Problems

One of the roots of the equation $x^2+ax+b=0$ is $1+\sqrt{3}$. Find $a$ and $b$ if you know that they are rational.

Author: A. Khrabrov

Do there exist integers $a$ and $b$ such that

a) the equation $x^2+ax+b=0$ does not have roots, and the equation $\lfloor x^2\rfloor +ax+b=0$ does have roots?

b) the equation $x^2+2ax+b=0$ does not have roots, and the equation $\lfloor x^2\rfloor +2ax+b=0$ does have roots?

Note that here, square brackets represent integers and curly brackets represent non-integer values or 0.

The equations $$\begin{array}{}\text{(1)}& a{x}^2+bx+c=0\end{array}$$ and $$\begin{array}{}\text{(2)}& -a{x}^2+bx+c\end{array}$$ are given. Prove that if $x_1$ and $x_2$ are, respectively, any roots of the equations (1) and (2), then there is a root $x_3$ of the equation $\frac{1}{2}a{x}^2+bx+c$ such that either $x_1\le x_3\le x_2$ or $x_1\ge x_3\ge x_2$.

Is it true that, if $b>a+c>0$, then the quadratic equation $a{x}^2+bx+c=0$ has two roots?

Solving the problem: “What is the solution of the expression $x^{2000}+x^{1999}+x^{1998}+1000x^{1000}+1000x^{999}+1000x^{998}+2000x^3+2000x^2+2000x+3000$ ($x$ is a real number) if $x^2+x+1=0$?”, Vasya got the answer of 3000. Is Vasya right?