Školjka

%V0 Solve the system of equations: $$x+y+z=a$$ $$x^2+y^2+z^2=b^2$$ $$xy=z^2$$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

%V0 Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system $$x_5+x_2=yx_1$$ $$x_1+x_3=yx_2$$ $$x_2+x_4=yx_3$$ $$x_3+x_5=yx_4$$ $$x_4+x_1=yx_5$$ where $y$ is a parameter.

%V0 Consider the sytem of equations $$a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0$$ $$a_{21}x_1+a_{22}x_2+a_{23}x_3 =0$$ $$a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0$$ with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions: a) $a_{11}, a_{22}, a_{33}$ are positive numbers; b) the remaining coefficients are negative numbers; c) in each equation, the sum ofthe coefficients is positive. Prove that the given system has only the solution $x_1=x_2=x_3=0$.

%V0 Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:$$$\begin{align*} ax_1^2+bx_1+c &= x_{2} \\ ax_2^2+bx_2 +c &= x_3 \\ \vdots \\ ax_n^2+bx_n+c &= x_1 \end{align*}$$$ Prove that the system has zero, one or more than one real solutions if $(b-1)^2-4ac$ is negative, equal to zero or positive respectively.

%V0 We consider the following system with $q=2p$: $$$\begin{align*} a_{11}x_{1}+\ldots&+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots&+a_{2q}x_{q}=0,\\ &\vdots \\ a_{p1}x_{1}+\ldots&+a_{pq}x_{q}=0,\\ \end{align*}$$$ in which every coefficient is an element from the set $\{-1,\,0,\,1\}$ $.$ Prove that there exists a solution $x_{1}, \ldots, x_{q}$ for the system with the properties: a.) all $x_{j}, j=1,\ldots,q$ are integers; b.) there exists at least one j for which $x_{j} \neq 0$; c.) $|x_{j}| \leq q$ for any $j=1, \ldots ,q$.

%V0 Prove that every integer $k$ greater than 1 has a multiple that is less than $k^4$ and can be written in the decimal system with at most four different digits.