Školjka

%V0 Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let $$\displaystyle c_n = \sum^8_{k=1} a^n_k$$ for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0$.

%V0 In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$ $$r_n \geq Cn^{\beta}, n = 1,2, \ldots$$

%V0 Define sequence $(a_n)$ by $\sum_{d|n} a_d = 2^n.$ Show that $n|a_n.$

%V0 Let $p(x)$ be a cubic polynomial with rational coefficients. $q_1$, $q_2$, $q_3$, ... is a sequence of rationals such that $q_n = p(q_{n + 1})$ for all positive $n$. Show that for some $k$, we have $q_{n + k} = q_n$ for all positive $n$.

%V0 Let $a_n$ be the last nonzero digit in the decimal representation of the number $n!.$ Does the sequence $a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?

%V0 Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Pick any $x_1$ in $[0, 1)$ and define the sequence $x_1, x_2, x_3, \ldots$ by $x_{n+1} = 0$ if $x_n = 0$ and $x_{n+1} = \frac{1}{x_n} - \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that $$x_1 + x_2 + \ldots + x_n < \frac{F_1}{F_2} + \frac{F_2}{F_3} + \ldots + \frac{F_n}{F_{n+1}},$$ where $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for $n \geq 1.$