Found problems: 5923
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}^{\ast}$ be a strictly increasing function. Prove that:
[list=a]
[*]There exists a decreasing sequence of positive real numbers, $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $y_{n}\leq2y_{f(n)}$, for all $n\in\mathbb{N}$.
[*]If $(x_{n})_{n\in\mathbb{N}}$ is a decreasing sequence of real numbers, converging to $0$, then there exists a decreasing sequence of real numbers $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $x_{n}\leq y_{n} \leq2y_{f(n)}$, for all $n\in\mathbb{N}$.[/list]
Suppose $f$ is a polynomial in $\mathbb{Z}[X]$ and m is integer .Consider the sequence $a_i$ like this $a_1=m$ and $a_{i+1}=f(a_i)$ find all polynomials $f$ and alll integers $m$ that for each $i$:
\[ a_i | a_{i+1}\]
Let $p > 3$ be a prime such that $p\equiv 3 \pmod 4.$ Given a positive integer $a_0$ define the sequence $a_0, a_1, \ldots $ of integers by $a_n = a^{2^n}_{n-1}$ for all $n = 1, 2,\ldots.$ Prove that it is possible to choose $a_0$ such that the subsequence $a_N , a_{N+1}, a_{N+2}, \ldots $ is not constant modulo $p$ for any positive integer $N.$
Prove that there exist two strictly increasing sequences $a_{n}$ and $b_{n}$ such that $a_{n}(a_{n} +1)$ divides $b_{n}^2 +1$ for every natural $n$.
Prove that among the elements of the sequence $\left\{ \left\lfloor n\sqrt{2003} \right\rfloor \right\}_{n\geq 1}$ one can find a geometric progression having any number of terms, and having the ratio bigger than $k$, where $k$ can be any positive integer.
[i]Radu Gologan[/i]
Determine the increasing geometric progressions, with three integer terms, such that the sum of these terms is $57$
$a_{n}$ ($n$ is integer) is a sequence from positive reals that \[a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4\] Prove $a_{n}$ is constant.
Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$.
Let $D$ be an infinite in both sides sequence of $0$s and $1$s. For each positive integer $n$ we denote with $a_n$ the number of different subsequences of $0$s and $1$s in $D$ of length $n$. Does there exist a sequence $D$ for which for each $n\geq 22$ the number $a_n$ is equal to the $n$-th prime number?
Let $\alpha > 0$ be a real number. Compute the limit of the sequence $\{x_n\}_{n\geq 1}$ defined by $$x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha}\\ 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases}$$
Consider an arithmetic progression $a_0,\ldots,a_n$ with $n\ge2$. Prove that $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$
Given is a sequence $a_1, a_2, \ldots$, such that $a_1=1$ and $a_{n+1}=\frac{9a_n+4}{a_n+6}$ for any $n \in \mathbb{N}$. Which terms of this sequence are positive integers?
Show that the sequence $\{a_n\}_{n\geq1}$ defined by $a_n = [n \sqrt 2]$ contains an infinite number of integer powers of $2$. ($[x]$ is the integer part of $x$.)
$x_1=\frac{1}{2}$ and $x_{k+1}=\frac{x_k}{x_1^2+...+x_k^2}$
Prove that $\sqrt{x_k^4+4\frac{x_{k-1}}{x_{k+1}}}$ is rational
In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed.
Let $c$ be a positive real and $a_1, a_2, \dots$ be a sequence of nonnegative integers satisfying the following conditions for every positive integer $n$:
[b](i)[/b]$\frac{2^{a_1}+2^{a_2}+\cdots+2^{a_n}}{n}$ is an integer;
[b](ii)[/b]$\textbullet 2^{a_n}\leq cn$.
Prove that the sequence $a_1, a_2, \dots$ is eventually constant.
Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.
In a language$,$ an alphabet with $25$ letters is used$;$ words are exactly all sequences of $($ not necessarily different $)$ letters of length $17.$ Two ends of a paper strip are glued so that the strip forms a ring$;$ the strip bears a sequence of $5^{18}$ letters$.$ Say that a word is singular if one can cut a piece bearing exactly that word from the strip$,$ but one cannot cut out two such non-overlapping pieces$.$ It is known that one can cut out $5^{16}$ non-overlapping pieces each containing the same word$.$ Determine the largest possible number of singular words$.$
[i](Bogdanov I.)[/i]
Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether
$$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$
Let $n$ be an integer greater than 2. A positive integer is said to be [i]attainable [/i]if it is 1 or can be obtained from 1 by a sequence of operations with the following properties:
1.) The first operation is either addition or multiplication.
2.) Thereafter, additions and multiplications are used alternately.
3.) In each addition, one can choose independently whether to add 2 or $n$
4.) In each multiplication, one can choose independently whether to multiply by 2 or by $n$.
A positive integer which cannot be so obtained is said to be [i]unattainable[/i].
[b]a.)[/b] Prove that if $n\geq 9$, there are infinitely many unattainable positive integers.
[b]b.)[/b] Prove that if $n=3$, all positive integers except 7 are attainable.
Determine all geometric progressions such that the product of the three first terms is $64$ and the sum of them is $14$.
Let $a_1, a_2, \cdots$, be a sequence with the following properties.
I. $a_1 = 1$, and
II. $a_{2n}=n\cdot a_n$ for any positive integer $n$.
What is the value of $a_{2^{100}}$?
$ \textbf{(A)}\; 1\qquad
\textbf{(B)}\; 2^{99}\qquad
\textbf{(C)}\; 2^{100}\qquad
\textbf{(D)}\; 2^{4950}\qquad
\textbf{(E)}\; 2^{9999} $
$x_1, x_2, ... , x_8$ is a permutation of $1, 2, ..., 8$. A move is to take $x_3$ or $x_8$ and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at $1, 2, ..., 8$.
Is it possible for Pingu to choose $2025$ positive integers $a_1, ..., a_{2025}$ such that:
1. The sequence $a_i$ is increasing;
2. $\gcd(a_1,a_2)>\gcd(a_2,a_3)>...>\gcd(a_{2024},a_{2025})>\gcd(a_{2025},a_1)>1$?
[i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]
The terms of a sequence $(T_n)$ satisfy $T_n T_{n+1} =n$ for all positive integers $n$ and
$$\lim_{n\to \infty} \frac{ T_{n} }{ T_{n+1}}=1.$$
Show that $ \pi T_{1}^{2}=2.$