Found problems: 5923
Define a sequence $\left\{a_{i}\right\}$ by $a_{1}=3$ and $a_{i+1}=3^{a_{i}}$ for $i \geq 1.$ Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_{i} ?$
Let a sequence $\left\{ {{x_n}} \right\}$ defined by:
\[\left\{ \begin{array}{l}
{x_0} = - 2 \\
{x_n} = \frac{{1 - \sqrt {1 - 4{x_{n - 1}}} }}{2},\forall n \ge 1 \\
\end{array} \right.\]
Denote $u_n=n.x_n$ and ${v_n} = \prod\limits_{i = 0}^n {\left( {1 + x_i^2} \right)} $. Prove that $\left\{ {{u_n}} \right\}$, $\left\{ {{v_n}} \right\}$ have finite limit.
Let $\mathcal P$ be the set of all points in $\mathbb R^n$ with rational coordinates. For the points $A,B \in \mathcal l{P}$, one can move from $A$ to $B$ if the distance $AB$ is $1$. Prove that every point in $\mathcal l{ P}$ can be reached from any other point in $\mathcal{P}$ by a finite sequence of moves if and only if $n \geq 5$.
Define a sequence $a_0$, $a_1$, $a_2$, $,...$ recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = a_{n+1} + xa_n$ for each $n \ge 0$ and some real number $x$. The infinite series $$ \sum^{\infty}_{n=0}\frac{a_n}{10^n} = 1.$$ Compute $x$.
Let $d_1, d_2, \ldots, d_n$ be given integers. Show that there exists a graph whose sequence of degrees is $d_1, d_2, \ldots, d_n$ and which contains an perfect matching if, and only if, there exists a graph whose sequence of degrees is $d_2, d_2, \ldots, d_n$ and a graph whose sequence of degrees is $d_1-1, d_2-1, \ldots, d_n-1$.
Let us define the sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,...$. with the following conditions
$a_1 = 3, b_1 = 1$ and $a_{n +1} =\frac{a_n^2+b_n^2}{2}$ and $b_{n + 1}= a_n \cdot b_n$ for each $n = 1, 2,...$.
Find all different prime factors οf the number $a_{2000} + b_{2000}$.
Denote by $B(c,r)$ the open disk of center $c$ and radius $r$ in the plane. Decide whether there exists a sequence $\{z_n\}^\infty_{n=1}$ of points in $\mathbb R^2$ such that the open disks $B(z_n,1/n)$ are pairwise disjoint and the sequence $\{z_n\}^\infty_{n=1}$ is convergent.
Consider the sequence $(a_n)$ satisfying $a_1=\dfrac{1}{2},a_{n+1}=\sqrt[3]{3a_{n+1}-a_n}$ and $0\le a_n\le 1,\forall n\ge 1.$
a. Prove that the sequence $(a_n)$ is determined uniquely and has finite limit.
b. Let $b_n=(1+2.a_1)(1+2^2a_2)...(1+2^na_n), \forall n\ge 1.$
Prove that the sequence $(b_n)$ has finite limit.
Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying
\[f_k(x)=x+k\varphi(x)\]
where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with
[list]
[*] $a_1=c$, and
[*] $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$
[/list]
Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if
(a) $k=2025$ ?
(b) $k=2065$ ?
[i]Proposed by chorn[/i]
Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions:
$(i)$ $a_0=a_n=1$;
$(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$;
$(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
$(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and $a_i$ divides $a_{i-1}+a_{i+1}$.
Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.
For each positive integer $ n$, the mean of the first $ n$ terms of a sequence is $ n$. What is the $ 2008$th term of the sequence?
$ \textbf{(A)}\ 2008 \qquad
\textbf{(B)}\ 4015 \qquad
\textbf{(C)}\ 4016 \qquad
\textbf{(D)}\ 4,030,056 \qquad
\textbf{(E)}\ 4,032,064$
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself?
$\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$
Gorf the frog is standing on the first lily pad in a row of lily pads numbered from $1$ to $20$ from left to right. On a single jump, Gorf is able to jump either $1,2,$ or $3$ lily pads to the right. Unfortunately all the prime-numbered lily pads are contaminated with a deadly poison. How many sequences of jumps are there that allow Gorf to jump to the twentieth lily pad, while avoiding the poison?
A family $\mathcal{F}$ of sets is called [i]really neat[/i] if for any $A,B\in \mathcal{F}$, there is a set $C\in \mathcal{F}$ such that $A\cup B = A\cup C=B\cup C$. Let
$$f(n)=\min \left\{ \max_{A\in \mathcal{F}} |A| \colon \mathcal{F} \text{ is really neat and } |\cup \mathcal{F}| =n\right\} .$$
Prove that the sequence $f(n)/n$ converges and find its limit.
Consider three sequences $(a_n)_{n=1}^{^\infty}$, $(b_n)_{n=1}^{^\infty}$ , $(c_n)_{n=1}^{^\infty}$, each of which has pairwisedistinct terms. Prove that there exist two indices $k$ and $l$ for which $k < l$, $$a_k < a_l
, b_k < b_l , \,\,\, and \,\,\, c_k < c_l.$$
We define the sequence $x_n$ so that
\[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\]
Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.
A sequence of integers $a_{1}, a_{2}, a_{3}, \cdots$ is defined as follows: $a_{1}=1$, and for $n \ge 1$, $a_{n+1}$ is the smallest integer greater than $a_{n}$ such that $a_{i}+a_{j} \neq 3a_{k}$ for any $i, j, $ and $k$ in $\{1, 2, 3, \cdots, n+1 \}$, not necessarily distinct. Determine $a_{1998}$.
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows:
we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.
Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.
Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that
\[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\]
where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$
The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\]
Show that $ a_{2008} < 5$.
Find all integers $k$, such that there exists an integer sequence ${\{a_n\}}$ satisfies two conditions below
(1) For all positive integers $n$,$a_{n+1}={a_n}^3+ka_n+1$
(2) $|a_n| \leq M$ holds for some real $M$
Determine if there exists a strictly increasing sequence of positive integers $a_1$, $a_2$, ... such that $a_n \le n^3$ for every positive integer $n$ and that every positive integer can be written uniquely as the difference of two terms in the sequence.
A sequence of real numbers $\{a_n\}_{n = 1}^\infty (n=1,2,...)$ has the following property:
\begin{align*}
6a_n+5a_{n-2}=20+11a_{n-1}\ (\text{for }n\geq3).
\end{align*}
The first two elements are $a_1=0, a_2=1$. Find the integer closest to $a_{2011}$.
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 $a_{2022}\leq 1$.
We choose random a unitary polynomial of degree $n$ and coefficients in the set $1,2,...,n!$. Prove that the probability for this polynomial to be special is between $0.71$ and $0.75$, where a polynomial $g$ is called special if for every $k>1$ in the sequence $f(1), f(2), f(3),...$ there are infinitely many numbers relatively prime with $k$.