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 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 Nađite sva rješenja $k, l, m \in \mathbb{N}$ jednadžbe: $$k!l! = k! + l! + m!\text{.}$$ ($n!$ označava umnožak prirodnih brojeva od $1$ do $n$.)

%V0 Neka je $a$ neki realni broj. Riješi sustav jednadžbi $$ \begin{array}{lcr} (x_1+x_2+x_3)\cdot x_4 &=& a\\ (x_1+x_2+x_4)\cdot x_3 &=& a\\ (x_1+x_3+x_4)\cdot x_2 &=& a\\ (x_2+x_3+x_4)\cdot x_1 &=& a. \end{array} $$

%V0 Consider the system $x + y = z + u$, $2xy = zu$. Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.

%V0 Each positive integer $a$ is subjected to the following procedure, yielding the number $d = d\left(a\right)$: (a) The last digit of $a$ is moved to the first position. The resulting number is called $b$. (b) The number $b$ is squared. The resulting number is called $c$. (c) The first digit of $c$ is moved to the last position. The resulting number is called $d$. (All numbers are considered in the decimal system.) For instance, $a = 2003$ gives $b = 3200$, $c = 10240000$ and $d = 02400001 = 2400001 = d\left(2003\right)$. Find all integers a such that $d\left( a\right) =a^2$.