Školjka

%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.

%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 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 Find all real solutions of the system of equations: $$\sum^n_{k=1} x^i_k = a^i$$ for $i = 1,2, \ldots, n.$

%V0 In what case does the system of equations $\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$ have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.

%V0 Solve the system of equations: $\begin{matrix} x^2 + x - 1 = y \\ y^2 + y - 1 = z \\ z^2 + z - 1 = x. \end{matrix}$