Školjka

%V0 Riješite sustav jednadžbi $$$\begin{equation*} \setlength{\arraycolsep}{2pt} \begin{array}{lclclcl} 2x_{1} &- &5x_{2} &+ &3x_{3} &= &0\\ 2x_{2} &- &5x_{3} &+ &3x_{4} &= &0\\ &&&&&\vdots\\ 2x_{1993} &- &5x_{1994} &+ &3x_{1} &= &0\\ 2x_{1994} &- &5x_{1} &+ &3x_{2} &= &0 \text{.} \end{array} \end{equation*}$$$

%V0 Odredite sva realna rješenja sustava jednadžbi $$$\begin{align*} x^2 - y^2 &= 2 \left( xz + yz + x + y \right)\\ y^2 - z^2 &= 2 \left( yx + zx + y + z \right)\\ z^2 - x^2 &= 2 \left( zy + xy + z + x \right) \text{.} \end{align*}$$$

%V0 Nađite realna rješenja sustava jednadžbi: $$$\begin{gather*} x + y + z = 2\\ \left(x + y\right)\left(y + z\right) + \left(y + z\right)\left(z + x\right) + \left(z + x\right)\left(x + y\right) = 1\\ x^{2}\left(y + z\right) + y^2\left(z+x\right) + z^2\left(x+y\right) = -6 \text{.} \end{gather*} $$$

%V0 Ako je $$$\begin{align*} x+y&+z=a,\\ x^2+y^2&+z^2=b^2,\\ x^{-1}+y^{-1}&+z^{-1}=c^{-1}, \end{align*}$$$ odredite $x^3+y^3+z^3$.

%V0 Realni brojevi $a$ i $b$ zadovoljavaju ove jednakosti: $$ a^3-3ab^2=44,\quad b^3-3a^2b=8. $$ Koliko je $a^2+b^2$?

%V0 Odredite realne brojeve $x, y, z, w$ takve da vrijedi: $$x+y+z=6w$$ $$x^2+y^2+z^2=12w^2$$ $$2xyz+3y^2-4z=4.$$