Found problems: 5923
Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that
\[ a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n},\]
where $ x \plus{} A$ denotes the set $ \{x \plus{} a \vert a \in A \}$.
Define the sequence of real numbers $\{x_n\}_{n \geq 1}$, where $x_1$ is any real number and \[x_n = 1 - x_1x_2\ldots x_{n-1} \text{ for all } n > 1.\] Show that $x_{2011} > \frac{2011}{2012}$.
Let $n \neq 0$. For every sequence of integers \[ A = a_0,a_1,a_2,\dots, a_n \] satisfying $0 \le a_i \le i$, for $i=0,\dots,n$, define another sequence \[ t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n) \] by setting $t(a_i)$ to be the number of terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $B$ such that $t(B) = B$.
A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that:
[list=1][*]the first fly is caught after a resting period of one minute;
[*]the resting period before catching the $2m^\text{th}$ fly is the same as the resting period before catching the $m^\text{th}$ fly and one minute shorter than the resting period before catching the $(2m+1)^\text{th}$ fly;
[*]when the chameleon stops resting, he catches a fly instantly.[/list]
[list=a][*]How many flies were caught by the chameleon before his first resting period of $9$ minutes in a row?
[*]After how many minutes will the chameleon catch his $98^\text{th}$ fly?
[*]How many flies were caught by the chameleon after 1999 minutes have passed?[/list]
For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and
\[ f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\
2^{\frac {x \plus{} 1}{2}} & \text{if \ \(x\) is odd}. \end{cases}
\]
Construct the sequence $ x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $ n$ such that $ x_n \equal{} 1992,$ and determine whether $ n$ is unique.
Consider the sequence $a_1, a_2, \dots$ of positive integers such that $a_1=2$ and $a_{n+1}=a_n^4+a_n^3-3a_n^2-a_n+2$, for all $n\geqslant 1$. Prove that there exist infinitely many prime numbers that don't divide any term of the sequence.
[i]Proposed by Pavel Ciurea[/i]
Study the convergence of a sequence $ \left( x_n\right)_{n\ge 0} $ for which $ x_0\in\mathbb{R}\setminus\mathbb{Q} , $ and $ x_{n+1}\in \left\{ \frac{x_n+1}{x_n} , \frac{x_n+2}{2x_n-1}\right\} , $ for all $ n\ge 1. $
Consider a sequence of 100 positive integers. Each member of the sequence, starting from the second one, is derived by either multiplying the previous number by 2 or dividing it by 16. Is it possible for the sum of these 100 numbers to be equal to $2^{2023}$?
[i]Proposed by Nika Glunchadze, Georgia[/i]
There are $2^n$ words of length $n$ over the alphabet $\{0, 1\}$. Prove that the following algorithm generates the sequence $w_0, w_1, \ldots, w_{2^n-1}$ of all these words such that any two consecutive words differ in exactly one digit.
(1) $w_0 = 00 \ldots 0$ ($n$ zeros).
(2) Suppose $w_{m-1} = a_1a_2 \ldots a_n,\quad a_i \in \{0, 1\}$. Let $e(m)$ be the exponent of $2$ in the representation of $n$ as a product of primes, and let $j = 1 + e(m)$. Replace the digit $a_j$ in the word $w_{m-1}$ by $1 - a_j$. The obtained word is $w_m$.
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
Let $ \lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form \[ \mu= \sum_{i=1}^{\infty}n_i\lambda_i ,\] where $ n_i \geq 0$ are integers and all but finitely many $ n_i$ are $ 0$. Let \[ L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .\] (In the latter sum, each $ \mu$ occurs as many times as its number of representations in the above form.) Prove that if \[ \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,\] then \[ \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.\]
[i]G. Halasz[/i]
Consider all right triangles with integer side lengths that form an arithmetic sequence. Compute the $2014$th smallest perimeter of all such right triangles.
A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube $n$ has moved to the position originally occupied by $27-n$ (for each $n = 1, 2, \ldots , 26$)?
Rachel has two indistinguishable tokens, and places them on the first and second square of a $1 \times 6$ grid of squares, She can move the pieces in two ways:
$\bullet$ If a token has free square in front of it, then she can move this token one square to the right.
$\bullet$ If the square immediately to the right of a token is occupied by the other token, then she can “leapfrog” the first token; she moves the first token two squares to the right, over the other token, so that it is on the square immediately to the right of the other token.
If a token reaches the $6$th square, then it cannot move forward any more, and Rachel must move the other one until it reaches the $5$th square. How many different sequences of moves for the tokens can Rachel make so that the two tokens end up on the $5$th square and the 6th square?
[b]p1.[/b] In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form a strictly increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to $94$. How many possible combinations of test scores could they have had? (Test scores at Greendale range between $0$ and $100$, inclusive.)
[b]p2.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that
$$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B1B1B1...}) + 1$$
Find the sum of all possible values of $10A + B$.
[b]p3.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $\overline{AC}$ and $\overline{BC}$, respectively, such that triangles $\vartriangle ADB$ and $\vartriangle CDE$ are similar and $DE =EB$. If $\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}$ with $a$, $b$ positive integers and $a$ squarefree, then find $a + b$.
[b]p4.[/b] Five bowling pins $P_1, P_2, ..., P_5$ are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is $\frac{a}{b}$ where $a$ and $b$ are relatively prime, find $a + b$. (Pins $P_i$ and $P_j$ are adjacent if and only if $|i - j| = 1$.)
[b]p5.[/b] How many terms in the expansion of $$(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})$$ have coeffcients equal to $1011$?
[b]p6.[/b] Suppose $f(x)$ is a monic quadratic polynomial with distinct nonzero roots $p$ and $q$, and suppose $g(x)$ is a monic quadratic polynomial with roots $p + \frac{1}{q}$ and $q + \frac{1}{p}$ . If we are given that $g(-1) = 1$ and $f(0)\ne -1$, then there exists some real number $r$ that must be a root of $f(x)$. Find $r$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Sequence of integers $\{u_n\}_{n \in \mathbb{N}_0}$ is given as: $u_0=0$, $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for all $n \in \mathbb{N}_0$
$a)$ Find $u_{1998}$
$b)$ If $p$ is a positive integer and $m=(2^p-1)^2$, find $u_m$
For every sequence $\{a_n\}~(n\in\mathbb N)$ we define the sequences $\{\Delta a_n\}$ and $\{\Delta^2a_n\}$ by the following formulas:
\begin{align*}\Delta a_n&=a_{n+1}-a_n,\\\Delta^2a_n&=\Delta a_{n+1}-\Delta a_n.\end{align*}Further, for all $n\in\mathbb N$ for which $\Delta a_n^2\ne0$, define
$$a_n'=a_n-\frac{(\Delta a_n)^2}{\Delta^2a_n}.$$
(a) For which sequences $\{a_n\}$ is the sequence $\{\Delta^2a_n\}$ constant?
(b) Find all sequences $\{a_n\}$, for which the numbers $a_n'$ are defined for all $n\in\mathbb N$ and for which the sequence $\{a_n'\}$ is constant.
(c) Assume that the sequence $\{a_n\}$ converges to $a=0$, and $a_n\ne a$ for all $n\in\mathbb N$ and the sequence $\{\tfrac{a_{n+1}-a}{a_n-a}\}$ converges to $\lambda\ne1$.
i. Prove that $\lambda\in[-1,1)$.
ii. Prove that there exists $n_0\in\mathbb N$ such that for all integers $n\ge n_0$ we have $\Delta^2a_n\ne0$.
iii. Let $\lambda\ne0$. For which $k\in\mathbb Z$ is the sequence $\{\tfrac{a_n'}{a_{n+k}}\}$ not convergent?
iv. Let $\lambda=0$. Prove that the sequences $\{a_n'/a_n\}$ and $\{a_n'/a_{n+1}\}$ converge to $0$. Find an example of $\{a_n\}$ for which the sequence $\{a_n'/a_{n+2}\}$ has a non-zero limit.
(d) What happens with part (c) if we remove the condition $a=0$?
Consider the sequence of consecutive even numbers starting from 0, arranged in a staggered format, where each row contains one more number than the previous row.
The beginning of this arrangement is shown below:
$0$
$2\; 4$
$6\;\underline{8}\;10$
$12\: 14\: 16\: 18$
$20\: 22 \: 24 \: 26\: 28 $
The number in the middle of the third row is 8. What is the number in the middle of the 101-st row?
A sequence $(x_n)_{n= 1}^{\infty}$ satisfies $x_1 = 1$ and for each $n > 1, x_n = \pm (n-1)x_{n-1} \pm (n-2)x_{n-2} \pm ... \pm 2x_2 \pm x_1$. Prove that the signs ” $\pm$” can be chosen so that $x_n \ne 12$ holds only for finitely many $n$.
Given a permutation ($a_0, a_1, \ldots, a_n$) of the sequence $0, 1,\ldots, n$. A transportation of $a_i$ with $a_j$ is called legal if $a_i=0$ for $i>0$, and $a_{i-1}+1=a_j$. The permutation ($a_0, a_1, \ldots, a_n$) is called regular if after a number of legal transportations it becomes ($1,2, \ldots, n,0$).
For which numbers $n$ is the permutation ($1, n, n-1, \ldots, 3, 2, 0$) regular?
$\textbf{N3:}$ For any positive integer $n$, define $rad(n)$ to be the product of all prime divisors of $n$ (without multiplicities), and in particular $rad(1)=1$. Consider an infinite sequence of positive integers $\{a_n\}_{n=1}^{\infty}$ satisfying that
\begin{align*} a_{n+1} = a_n + rad(a_n), \: \forall n \in \mathbb{N} \end{align*}
Show that there exist positive integers $t,s$ such that $a_t$ is the product of the $s$ smallest primes.
[i]Proposed by ltf0501[/i]
Evaluate: $\sum^{\infty}_{k=1} \tfrac{k}{a^{k-1}}$ for all $|a|<1$.
A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time:
[list]
[*] Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and no longer friends with $C$. All other friendship statuses are unchanged.
[/list]
Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
[i]Proposed by Adrian Beker, Croatia[/i]
Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]