Školjka

%V0 Determine all the triples $(a, b, c)$ of positive real numbers such that the system $$ax + by -cz = 0,$$ $$a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,$$ is compatible in the set of real numbers, and then find all its real solutions.

%V0 Find all solutions of the following system of $n$ equations in $n$ variables: $$$\begin{align*} x_{1}|x_{1}|-(x_{1}-a)&|x_{1}-a| = x_{2}|x_{2}|, \\ x_{2}|x_{2}|-(x_{2}-a)&|x_{2}-a| = x_{3}|x_{3}|, \\ &\vdots \\ x_{n}|x_{n}|-(x_{n}-a)&|x_{n}-a| = x_{1}|x_{1}| \\ \end{align*}$$$ where $a$ is a given number.

%V0 Find all solutions of the following system of $n$ equations in $n$ variables: $$$\begin{align*} x_{1}|x_{1}|-(x_{1}-a)&|x_{1}-a| = x_{2}|x_{2}|, \\ x_{2}|x_{2}|-(x_{2}-a)&|x_{2}-a| = x_{3}|x_{3}|, \\ &\vdots \\ x_{n}|x_{n}|-(x_{n}-a)&|x_{n}-a| = x_{1}|x_{1}| \\ \end{align*}$$$ where $a$ is a given number.

%V0 Let $a, b, c$ be positive numbers with $\sqrt a +\sqrt b +\sqrt c = \frac{\sqrt 3}{2}$. Prove that the system of equations $$\sqrt{y-a}+\sqrt{z-a}=1,$$ $$\sqrt{z-b}+\sqrt{x-b}=1,$$ $$\sqrt{x-c}+\sqrt{y-c}=1$$ has exactly one solution $(x, y, z)$ in real numbers.

%V0 Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: $$\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.$$ Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

%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.