Found problems: 1239
Find all sequences $(a_n)$ of positive integers satisfying the equality $a_n=a_{a_{n-1}}+a_{a_{n+1}}$
a) for all $n\ge 2$
b) for all $n \ge 3$
(I. Gorodnin)
The sequence $a_1,a_2…$ , is defined by the equations $a_1=1$ and $a_n=n.a_{[n/2]}$ for $n>1$. Prove that $a_n>n^2$ for $n>11$.
Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by
$$f_n(x)=\int^x_0f_{n-1}(t)dt$$
for all integers $n\ge1$.
a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$.
b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.
An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ .
(a) Prove that there are infinitely many numbers in the sequence equal to $0$.
(b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$.
For every non-negative integer $n$ , let $s_n$ be the sum of digits in the decimal expansion of $2^n$. Is the sequence $(s_n)_{n \in \mathbb{N}}$ eventually increasing ?
Each term of an infinite sequence $a_1 ,a_2 ,a_3 , \dots$ is equal to 0 or 1. For each positive integer $n$,
$$a_n + a_{n+1} \neq a_{n+2} + a_{n+3},\, \text{and}$$
$$a_n + a_{n+1} + a_{n+2} \neq a_{n+3} + a_{n+4} + a_{n+5}.$$
Prove that if $a_1 = 0$, then $a_{2020} = 1$.
Let $F_1, F_2, F_3 \cdots$ be the Fibonacci sequence, the sequence of positive integers satisfying $$F_1 =F_2=1$$ and $$F_{n+2} = F_{n+1} + F_n$$ for all $n \ge 1$.
Does there exist an $n \ge 1$ such that $F_n$ is divisible by $2014$? Prove your answer.
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 $u_{100} + u_{101}$.
Define sequence $a_{0}, a_{1}, a_{2}, \ldots, a_{2018}, a_{2019}$ as below:
$
a_{0}=1
$
$a_{n+1}=a_{n}-\frac{a_{n}^{2}}{2019}$, $n=0,1,2, \ldots, 2018$
Prove $a_{2019} < \frac{1}{2} < a_{2018}$
There are $30$ children, $a_1,a_2,...,a_{30}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.
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 $n$.
A triangle with sides $a,b,c$ and perimeter $2p$ is given. Is possible, a new triangle with sides $p-a$, $p-b$, $p-c$ is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely?
The sequences $(x_n),(y_n),(z_n),n\in\mathbb N$, are defined by the relations
$$x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},$$where $x_1=2$, $y_1=4$, and $x_1y_1z_1=x_1+y_1+z_1$.
(a) Show that $x_n^2\ne1$, $y_n^2\ne1$, $z_n^2\ne1$ for all $n$;
(b) Does there exist a $k\in\mathbb N$ for which $x_k+y_k+z_k=0$?
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.
Define a complexity of a set $a_1,a_2,\dots,$ consisting of 0 and 1 to be the smallest positive integer $k$ such that for some positive integers $\epsilon_1,\epsilon_2,\dots, \epsilon_k$ each number of the sequence $a_n$, $n>k$, has the same parity as $\epsilon_1 a_{n-1}+\epsilon_2 a_{n-2}+\dots+\epsilon_k a_{n-k}$. Sequence $a_1,a_2,\dots,$ has a complexity of $1000$. What is the complexity of sequence $1-a_1,1-a_2,\dots,$.
[I]Proposed by A. Kirichenko[/i]
Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$
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
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
Suppose a sequence of reals $\{a_n\}_{n\geq 0}$ satisfies $a_0 = 0$, $\frac{100}{101} <a_{100}<1$, and
$$2a_n - a_{n-1} -a_{n+1} \leq 2 (1-a_n )^3$$
for every $n\geq 1$.
(1) Define a sequence $b_n = a_n - \frac{n}{n+1}$. Prove that $b_n\leq b_{n+1}$ for any $n\geq 100$.
(2) Determine whether infinite series $\sum_{n=1}^\infty \frac{a_n}{n^2}$ converges or diverges.
Given a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that for any positive integers $k$, $l$ we have $k+l ~ | ~ a_k + a_l$. Prove that for all positive integers $k > l$, $a_k - a_l$ is divisible by $k-l$.
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its
sides.