For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$. (a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$. (b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.

Let $a_n$ be a sequence given by $a_1 = 18$, and $a_n = a_{n-1}^2+6a_{n-1}$, for $n>1$. Prove that this sequence contains no perfect powers.

Suppose that $(a_n)_{n\geq 1}$ is a sequence of real numbers satisfying $a_{n+1} = \frac{3a_n}{2+a_n}$. (i) Suppose $0 < a_1 <1$, then prove that the sequence $a_n$ is increasing and hence show that $\lim_{n \to \infty} a_n =1$. (ii) Suppose $ a_1 >1$, then prove that the sequence $a_n$ is decreasing and hence show that $\lim_{n \to \infty} a_n =1$.

Given a sequence $a_n$: \[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots \] (one '1', two '2' and so on) and another sequence $b_n$ such that $a_{b_n}=b_{a_n}$ for all positive integers $n$. It is known that $b_k=1$ for some $k>100$. Prove that $b_m=1$ for all $m>k$.

For $n = 1,2,3,...$. $a_n$ is defined by: $$a_n =\frac{1 \cdot 4 \cdot 7 \cdot ... (3n-2)}{2 \cdot 5 \cdot 8 \cdot ... (3n-1)}$$ Prove that for every $n$ holds that $$\frac{1}{\sqrt{3n+1}}\le a_n \le \frac{1}{\sqrt[3]{3n+1}}$$

Let $(a_n)_{n\geq 0}$ be a sequence defined by $a_0\geq 0$ and the recurrence relation $$a_{n+1}=\frac{a_n^2-1}{n+1},$$ for all $n\geq 0$. Prove that here exists a real number $a> 0$ such that: [list] [*] if $a_0\geq a,$ $\lim_{n\rightarrow\infty}a_n = \infty$; [*] if $a_0\in [0,a),$ $\lim_{n\rightarrow\infty}a_n = 0$.

Consider the sequence $\{1,3,13,31,...\}$ that is obtained by following diagonally the following array of numbers in a spiral. Find the number in the $100$th position of that sequence. [img]https://cdn.artofproblemsolving.com/attachments/b/d/3531353472a748e3e0b1497a088472691f67fd.png[/img]

Given a sequence $(x_n )$ such that $\lim_{n\to \infty} x_n - x_{n-2}=0,$ prove that $$\lim_{n\to \infty} \frac{x_n -x_{n-1}}{n}=0.$$

A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.

We define the following sequence: $a_0=a_1=1$, $a_{n+1}=14a_n-a_{n-1}$. Prove that $2a_n-1$ is a perfect square.

Define the sequence $A_1, A_2, A_3, \dots$ by $A_1 = 1$ and for $n=1,2,3,\dots$ $$A_{n+1}=\frac{A_n+2}{A_n +1}.$$ Define the sequences $B_1, B_2, B_3,\dots$ by $B_1=1$ and for $n=1,2,3,\dots$ $$B_{n+1}=\frac{B_n^2 +2}{2B_n}.$$ Prove that $B_{n+1}=A_{2^n}$ for all non-negative integers $n$.

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

Let be a natural number $ m\ge 2. $ [b]a)[/b] Let be $ m $ pairwise distinct rational numbers. Prove that there is an ordering of these numbers such that these are terms of an arithmetic progression. [b]b)[/b] Given that for any $ m $ pairwise distinct real numbers there is an ordering of them such that they are terms of an arithmetic sequence, determine the number $ m. $ [i]Bogdan Blaga[/i]

For every non-negative integer $i$, define the number $M(i)$ as follows: write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$. (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ ) (a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $. Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ . (b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ . Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$. (A Kanel)

Define the sequence $x_1,x_2,\dots$ by the initial terms $x_1=2, x_2=4$, and the recurrence relation \[x_{n+2}=3x_{n+1}-2x_n+\frac{2^n}{x_n} \quad \text{for} \quad n \ge 1.\] Prove that $\lim_{n \to \infty} \frac{x_n}{2^n}$ exists and satisfies \[\frac{1+\sqrt{3}}{2} \le \lim_{n \to \infty} \frac{x_n}{2^n} \le \frac{3}{2}.\]

[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ [b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

Let be two sequences $ \left( a_n \right)_{n\ge 0} , \left( b_n \right)_{n\ge 0} $ satisfying the following system: $$ \left\{ \begin{matrix} a_0>0,& \quad a_{n+1} =a_ne^{-a_n} , &\quad\forall n\ge 0 \\ b_{0}\in (0,1) ,& \quad b_{n+1} =b_n\cos \sqrt{b_n} ,& \quad\forall n\ge 0 \end{matrix} \right. $$ Calculate $ \lim_{n\to\infty} \frac{a_n}{b_n} . $ [i]Florian Dumitrel[/i]

$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$

Let $\frac{2}{\sqrt5+1}\leq p < 1$, and let the real sequence $\{ a_n \}$ have the following property: for every sequence $\{ e_n \}$ of $0$'s and $\pm 1$'s for which $\sum_{n=1}^\infty e_np^n=0$, we also have $\sum_{n=1}^\infty e_na_n=0$. Prove that there is a number $c$ such that $a_n=cp^n$ for all $n$. [Z. Daroczy, I. Katai]

The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$. Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

[b]a)[/b] Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of real numbers having the property that $ \left| x_{n+1} -x_n \right|\leqslant 1/2^n, $ for any $ n\geqslant 1. $ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent. [b]b)[/b] Create a sequence $ \left( y_n \right)_{n\ge 1} $ of real numbers that has the following properties: $ \text{(i) } \lim_{n\to\infty } \left( y_{n+1} -y_n \right) = 0 $ $ \text{(ii) } $ is bounded $ \text{(iii) } $ is divergent [i]Eugen Popa[/i]

We say that we extend a finite sequence of positive integers $(a_1,\dotsc,a_n)$ if we replace it by \[(1,2,\dotsc,a_1-1,a_1,1,2,\dotsc,a_2-1,a_2,1,2,\dotsc,a_3-1,a_3,\dotsc,1,2,\dotsc,a_n-1,a_n)\] i.e., each element $k$ of the original sequence is replaced by $1,2,\dotsc,k$. Géza takes the sequence $(1,2,\dotsc,9)$ and he extends it $2017$ times. Then he chooses randomly one element of the resulting sequence. What is the probability that the chosen element is $1$?

Let $\alpha$ be a real number with $\alpha>1$. Let the sequence $(a_n)$ be defined as $$a_n=1+\sqrt[\alpha]{2+\sqrt[\alpha]{3+\ldots+\sqrt[\alpha]{n+\sqrt[\alpha]{n+1}}}}$$ for all positive integers $n$. Show that there exists a positive real constant $C$ such that $a_n<C$ for all positive integers $n$.