dfrac{x^2-2x+2}{x-1}+\dfrac{x^2-8x+20}{x-4}=\dfrac{x^2-6x+16}{x-3}+\dfrac{x^2-4x+2}{x-2}\)

I know the answer is 10 000 000 but I don't know how to solve this problem.