Suppose we have four real numbers $a,b,c,d$ such that $a$ is nonzero, $a,b,c$ form a geometric sequence, in that order, and $b,c,d$ form an arithmetic sequence, in that order. Compute the smallest possible value of $\frac{d}{a}.$ (A geometric sequence is one where every succeeding term can be written as the previous term multiplied by a constant, and an arithmetic sequence is one where every succeeeding term can be written as the previous term added to a constant.)

A sequence of $2n + 1$ non-negative integers $a_1, a_2, ..., a_{2n + 1}$ is given. There's also a sequence of $2n + 1$ consecutive cells enumerated from $1$ to $2n + 1$ from left to right, such that initially the number $a_i$ is written on the $i$-th cell, for $i = 1, 2, ..., 2n + 1$. Starting from this initial position, we repeat the following sequence of steps, as long as it's possible: [i]Step 1[/i]: Add up the numbers written on all the cells, denote the sum as $s$. [i]Step 2[/i]: If $s$ is equal to $0$ or if it is larger than the current number of cells, the process terminates. Otherwise, remove the $s$-th cell, and shift shift all cells that are to the right of it one position to the left. Then go to Step 1. Example: $(1, 0, 1, \underline{2}, 0) \rightarrow (1, \underline{0}, 1, 0) \rightarrow (1, \underline{1}, 0) \rightarrow (\underline{1}, 0) \rightarrow (0)$. A sequence $a_1, a_2,. . . , a_{2n+1}$ of non-negative integers is called balanced, if at the end of this process there’s exactly one cell left, and it’s the cell that was initially enumerated by $(n + 1)$, i.e. the cell that was initially in the middle. Find the total number of balanced sequences as a function of $n$. [i]Proposed by Viktor Simjanoski, North Macedonia[/i]

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 $n$.

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 $\{a_n\}$ be a sequence of integers satisfying the following conditions. [list] [*] $a_1=2021^{2021}$ [*] $0 \le a_k < k$ for all integers $k \ge 2$ [*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$. [/list] Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.

Show that the recursion $n=x_n(x_{n-1}+x_n+x_{n+1})$, $n=1,2,\ldots$, $x_0=0$ has exaclty one nonnegative solution. (translated by L. Erdős)

Let be given the sequence $(x_n)$ defined by $x_1 = 1$ and $x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor$ for all $n = 1,2,3,...,$ where $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have $$x_nx_{n+2} - x^2_{n+1} = 4^{n-1}$$ Trần Nam Dũng

Find the number of sequences $a_1,a_2,\dots,a_{100}$ such that $\text{(i)}$ There exists $i\in\{1,2,\dots,100\}$ such that $a_i=3$, and $\text{(ii)}$ $|a_i-a_{i+1}|\leq 1$ for all $1\leq i<100$.

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

a) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$, one can find $100$ members for which $a_k > k$. b) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$ there are infinitely many such numbers $a_k$ such that $a_k > k$. (A Andjans, Riga) PS. (a) for juniors (b) for seniors

Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$

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]

It is known that the general term $\{a_n\}$ of the sequence is $a_n =(\sqrt3 +\sqrt2)^{2n}$ ($n \in N*$), let $b_n= a_n +\frac{1}{a_n}$ . (1) Find the recurrence relation between $b_{n+2}$, $b_{n+1}$, $b_n$. (2) Find the unit digit of the integer part of $a_{2011}$.

Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: \[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\] Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ . [i]Proposed by Marcin Kuczma, Poland[/i]

Let $ a > 2$ be given, and starting $ a_0 \equal{} 1, a_1 \equal{} a$ define recursively: \[ a_{n\plus{}1} \equal{} \left(\frac{a^2_n}{a^2_{n\minus{}1}} \minus{} 2 \right) \cdot a_n.\] Show that for all integers $ k > 0,$ we have: $ \sum^k_{i \equal{} 0} \frac{1}{a_i} < \frac12 \cdot (2 \plus{} a \minus{} \sqrt{a^2\minus{}4}).$

The sequence $a_0, a_1, a_2, ...$ is defined by $a_{n+1} = a^2_n + (a_n - 1)^2$ for $n \ge 0$. Find all rational numbers $a_0$ for which there exist four distinct indices $k, m, p, q$ such that $a_q - a_p = a_m - a_k$.

For a sequence $a_1<a_2<\cdots<a_n$ of integers, a pair $(a_i,a_j)$ with $1\leq i<j\leq n$ is called [i]interesting[/i] if there exists a pair $(a_k,a_l)$ of integers with $1\leq k<l\leq n$ such that $$\frac{a_l-a_k}{a_j-a_i}=2.$$ For each $n\geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$.

Arithmetical progression $a_1, a_2, a_3, a_4,...$ contains $a_1^2 , a_2^2$ and $a_3^2$ at some positions. Prove that all terms of this progression are integers.

Define a complexity of a set $a_1,a_2,\dots,$ consisting of 0 and 1 to be the smallest positive integer $k$ such that for some positive integers $\epsilon_1,\epsilon_2,\dots, \epsilon_k$ each number of the sequence $a_n$, $n>k$, has the same parity as $\epsilon_1 a_{n-1}+\epsilon_2 a_{n-2}+\dots+\epsilon_k a_{n-k}$. Sequence $a_1,a_2,\dots,$ has a complexity of $1000$. What is the complexity of sequence $1-a_1,1-a_2,\dots,$. [I]Proposed by A. Kirichenko[/i]

The sum $$\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}$$ $ $ \\ can be expressed as a rational number $N$. Find the last 3 digits of $2021! \cdot N$.

Let $a_1, a_2, \ldots, a_{11}$ be integers. Prove that there exist numbers $b_1, b_2, \ldots, b_{11}$ such that [list] [*] $b_i$ is equal to $-1,0$ or $1$ for all $i \in \{1, 2,\dots, 11\}$. [*] all numbers can't be zero at a time. [*] the number $N=a_1b_1+a_2b_2+\ldots+a_{11}b_{11}$ is divisible by $2024$. [/list]

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, strictly decreasing function such that $f([0,1])\subseteq[0,1]$. [list=i] [*]For all positive integers $n{}$ prove that there exists a unique $a_n\in(0,1)$, solution of the equation $f(x)=x^n$. Moreover, if $(a_n){}$ is the sequence defined as above, prove that $\lim_{n\to\infty}a_n=1$. [*]Suppose $f$ has a continuous derivative, with $f(1)=0$ and $f'(1)<0$. For any $x\in\mathbb{R}$ we define \[F(x)=\int_x^1f(t) \ dt.\]Let $\alpha{}$ be a real number. Study the convergence of the series \[\sum_{n=1}^\infty F(a_n)^\alpha.\] [/list]

Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.

For every positive integer $k$ let $a_{k,1},a_{k,2},\ldots$ be a sequence of positive integers. For every positive integer $k$ let sequence $\{a_{k+1,i}\}$ be the difference sequence of $\{a_{k,i}\}$, i.e. for all positive integers $k$ and $i$ the following holds: $a_{k,i+1}-a_{k,i}=a_{k+1,i}$. Is it possible that every positive integer appears exactly once among numbers $a_{k,i}$? [i]Proposed by Dávid Matolcsi, Berkeley[/i]

Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $