Školjka

%V0 Let $a$, $b$, $c$ be real numbers such that for every two of the equations $$x^2+ax+b=0, \quad x^2+bx+c=0, \quad x^2+cx+a=0$$ there is exactly one real number satisfying both of them. Determine all possible values of $a^2+b^2+c^2$.

%V0 Let $n \in \mathbb{Z}^+$ and let $a, b \in \mathbb{R}.$ Determine the range of $x_0$ for which $$\sum^n_{i=0} x_i = a \text{ and } \sum^n_{i=0} x^2_i = b,$$ where $x_0, x_1, \ldots , x_n$ are real variables.

%V0 Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $\displaystyle P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$

%V0 Let $x, y, z$ be real numbers each of whose absolute value is different from $\displaystyle \frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that $$\frac{3x-x^{3}}{1-3x^{2}}+\frac{3y-y^{3}}{1-3y^{2}}+\frac{3z-z^{3}}{1-3z^{2}}=\frac{3x-x^{3}}{1-3x^{2}}\cdot\frac{3y-y^{3}}{1-3y^{2}}\cdot\frac{3z-z^{3}}{1-3z^{2}}$$

%V0 $(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system $$x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots$$

%V0 $(FRA 5)$ Let $\alpha(n)$ be the number of pairs $(x, y)$ of integers such that $x+y = n, 0 \le y \le x$, and let $\beta(n)$ be the number of triples $(x, y, z)$ such that$x + y + z = n$ and $0 \le z \le y \le x.$ Find a simple relation between $\alpha(n)$ and the integer part of the number $\frac{n+2}{2}$ and the relation among $\beta(n), \beta(n -3)$ and $\alpha(n).$ Then evaluate $\beta(n)$ as a function of the residue of $n$ modulo $6$. What can be said about $\beta(n)$ and $1+\frac{n(n+6)}{12}$? And what about $\frac{(n+3)^2}{6}$? Find the number of triples $(x, y, z)$ with the property $x+ y+ z \le n, 0 \le z \le y \le x$ as a function of the residue of $n$ modulo $6.$What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?