Algebra
# Completing The Square Generalized

As \(x, y, z\) ranges over all possible real numbers, what is the minimum value of

\[ 3x^2+12y^2+27z^2-4xy-12yz-6xz-8y-24z+100 ? \]

This problem has been proposed by Hui.

\[ \begin{cases}x^4+2x^3-y=-\frac{1}{4}+\sqrt{3}\\ y^4+2y^3-x=-\frac{1}{4}-\sqrt{3} \end{cases}\]

All the ordered pairs of real numbers that satisfy the system of equations above are \((x_1,y_1),(x_2,y_2),...(x_n,y_n)\). Find the value of \(x_1+x_2+...+x_n+y_1+...y_n\) correct upto two decimal places.

- If you think there are infinite solutions then answer 777 and if you think no real solutions answer 666.
- Ordered pair means (11,12),(12,11) are considered different.

\[P(x)=(x^2+5000x-1)^2+(2x+5000)^2\]

Let \(P(x)\) be a polynomial, then find the sum of all the real solutions to the equation \[P(x)=\text{min}(P(x)).\]