Found problems: 5923
Define a sequence $(a_n)$ for $n\ge1$ by $a_1=2$ and $a_{n+1}=a_n^{1+n^{-3/2}}$. Is $(a_n)$ convergent (i.e. $\lim_{n\to\infty}a_n<\infty$)?
A sequence of polynomials is defined by the recursion $P_1(x) = x+1$ and$$P_{n}(x) = \frac{(P_{n-1}(x)+1)^5 - (P_{n-1}(-x)+1)^5}{2}$$for all $n \geq 2$. Find the remainder when $P_{2022}(1)$ is divided by $1000$.
[i]Proposed by [b]treemath[/b][/i]
The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$
\[a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n\]
Show that $2003$ is not divisor of any of the terms in these two sequences.
Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties:
[list=disc]
[*]every term in the sequence is less than or equal to $2^{2023}$, and
[*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]
[/list]
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
Consider the sequence $\{1,3,13,31,...\}$ that is obtained by following diagonally the following array of numbers in a spiral. Find the number in the $100$th position of that sequence.
[img]https://cdn.artofproblemsolving.com/attachments/b/d/3531353472a748e3e0b1497a088472691f67fd.png[/img]
Consider two infinite sequences $a_0,a_1,a_2,\dots$ and $b_0,b_1,b_2,\dots$ of real numbers such that $a_0=0$, $b_0=0$ and
\[a_{k+1}=b_k, \quad b_{k+1}=\frac{a_kb_k+a_k+1}{b_k+1}\]
for each integer $k \ge 0$. Prove that $a_{2024}+b_{2024} \ge 88$.
For each positive integer $m> 1$, let $P (m)$ be the product of all prime numbers that divide $m$.
Define the sequence $a_1, a_2, a_3,...$ as followed:
$a_1> 1$ is an arbitrary positive integer,
$a_{n + 1} = a_n + P (a_n)$ for each positive integer $n$.
Prove that there exist positive integers $j$ and $k$ such that $a_j$ is the product of the first $k$ prime numbers.
Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.)
[b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often.
[b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of real numbers of the interval $ [1,\infty) . $ Suppose that the sequence $ \left( \left[ x_n^k\right]\right)_{n\ge 1} $ is convergent for all natural numbers $ k. $ Prove that $ \left( x_n\right)_{n\ge 1} $ is convergent.
Here, $ [\beta ] $ means the greatest integer smaller than $ \beta . $
For any positive integer $x$, we set
$$
g(x) = \text{ largest odd divisor of } x,
$$
$$
f(x) = \begin{cases}
\frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\
2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.}
\end{cases}
$$
Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.
Show that it is possible to color the set of integers \[M=\{ 1, 2, 3, \cdots, 1987 \},\] using four colors, so that no arithmetic progression with $10$ terms has all its members the same color.
Let $n$ be any positive integer and $M$ a set that contains $n$ positive integers. A sequence with $2^n$ elements is christmassy if every element of the sequence is an element of $M$. Prove that, in any christmassy sequence there exist some successive elements, the product of whom is a perfect square.
Sequences $x_n$ and $y_n$ satisfy the simultaneous relationships $x_k = x_{k+1} + y_{k+1}$ and $x_k > y_k$ for all $k \ge 1$. Furthermore, either $y_k = y_{k+1}$ or $y_k = x_{k+1}$. If $x_1 = 3 + \sqrt2$, $x_3 = 5 -\sqrt2$, and $y_1 = y_5$, evaluate $$(y_1)^2 + (y_2)^2 + (y_3)^2 + . . .$$
Given the sequence $(u_n)$ satisfying:$$\left\{ \begin{array}{l}
1 \le {u_1} \le 3\\
{u_{n + 1}} = 4 - \dfrac{{2({u_n} + 1)}}{{{2^{{u_n}}}}},\forall n \in \mathbb{Z^+}.
\end{array} \right.$$
Prove that: $1\le u_n\le 3,\forall n\in \mathbb{Z^+}$ and find the limit of $(u_n).$
For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$.
(a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$.
(b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.
Let $a_1,a_2,\ldots$ be a bounded sequence of reals. Is it true that the fact
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c$$implies $b=c$?
Define the sequence $a_1, a_2 \dots$ as follows: $a_1=1$ and for every $n\ge 2$,
\[
a_n = \begin{cases}
n-2 & \text{if } a_{n-1} =0 \\
a_{n-1} -1 & \text{if } a_{n-1} \neq 0
\end{cases}
\]
A non-negative integer $d$ is said to be {\em jet-lagged} if there are non-negative integers $r,s$ and a positive integer $n$ such that $d=r+s$ and that $a_{n+r} = a_n +s$. How many integers in $\{1,2,\dots, 2016\}$ are jet-lagged?
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.
Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$
[i]Proposed by Carl Schildkraut[/i]
Let $a_1,a_2,\dots,a_{100}$ be a sequence of integers. Initially, $a_1=1$, $a_2=-1$ and the remaining numbers are $0$. After every second, we perform the following process on the sequence: for $i=1,2,\dots,99$, replace $a_i$ with $a_i+a_{i+1}$, and replace $a_{100}$ with $a_{100}+a_1$. (All of this is done simultaneously, so each new term is the sum of two terms of the sequence from before any replacements.) Show that for any integer $M$, there is some index $i$ and some time $t$ for which $|a_i|>M$ at time $t$.
For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$
Let $n \ge 3$ and $\sigma \in S_n$ a permutation of the first $n$ positive integers. Prove that the numbers $\sigma (1), 2\sigma (2), 3\sigma(3), ... , n\sigma (n)$ cannot form an arithmetic, nor a geometric progression.