Found problems: 5923
The sequence of real numbers $\{a_n\}$, $n \in \mathbb{N}$ satisfies the following condition: $a_{n+1}=a_n(a_n+2)$ for any $n \in \mathbb{N}$. Find all possible values for $a_{2004}$.
How many of the fi rst $1000$ positive integers can be written as the sum of fi nitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$ ,$...$?
Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula
$$a_n =3(n^2 + n) + 7$$
Prove that this sequence has the following properties:
1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $,
2( No term of the sequence is the cube of an integer.
Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2. Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of
\[
\left|1-\sum_{i \in X} a_{i}\right|
\]
is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ with sum equal to 2 such that
\[
\sum_{i \in X} b_{i}=1.
\]
Find all sequences $a_1$, $a_2$, $\dots$ of nonnegative integers such that for all positive integers $n$, the polynomial \[1+x^{a_1}+x^{a_2}+\dots+x^{a_n}\] has at least one integer root. (Here $x^0=1$.)
[i]Kornpholkrit Weraarchakul[/i]
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
Given sequence $\{a_n\}$ satisfying:
$$ a_{n+1} = \frac{ lcm(a_n,a_{n-1})}{\gcd(a_n, a_{n-1})} $$
It is given that $a_{209} =209$ and $a_{361} = 361$. Find all possible values of $a_{2020}$.
Given unbounded strictly increasing sequence $a_1, a_2, ... , a_n, ...$ of positive numbers. Prove that
a) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:
$$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1$$
b) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:
$$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1985$$
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.
For any finite sequence $(x_1,\ldots,x_n)$, denote by $N(x_1,\ldots,x_n)$ the number of ordered index pairs $(i,j)$ for which $1 \le i<j\le n$ and $x_i=x_j$. Let $p$ be an odd prime, $1 \le n<p$, and let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be arbitrary residue classes modulo $p$. Prove that there exists a permutation $\pi$ of the indices $1,2,\ldots,n$ for which
\[N(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)})\le \min(N(a_1,a_2,\ldots,a_n),N(b_1,b_2,\ldots,b_n)).\]
Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate
$\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$
The real positive numbers $a_1, a_2,a_3$ are three consecutive terms of an arithmetic progression, and similarly, $b_1, b_2, b_3$ are distinct real positive numbers and consecutive terms of an arithmetic progression. Is it possible to use three segments of lengths $a_1, a_2, a_3$ as bases, and other three segments of lengths $b_1, b_2, b_3$ as altitudes, to construct three rectangles of equal area ?
From given positive numbers, the following infinite sequence is defined: $a_1$ is the sum of all
original numbers, $a_2$ is the sum of the squares of all original numbers, $a_3$ is the sum of the
cubes of all original numbers, and so on ($a_k$ is the sum of the $k$-th powers of all original
numbers).
a) Can it happen that $a_1 > a_2 > a_3 > a_4 > a_5$ and $a_5 < a_6 < a_7 < \ldots$? (4 points)
b) Can it happen that $a_1 < a_2 < a_3 < a_4 < a_5$ and $a_5 > a_6 > a_7 > \ldots$? (4 points)
[i](Alexey Tolpygo)[/i]
The first four terms of an infinite sequence $S$ of decimal digits are $1$, $9$, $8$, $2$, and succeeding terms are given by the final digit in the sum of the four immediately preceding terms. Thus $S$ begins $1$, $9$, $8$, $2$, $0$, $9$, $9$, $0$, $8$, $6$, $3$, $7$, $4$, $\cdots$. Do the digits $3$, $0$, $4$, $4$ ever come up consecutively in $S$?
Let $M=\{1,2,...,n\}$. Prove that the number of pairs $(A,a)$, where $A\subset M$ and $a$ is a permutation of $M$, for which $a(A)\cap A=\emptyset $, is equal to $n!.F_{n+1}$, where $F_{n+1}$ is the $n+1$ member of the Fibonacci sequence.
Let $n\ge2$ be a positive integer. Given a sequence $\left(s_i\right)$ of $n$ distinct real numbers, define the "class" of the sequence to be the sequence $\left(a_1,a_2,\ldots,a_{n-1}\right)$, where $a_i$ is $1$ if $s_{i+1} > s_i$ and $-1$ otherwise.
Find the smallest integer $m$ such that there exists a sequence $\left(w_i\right)$ of length $m$ such that for every possible class of a sequence of length $n$, there is a subsequence of $\left(w_i\right)$ that has that class.
[i]David Yang.[/i]
The sequence $a_0,a_1,a_2,...$ is defined by $a_0 = 0, a_1 = a_2 = 1$ and $a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)$ for all $n \in N$. Show that all $a_n$ are perfect squares .
Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations :
$$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and
$$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$
Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$
The sequence $a_1, a_2, a_3, \ldots$ satisfies $a_1=1$, and for all $n \ge 2$, it holds that
$$ a_n= \begin{cases}
a_{n-1}+3 ~~ \text{if} ~ n-1 \in \{ a_1,a_2,\ldots,,a_{n-1} \} ; \\
a_{n-1}+2 ~~ \text{otherwise}.
\end{cases} $$
Prove that for all positive integers n, we have
\[ a_n < n \cdot (1 + \sqrt{2}). \]
[i]Dominik Burek (Poland)[/i] (also known as [b][url=https://artofproblemsolving.com/community/user/100466]Burii[/url][/b])
Find all sequence of consecutive positive numbers which the sum of them is equal with $2019$.
Let $(x_n)_{n\geq 1}$ be the sequence defined by $x_1\in (0,1)$ and $x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}$ for all $n\geq 1$. Find the values of $\alpha\in\mathbb{R}$ for which the series $\sum_{n=1}^{\infty}x_n^{\alpha}$ is convergent.
Given the sequence $\{y_n\}_{n=1}^{\infty}$ defined by $y_1=y_2=1$ and
\[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\]
find all integers $k$ such that every term of the sequence is a perfect square.
Find a sequence of positive integer $f(n)$, $n \in \mathbb{N}$ such that
$(1)$ $f(n) \leq n^8$ for any $n \geq 2$,
$(2)$ for any pairwisely distinct natural numbers $a_1,a_2,\cdots, a_k$ and $n$, we have that
$$f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)$$
Prove that for any sequence $a_1, a_2, \ldots , a_{2002}$ of non-negative integers written in the usual decimal notation with $a_1 > 0$ there exists an integer $n$ such that $n^2$ starts with digits $a_1, a_2, \ldots , a_{2002}$ (in this order).