Školjka

%V0 Find all complex numbers $m$ such that polynomial $$x^3 + y^3 + z^3 + mxyz$$ can be represented as the product of three linear trinomials.

%V0 Find all solutions $(x_1,\,x_2,\,\ldots ,\,x_n)$ of the equation $$1 + \frac{1}{x_1} + \frac{x_1+1}{x_1x_2} + \frac{(x_1+1)(x_2+1)}{x_1x_2x_3} + \cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x_1x_2 \ldots x_n} = 0\text{.}$$

%V0 Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E,F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B,C,D$ of trapezoids that satisfy the following: (i) $|AB| \leq a$ ($a$ fixed); (ii) $|EF| = l$ ($l$ fixed); (iii) the sum of squares of the nonparallel sides of the trapezoid is constant. Remark Remark. The constants are chosen so that such trapezoids exist.

%V0 If $a_i \ (i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation $$\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n$$ has at least $n - 1$ real roots.

%V0 Two ships sail on the sea with constant speeds and fixed directions. It is known that at $9:00$ the distance between them was $20$ miles; at $9:35$, $15$ miles; and at $9:55$, $13$ miles. At what moment were the ships the smallest distance from each other, and what was that distance ?

%V0 The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.