Let be real numbers each of whose absolute value is different from such that . Prove that
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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
$$\frac{3x-x^{3}}{1-3x^{2}}+\frac{3y-y^{3}}{1-3y^{2}}+\frac{3z-z^{3}}{1-3z^{2}}=\frac{3x-x^{3}}{1-3x^{2}}\cdot\frac{3y-y^{3}}{1-3y^{2}}\cdot\frac{3z-z^{3}}{1-3z^{2}}$$
Let , , be real numbers such that for every two of the equations there is exactly one real number satisfying both of them. Determine all possible values of .
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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 $a^2+b^2+c^2$.
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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.
Let be a polynomial with real coefficients such that if . Prove that there exist polynomials and with nonnegative coefficients such that if
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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 $x > 0.$
Knowing that the system has a real solution for which , find the value of for that solution.
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Knowing that the system
$$x + y + z = 3,$$ $$x^3 + y^3 + z^3 = 15,$$ $$x^4 + y^4 + z^4 = 35,$$
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.
Given reals , find an expression for the solution of the system
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$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system $$x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots$$
Školjka 
be real numbers each of whose absolute value is different from 
such that 
. Prove that
, 
, 
be real numbers such that for every two of the equations 
there is exactly one real number satisfying both of them. Determine all possible values of 
. 
and let 
Determine the range of 
for which 
are real variables. 
be a polynomial with real coefficients such that 
if 
. Prove that there exist polynomials 
and 
with nonnegative coefficients such that 
if 
, find the value of 
for that solution. 
Prove that for 
, the identity 
holds. 
Given reals 
, find an expression for the solution of the system 