Found problems: 5923
Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2003^nA\right\}\right)_{n\ge 1} . $
[b]a)[/b] Prove that each term of the above sequence hasn't a finite cardinal.
[b]b)[/b] Determine the intersection of the third element of the above sequence with the $ 2003\text{rd} $ element.
[i]Gheorghe Iurea[/i]
[hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1943241p13387495[/url].[/hide]
Let $ n $ be a positive integer. A sequence $(a_1,a_2,\ldots,a_n)$ of length is called $balanced$ if for every $ k $ $(1\leq k\leq n)$ the term $ a_k $ is equal with the number of distinct numbers from the subsequence $(a_1,a_2,\ldots,a_k).$
a) How many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ do exist?
b) For every positive integer $m$ find how many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ exist such that $a_n=m.$
Let $N$ be the number of sequences $a_1, a_2, . . . , a_{10}$ of ten positive integers such that
(i) the value of each term of the sequence at most $30$,
(ii) the arithmetic mean of any three consecutive terms of the sequence is an integer, and
(iii) the arithmetic mean of any fi ve consecutive terms of the sequence is an integer.
Compute $\sqrt{N}$.
Let $n$ be a positive integer. Define a sequence by $a_0 = 1$, $a_{2i+1} = a_i$, and $a_{2i+2} = a_i + a_{i+1}$ for each $i \ge 0$. Determine, with proof, the value of $a_0 + a_1 + a_2 + \dots + a_{2^n-1}$.
[i]Proposed by Kevin Ren.[/i]
In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most $100$ cities at distance exactly three from it. Prove that there is no city such that more than $2550$ other cities have distance exactly four from it.
A sequence of natural numbers $a_1,a_2,...$ satisfies $a_1 = 1, a_{n+2} = 2a_{n+1} - a_n +2$ for $n \in N$.
Prove that for every natural $n$ there exists a natural $m$ such that $a_na_{n+1} = a_m$.
Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]
Consider the sequence $1,9,8,3,4,3,…$ in which $a_{n+4}$ is the units digit of $a_n+a_{n+3}$, for $n$ positive integer. Prove that $a^2_{1985}+a^2_{1986}+…+a^2_{2000}$ is a multiple of $2$.
The natural number $m\geq 2$ is given.Sequence of natural numbers $(b_0,b_1,\ldots,b_m)$ is called concave if $b_k+b_{k-2}\le2b_{k-1}$ for all $2\le k\le m.$ Prove that there exist not greater than $2^m$ concave sequences starting with $b_0 =1$ or $b_0 =2$
For which $n \ge 2$ can one find a sequence of distinct positive integers $a_1, a_2, \ldots , a_n$ so that the sum
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots +\frac{a_n}{a_1}$$
is an integer?
[i](3 points)[/i]
Given an integer $n$ ,For a sequence of $X$ with the number of $n$ and $Y$ with the number of $100n$ , we call it a [b]spring [/b] . We have two following rules
$\blacksquare$ Choose four adjacent character , if it is $YXXY$ , than it can be changed into $XYYX$
$\blacksquare $ Choose. four adjacent character , if it is $XYYX $ , than it can be changed into $YXXY$
If [b]spring [/b] $A$ can become $B$ using the rules , than we call they are [b][color=#3D85C6]similar [/color][/b]
Thy to find the maximum of $C$ such that there exists $C$ distinct [b]springs[/b] and they are [b][color=#3D85C6]similar[/color][/b]
Let $ a,b,c,d$ be four distinct positive integers in arithmetic progression. Prove that $ abcd$ is not a perfect square.
Let $A > 0$ and $B = (3 + 2\sqrt{2})A$. Prove that in the finite sequence $a_k = \lfloor k / \sqrt{2} \rfloor$ for $k \in (A, B) \cap \mathbb{Z}$, the number of even and odd terms differs by at most $2$.
An underground burrow consists of an infinite sequence of rooms labeled by the integers $(\dots, -3, -2, -1, 0, 1, 2, 3,\dots)$. Initially, some of the rooms are occupied by one or more rabbits. Each rabbit wants to be alone. Thus, if there are two or more rabbits in the same room (say, room $m$), half of the rabbits (rounding down) will flee to room $m-1$, and half (also rounding down) to room $m+1$. Once per minute, this happens simultaneously in all rooms that have two or more rabbits. For example, if initially all rooms are empty except for $5$ rabbits in room $\#12$ and $2$ rabbits in room $\#13$, then after one minute, rooms $\text{\#11--\#14}$ will contain $2$, $2$, $2$, and $1$ rabbits, respectively, and all other rooms will be empty.
Now suppose that initially there are $k+1$ rabbits in room $k$ for each $k=0, 1, 2, \ldots, 9, 10$, and all other rooms are empty.
[list=a]
[*]Show that eventually the rabbits will stop moving.
[*] Determine which rooms will be occupied when this occurs.
[/list]
Do there exists many infinitely points like $(x_1,y_1),(x_2,y_2),...$ such that for any sequences like {$b_1,b_2,...$} of real numbers there exists a polynomial $P(x,y)\in R[x,y]$ such that we have for all $i$ :
$P(x_{i},y_{i})=b_{i}$
Let $\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx$.
(a) Prove that $s_n \leq \dfrac 2n$ for all odd $n$.
(b) Find all the limit points of the sequence $s_1, s_2, s_3, \dots$.
[i]Proposed by Cristi Calin[/i]
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
A word of length $n$ is an ordered sequence $x_1x_2\ldots x_n$ where $x_i$ is a letter from the set $\{ a,b,c \}$. Denote by $A_n$ the set of words of length $n$ which do not contain any block $x_ix_{i+1}, i=1,2,\ldots ,n-1,$ of the form $aa$ or $bb$ and by $B_n$ the set of words of length $n$ in which none of the subsequences $x_ix_{i+1}x_{i+2}, i=1,2,\ldots n-2,$ contains all the letters $a,b,c$.
Prove that $|B_{n+1}|=3|A_n|$.
[i]Vasile Pop[/i]
Let $f:[0,\infty)\to\mathbb R$ be differentiable and satisfy
$$f'(x)=-3f(x)+6f(2x)$$for $x>0$. Assume that $|f(x)|\le e^{-\sqrt x}$ for $x\ge0$. For $n\in\mathbb N$, define
$$\mu_n=\int^\infty_0x^nf(x)dx.$$
$a.$ Express $\mu_n$ in terms of $\mu_0$.
$b.$ Prove that the sequence $\frac{3^n\mu_n}{n!}$ always converges, and the the limit is $0$ only if $\mu_0$.
An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{n+1}=a_{n}+\lfloor \sqrt{a_{n}}\rfloor.\] Show that $a_{n}$ is a square if and only if $n=2^{k}+k-2$ for some $k \in \mathbb{N}$.
[b]p1.[/b] Find the last digit of $2022^{2022}$.
[b]p2.[/b] Let $a_1 < a_2 <... < a_8$ be eight real numbers in an increasing arithmetic progression. If $a_1 + a_3 + a_5 + a_7 = 39$ and $a_2 + a_4 + a_6 + a_8 = 40$, determine the value of $a_1$.
[b]p3.[/b] Patrick tries to evaluate the sum of the first $2022$ positive integers, but accidentally omits one of the numbers, $N$, while adding all of them manually, and incorrectly arrives at a multiple of $1000$. If adds correctly otherwise, find the sum of all possible values of $N$.
[b]p4.[/b] A machine picks a real number uniformly at random from $[0, 2022]$. Andrew randomly chooses a real number from $[2020, 2022]$. The probability that Andrew’s number is less than the machine’s number is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[b]p5.[/b] Let $ABCD$ be a square and $P$ be a point inside it such that the distances from $P$ to sides $AB$ and $AD$ respectively are $2$ and $4$, while $PC = 6$. If the side length of the square can be expressed in the form $a +\sqrt{b}$ for positive integers $a, b$, then determine $a + b$.
[b]p6.[/b] Positive integers $a_1, a_2, ..., a_{20}$ sum to $57$. Given that $M$ is the minimum possible value of the quantity $a_1!a_2!...a_{20}!$, find the number of positive integer divisors of $M$.
[b]p7.[/b] Jessica has $16$ balls in a box, where $15$ of them are red and one is blue. Jessica draws balls out the box three at a time until one of the three is blue. If she ever draws three red marbles, she discards one of them and shuffles the remaining two back into the box. The expected number of draws it takes for Jessica to draw the blue ball can be written as a common fraction $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.
[b]p8.[/b] The Lucas sequence is defined by these conditions: $L_0 = 2$, $L_1 = 1$, and $L_{n+2} =L_{n+1} +L_n$ for all $n \ge 0$. Determine the remainder when $L^2_{2019} +L^2_{2020}$ is divided by $L_{2023}$.
[b]p9.[/b] Let $ABCD$ be a parallelogram. Point $P$ is selected in its interior such that the distance from $P$ to $BC$ is exactly $6$ times the distance from $P$ to $AD$, and $\angle APB = \angle CPD = 90^o$. Given that $AP = 2$ and $CP = 9$, the area of $ABCD$ can be expressed as $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.
[b]p10.[/b] Consider the polynomial $P(x) = x^{35} + ... + x + 1$. How many pairs $(i, j)$ of integers are there with $0 \le i < j \le 35$ such that if we flip the signs of the $x^i$ and $x^j$ terms in $P(x)$ to form a new polynomial $Q(x)$, then there exists a nonconstant polynomial $R(x)$ with integer coefficients dividing both $P(x)$ and $Q(x)$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $ a$ and $ k$ be positive integers. Let $ a_i$ be the sequence defined by
$ a_1 \equal{} a$ and
$ a_{n \plus{} 1} \equal{} a_n \plus{} k\pi(a_n)$
where
$ \pi(x)$ is the product of the digits of $ x$ (written in base ten)
Prove that we can choose $ a$ and $ k$ such that the infinite sequence $ a_i$ contains exactly $ 100$ distinct terms
Consider the following sequence of positive real numbers $\dots<a_{-2}<a_{-1}<a_0<a_1<a_2<\dots$ infinite in both directions. For each positive integer $k$ let $b_k$ be the least integer such that the ratio between the sum of $k$ consecutive terms and the greatest of these $k$ terms is less than or equal to $b_k$(This fact occurs for any sequence of $k$ consecutive numbers). Prove that the sequence $b_1,b_2,b_3,...$ coincides with the sequence $1,2,3,...$ or is eventually constant.
Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.
The real sequence $\{a_n|n=0,1,2,3,...\}$ defined $a_0=1$ and
\[ a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ). \]
Denote
\[ A_n=\frac{3}{3 \cdot a_n^2-1}. \]
Prove that $A_n$ is a perfect square and it has at least $n$ distinct prime divisors.