Školjka

%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 $(FRA 5)$ 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 $\frac{(n+6)(2n^2+9n+12)}{72}$?

%V0 Solve the system of equations $$|a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1$$ $$|a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1$$ $$|a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1$$ $$|a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1$$ where $a_1, a_2, a_3, a_4$ are four different real numbers.

%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 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 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.