Školjka

%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 $n \geq 1.$

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

%V0 Let $\{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: $$a_k - 2 \cdot a_{k + 1} + a_{k + 2} \geq 0$$ and $\sum^k_{j = 1} a_j \leq 1$ for all $k = 1,2, \ldots$. Prove that: $$0 \leq (a_{k} - a_{k + 1}) < \frac {2}{k^2}$$ for all $k = 1,2, \ldots$.

%V0 Let $u_1, u_2, \ldots, u_m$ be $m$ vectors in the plane, each of length $\leq 1,$ with zero sum. Show that one can arrange $u_1, u_2, \ldots, u_m$ as a sequence $v_1, v_2, \ldots, v_m$ such that each partial sum $v_1, v_1 + v_2, v_1 + v_2 + v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $\sqrt {5}.$