The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$ {A. Andjans, Riga)

Let $\{u_n\}_ {n\ge 1}$ be given sequence satisfying the conditions: $u_1 = 0$, $u_2 = 1$, $u_{n+1} = u_{n-1} + 2n - 1$ for $n \ge 2$. 1) Calculate $u_5$. 2) Calculate $u_{100} + u_{101}$.

A sequence ($a_n$) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$. Prove that the inequality $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$.

Let $a_1=1,$ $a_2=2,$ $a_n=2a_{n-1}+a_{n-2},$ $n=3,4,\cdots.$ Prove that for any integer $n\geq5,$ $a_n$ has at least one prime factor $p,$ such that $p\equiv 1\pmod{4}.$

A very well known family of mathematicians has three children called [i]Antonia, Bernhard[/i] and [i]Christian[/i]. Each evening one of the children has to do the dishes. One day, their dad decided to construct of plan that says which child has to do the dishes at which day for the following $55$ days. Let $x$ be the number of possible such plans in which Antonia has to do the dishes on three consecutive days at least once. Furthermore, let $y$ be the number of such plans in which there are three consecutive days in which Antonia does the dishes on the first, Bernhard on the second and Christian on the third day. Determine, whether $x$ and $y$ are different and if so, then decide which of those is larger.

Prove the inequality $$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$ for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$

A sequence $x_0, x_1, x_2, . . .$ is given by $x_0 = 8$ and $x_{n+1} =\frac{1 + x_n}{1- x_n}$ for $n = 0, 1, 2, . . . .$ Determine the number $x_{2013}$.

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

For each positive real number $r$, define $a_0(r) = 1$ and $a_{n+1}(r) = \lfloor ra_n(r) \rfloor$ for all integers $n \ge 0$. (a) Prove that for each positive real number $r$, the limit \[ L(r) = \lim_{n \to \infty} \frac{a_n(r)}{r^n} \] exists. (b) Determine all possible values of $L(r)$ as $r$ varies over the set of positive real numbers. [i]Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.[/i]

$N$ denotes the set of all natural numbers. Define a function $T: N \to N$ such that $T (2k) = k$ and $T (2k + 1) = 2k + 2$. We write $T^2 (n) = T (T (n))$ and in general $T^k (n) = T^{k-1} (T (n))$ for all $k> 1$. (a) Prove that for every $n \in N$, there exists $k$ such that $T^k (n) = 1$. (b) For $k \in N$, $c_k$ denotes the number of elements in the set $\{n: T^k (n) = 1\}$. Prove that $c_{k + 2} = c_{k + 1} + c_k$, for $1 \le k$.

Given a sequence $ (c_n) $ of natural numbers defined recursively: $ c_1 = 2 $, $ c_{n+1} = \left[ \frac{3}{2}c_n\right] $. Prove that there are infinitely many even numbers and infinitely many odd numbers among the terms of this sequence.

For a sequence of integers $a_1,a_2,a_3,...$ with $0<a_1<a_2<a_3<...$ applies: $$a_n=4a_{n-1}-a_{n-2} \,\,\, for \,\,\, n > 2$$ It is further given that $a_4 = 194$. Calculate $a_5$.

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]