Found problems: 1239
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$:
$p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.
We define the following sequence: $a_0=a_1=1$, $a_{n+1}=14a_n-a_{n-1}$. Prove that
$2a_n-1$ is a perfect square.
For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix}
a^2+1 & a \\
a & 1
\end{pmatrix}^{n-2}$
Consider an infinite sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, \ldots, \underbrace{n,\ldots,n}_{n\text{ times}},\ldots.$$
Find the 1000th term of the sequence.
Find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the following conditions:
(i) For any positive integer $n$, $b_n \leq c_n$;
(ii) For any positive integer $n$, $b_{n+1}$ and $c_{n+1}$ is the two roots of the equation $x^2+b_nx+c_n=0$.
$a_1,a_2,\dots,a_k$ and $b_1,b_2,\dots,a_n$ are integer sequences with $a_1=b_1=0$, $a_k=26,b_n=25$, and $k+n=23$. It is known that $a_{i+1}-a_i=2\enspace\text{or}\enspace3$ and $b_{j+1}-b_j=2\enspace\text{or}\enspace3$ for all applicable $i$ and $j$, and the numbers $a_2,\dots,a_k,b_2,\dots,b_n$ are pairwise different. Determine the total number of such pairs of sequences.
Let $(a_n)_{n\in\mathbb N}$ be an increasing sequence of positive integers. A term $a_k$ in the sequence is said to be good if it a sum of some other terms (not necessarily distinct). Prove that all terms of the sequence, apart from finitely many of them, are good.
We say that a doubly infinite sequence
$. . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . .$
is subaveraging if $s_n = (s_{n−1} + s_{n+1})/4$ for all integers n.
(a) Find a subaveraging sequence in which all entries are different from each other. Prove that all
entries are indeed distinct.
(b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such
that $s_m = s_n$, then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .
Suppose there are $n\geq{5}$ different points arbitrarily arranged on a circle, the labels are $1, 2,\dots $, and $n$, and the permutation is $S$. For a permutation , a “descending chain” refers to several consecutive points on the circle , and its labels is a clockwise descending sequence (the length of sequence is at least $2$), and the descending chain cannot be extended to longer .The point with the largest label in the chain is called the "starting point of descent", and the other points in the chain are called the “non-starting point of descent” . For example: there are two descending chains $5, 2$and $4, 1$ in $5, 2, 4, 1, 3$ arranged in a clockwise direction, and $5$ and $4$ are their starting points of descent respectively, and $2, 1$ is the non-starting point of descent . Consider the following operations: in the first round, find all descending chains in the permutation $S$, delete all non-starting points of descent , and then repeat the first round of operations for the arrangement of the remaining points, until no more descending chains can be found. Let $G(S)$ be the number of all descending chains that permutation $S$ has appeared in the operations, $A(S)$ be the average value of $G(S)$of all possible n-point permutations $S$.
(1) Find $A(5)$.
(2)For $n\ge{6}$ , prove that $\frac{83}{120}n-\frac{1}{2} \le A(S) \le \frac{101}{120}n-\frac{1}{2}.$
Let $a_1,a_2,...$ be a sequence of natural numbers such that for each $n$, the product $(a_n - 1)(a_n- 2)...(a_n - n^2)$ is a positive integral multiple of $n^{n^2-1}$. Prove that for any finite set $P$ of prime numbers the following inequality holds:
$$\sum_{p\in P}\frac{1}{\log_p a_p}< 1$$
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 $a^n$.
Given a positive integer $m$, a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$-powerful if it satisfies
$$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$for all positive integers $n$.
(a) Show that a sequence is $30$-powerful if and only if at most one of its terms is non-zero.
(b) Find a sequence none of whose terms are zero but which is $2017$-powerful.
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions:
\[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\]
Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer.
Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.
Determine all possible values of $a_1$ for which there exists a sequence $a_1, a_2, \dots$ of rational numbers satisfying $$a_{n+1}^2-a_{n+1}=a_n$$
for all positive integers $n$.
The sequence $(Q_{n}(x))$ of polynomials is defined by
$$Q_{1}(x)=1+x ,\; Q_{2}(x)=1+2x,$$
and for $m \geq 1 $ by
$$Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x),$$
$$Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x).$$
Let $x_n$ be the largest real root of $Q_{n}(x).$ Prove that $(x_n )$ is an increasing sequence and that $\lim_{n\to \infty} x_n =0.$
Determine all sequences $a_1,a_2,a_3,\dots$ of positive integers that satisfy the equation
$$(n^2+1)a_{n+1} - a_n = n^3+n^2+1$$
for all positive integers $n$.
For any positive integer $m \geq 2$, let $p(m)$ be the smallest prime dividing $m$ and $P(m)$ be the largest prime dividing $m$. Let $C$ be a positive integer. Define sequences $\{a_n\}$ and $\{b_n\}$ by $a_0 = b_0 = C$ and, for each positive integer $k$ such that $a_{k-1}\geq 2$,
$$a_k=a_{k-1}-\frac{a_{k-1}}{p(a_{k-1})};$$
and, for each positive integer $k$ such that $b_{k-1}\geq 2$,
$$b_k=b_{k-1}-\frac{b_{k-1}}{P(b_{k-1})}$$
It is easy to see that both $\{a_n\}$ and $\{b_n\}$ are finite sequences which terminate when they reach the number $1$.
Prove that the numbers of terms in the two sequences are always equal.
The sequence $(a_n)$ is defined by $a_0 = 0$ and $a_{n+1} = [\sqrt[3]{a_n +n}]^3$ for $n \ge 0$.
(a) Find $a_n$ in terms of $n$.
(b) Find all $n$ for which $a_n = n$.
Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?
Let $f(x) = \frac{1}{1+x}$ where $x$ is a positive real number, and for any positive integer $n$,
let $g_n(x) = x + f(x) + f(f(x)) + ... + f(f(... f(x)))$, the last term being $f$ composed with itself $n$ times. Prove that
(i) $g_n(x) > g_n(y)$ if $x > y > 0$.
(ii) $g_n(1) = \frac{F_1}{F_2}+\frac{F_2}{F_3}+...+\frac{F_{n+1}}{F_{n+2}}$ , where $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} +F_n$ for $n \ge 1$.
Find all sequences of integer $x_1,x_2,..,x_n,...$ such that $ij$ divides $x_i+x_j$ for any distinct positive integer $i$, $j$.
The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for every integer $n$. Let $k$ be a fixed integer. A sequence $(a_n)$ of integers is said to be $\textit{phirme}$ if $a_n + a_{n+1} = F_{n+k}$ for all $n \geq 1$. Find all $\textit{phirme}$ sequences in terms of $n$ and $k$.
Given a function $ f:\mathbb{N}\longrightarrow\mathbb{N} , $ find the necessary and sufficient condition that makes the sequence
$$ \left(\left( 1+\frac{(-1)^{f(n)}}{n+1} \right)^{(-1)^{-f(n+1)}\cdot(n+2)}\right)_{n\ge 1} $$
to be monotone.