Školjka

%V0 $(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$

%V0 Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds: $$P_{n+1}(x) + P_{n-1}(x) = xP_n(x).$$ Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$ $$(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.$$

%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 $(a_n)^{\infty}_{n=1}$ be a sequence of integers with $a_{n} < a_{n+1}, \quad \forall n \geq 1.$ For all quadruple $(i,j,k,l)$ of indices such that $1 \leq i < j \leq k < l$ and $i + l = j + k$ we have the inequality $a_{i} + a_{l} > a_{j} + a_{k}.$ Determine the least possible value of $a_{2008}$.