$(FRA 2)$ 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 $k > 1.$

For each positive integer $n$, set $x_n=\binom{2n}{n}$. a. Prove that if $\frac{2017^k}{2}<n<2017^k$ for some positive integer $k$ then $2017$ divides $x_n$. b. Find all positive integer $h>1$ such that there exists positive integers $N,T$ such that $(x_n)_{n>N}$ is periodic mod $h$ with period $T$.

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

A sequence $ \left( x_n\right)_{n\ge 0} $ of real numbers satisfies $ x_0>1=x_{n+1}\left( x_n-\left\lfloor x_n\right\rfloor\right) , $ for each $ n\ge 1. $ Prove that if $ \left( x_n\right)_{n\ge 0} $ is periodic, then $ x_0 $ is a root of a quadratic equation. Study the converse.

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?

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

Let $a_1$, $a_2$, $ ...$ be a sequence of numbers, each either $1$ or $-1$. Show that if $$\frac{a_1}{3}+\frac{a_2}{3^2} + ... =\frac{p}{q}$$ for integers $p$ and $q$ such that $3$ does not divide $q$, then the sequence $a_1$, $a_2$, $ ...$ is periodic; that is, there is some positive integer $n$ such that $a_i = a_{n+i}$ for $i = 1$, $2$,$...$.

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.

A finite sequence $x_1,\dots,x_r$ of positive integers is a [i]palindrome[/i] if $x_i=x_{r+1-i}$ for all integers $1 \le i \le r$. Let $a_1,a_2,\dots$ be an infinite sequence of positive integers. For a positive integer $j \ge 2$, denote by $a[j]$ the finite subsequence $a_1,a_2,\dots,a_{j-1}$. Suppose that there exists a strictly increasing infinite sequence $b_1,b_2,\dots$ of positive integers such that for every positive integer $n$, the subsequence $a[b_n]$ is a palindrome and $b_{n+2} \le b_{n+1}+b_n$. Prove that there exists a positive integer $T$ such that $a_i=a_{i+T}$ for every positive integer $i$.

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc. (a) Prove the relation \[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \] where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that \[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\] (b) Prove that the sequence of circles $K_1,K_2, \ldots $ is periodic.

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

$a_1,a_2,\ldots$ is a sequence of [u]nonzero integer[/u] numbers that for all $n\in\mathbb{N}$, if $a_n=2^\alpha k$ such that $k$ is an odd integer and $\alpha$ is a nonnegative integer then: $a_{n+1}=2^\alpha-k$. Prove that if this sequence is periodic, then for all $n\in\mathbb{N}$ we have: $a_{n+2}=a_n$. (The sequence $a_1,a_2,\ldots$ is periodic iff there exists natural number $d$ that for all $n\in\mathbb{N}$ we have: $a_{n+d}=a_n$)