Školjka

%V0 Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

%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.

%V0 Determine all real values of the parameter $a$ for which the equation $$16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0$$ has exactly four distinct real roots that form a geometric progression.

%V0 Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$

%V0 A box contains $p$ white balls and $q$ black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?

%V0 (a) Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes $$a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.$$ (b) Find the rearrangement that minimizes $Q.$