Found problems: 1239
A sequence of real numbers is called a [i]Fibonacci [/i] sequence if $$t_{n+2} = t_{n+1} + t_n$$ for $n= 1,2,3,. .$ .
Two Fibonacci sequences are said to be [i]essentially different[/i] if the terms of one sequence cannot be obtained by multiplying the terms of the other by a constant. For example, the Fibonacci sequences $1,2,3,5,8,...$ and $1,3,4,7,11,...$ are essentially different, but the sequences $1,2,3,5,8,...$ and $2,4,6,10,16,...$ are not.
(a) Prove that there exist real numbers $p$ and $q$ such that the sequences $1,p,p^2,p^3,...$ and $1,q,q^2,q^3,...$ are essentially different Fibonacci sequences.
(b) Let $a_1,a_2,a_3,...$ and $b_1,b_2,b_3,...$ be essentially different Fibonacci sequences. Prove that for every Fibonacci sequence $t_1,t_2,t_3,...$, there exists exactly one number $\alpha$ and exactly one number $\beta$, such that: $$t_n = \alpha a_n + \beta b_n$$ for $n = 1,2,3,...$
(c) $t_1,t_2,t_3,...$, is the Fibonacci sequence with $t_1 = 1$ and $t_2= 2$. Express $t_n$ in terms of $n$.
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
Let $(x_n)_{n\in Z}$ and $(y_n)_{n\in Z}$ be two sequences of integers such that $|x_{n+2} - x_n| \le 2$ and $x_n + x_m = y_{n^2+m^2}$ for all $n, m \in Z$. Show that the sequence of $x_n$s takes at most $6$ distinct values.
(Paolo Leonetti)
Let $(a_n)$ be a sequence of positive real numbers. Show that
$$ \limsup_{n \to \infty} n \left(\frac{1 +a_{n+1}}{a_n } -1 \right) \geq 1$$
and prove that $1$ cannot be replaced by any larger number.
A nonempty set $ A$ of real numbers is called a $ B_3$-set if the conditions $ a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $ a_1 \plus{} a_2 \plus{} a_3 \equal{} a_4 \plus{} a_5 \plus{} a_6$ imply that the sequences $ (a_1, a_2, a_3)$ and $ (a_4, a_5, a_6)$ are identical up to a permutation. Let $A = \{a_0 = 0 < a_1 < a_2 < \cdots \}$, $B = \{b_0 = 0 < b_1 < b_2 < \cdots \}$ be infinite sequences of real numbers with $ D(A) \equal{} D(B),$ where, for a set $ X$ of real numbers, $ D(X)$ denotes the difference set $ \{|x\minus{}y|\mid x, y \in X \}.$ Prove that if $ A$ is a $ B_3$-set, then $ A \equal{} B.$
Let $(a_n)_{n=1}^{\infty}$ be a strictly increasing sequence such that inequality
$$a_n(a_n-2a_{n-1})+a_{n-1}(a_{n-1}-2a_{n-2})\geq 0$$
holds for all $n \geq 3$. Prove that for all $n\geq2$ the inequality
$$a_n \geq a_{n-1}+a_{n-2}+\dots+a_1$$
holds as well.
Prove that there exists such an infinite sequence $\{x_i\}$, that for all $m$ and all $k$ ($m\ne k$) holds the inequality $$|x_m-x_k|>1/|m-k|$$
A finite sequence of integers $a_0,,a_1,\dots,a_n$ is called [i]quadratic[/i] if for each $i\in\{1,2,\dots n\}$ we have the equality $|a_i-a_{i-1}|=i^2$.
$\text{(i)}$ Prove that for any two integers $b$ and $c$, there exist a positive integer $n$ and a quadratic sequence with $a_0=b$ and $a_n = c$.
$\text{(ii)}$ Find the smallest positive integer $n$ for which there exists a quadratic sequence with $a_0=0$ and $a_n=2021$.
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.
$\boxed{A6}$The sequence $a_0,a_1,...$ is defined by the initial conditions $a_0=1,a_1=6$ and the recursion $a_{n+1}=4a_n-a_{n-1}+2$ for $n>1.$Prove that $a_{2^k-1}$ has at least three prime factors for every positive integer $k>3.$
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
Let $(a_n)_{n \geq 1}$ be a sequence of real numbers. We define a sequence of real functions $(f_n)_{n \geq 0}$ such that for all $x \in \mathbb{R}$, the following holds:
\[
f_0(x) = 1 \quad \text{and} \quad f_n(x) = \int_{a_n}^{x} f_{n-1}(t) \, dt \quad \text{for } n \geq 1.
\]
Find all possible sequences $(a_n)_{n \geq 1}$ such that $f_n(0) = 0$ for all $n \geq 2$.\\
[b]Note:[/b] It is not necessarily true that $f_1(0) = 0$.
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
Let ${(a_n)}_{n=0}^{\infty}$ and ${(b_n)}_{n=0}^{\infty}$ be real squences such that $a_0=40$, $b_0=41$ and for all $n\geq 0$ the given equalities hold.
$$a_{n+1}=a_n+\frac{1}{b_n} \hspace{0.5 cm} \text{and} \hspace{0.5 cm} b_{n+1}=b_n+\frac{1}{a_n}$$
Find the least possible positive integer value of $k$ such that the value of $a_k$ is strictly bigger than $80$.
The sequence $(a_n)$ is defined as:
$$a_1=1007$$
$$a_{i+1}\geq a_i+1$$
Prove the inequality:
$$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$
For every positive integer $k$ let $a_{k,1},a_{k,2},\ldots$ be a sequence of positive integers. For every positive integer $k$ let sequence $\{a_{k+1,i}\}$ be the difference sequence of $\{a_{k,i}\}$, i.e. for all positive integers $k$ and $i$ the following holds: $a_{k,i+1}-a_{k,i}=a_{k+1,i}$. Is it possible that every positive integer appears exactly once among numbers $a_{k,i}$?
[i]Proposed by Dávid Matolcsi, Berkeley[/i]
Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$.
Show that there exists a sequence ($a_n$) in $F_p$ with the property that for every other sequence ($b_n$) in $F_p$, the inequality $a_n \le b_n$ holds for all $n$.
Let $\alpha\in(0,1]$ be a given real number and let a real sequence $\{a_n\}^\infty_{n=1}$ satisfy the inequality
$$a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots$$Prove that if $\{a_n\}$ is bounded, then it must be convergent.
Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.
Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le i\le m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$,
so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j \ge i$.
Similarly, we define, for $1\le j \le n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$.
E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$.
(a) Prove that $a_j = c_j $ for $1 \le j \le n$.
(b) Prove that for $1\le k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by
\[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right),
\] where $\lfloor x\rfloor$ denotes the integer part of $x$.
[b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often.
[b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often.
[i]Proposed by Johan Meyer, South Africa[/i]
Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and
$$ e_n=a + \prod_{k=0}^{n-1} e_k$$
for $n \geq 1$.
Prove that
(a) There exist infinitely many prime numbers that divide one element of the sequence.
(b) There exists one prime number that does not divide an element of the sequence.
(Theresia Eisenkölbl)
For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$
(a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$
(b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$
[I]United Kingdom[/i]
Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.