Found problems: 5923
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 $k \mid a_m$.
The cells of a square $100 \times 100$ table are colored in one of two colors, black or white. A coloring is called admissible if for any row or column, the number $b$ of black colored cells satisfies $50 \le b \le 60$. It is permitted to recolor a cell if by doing so the resulting configuration is still admissible. Prove that one can transition from any admissible coloring of the board to any other using a sequence of valid recoloring operations.
The Sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by $a_{0}=1, a_{1}=-4$ and $a_{n+2}=-4a_{n+1}-7a_{n}$ , for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50}-a_{49}a_{51}$.
Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [\minus{}1,0,1]$ for $ i\equal{}1,...,n,$ and
$ x_1\plus{}...\plus{}x_n \equiv k$ mod 4
a) Prove that $ f_1(n) \equal{} f_3(n)$ for all positive integers $ n$.
(b) Prove that
$ f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4$
for all positive integers $ n$.
The first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s, three $2$'s, and three $3$'s. The sequence continues in this fashion.
\[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\dots.\]
Find the $2014$th number in this sequence.
Two players, the builder and the destroyer, plays the following game. Builder starts and players chooses alternatively different elements from the set $\{0,1,\ldots,10\}.$ Builder wins if some four integer of those six integer he chose forms an arithmetic sequence. Destroyer wins if he can prevent to form such an arithmetic four-tuple. Which one has a winning strategy?
The Baltic Sea has $2016$ harbours. There are two-way ferry connections between some of them. It is impossible to make a sequence of direct voyages $C_1 - C_2 - ... - C_{1062}$ where all the harbours $C_1, . . . , C_{1062}$ are distinct. Prove that there exist two disjoint sets $A$ and $B$ of $477$ harbours each, such that there is no harbour in $A$ with a direct ferry connection to a harbour in $B.$
$\boxed{A5}$Let $n\in{N},n>2$,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}.$Prove that
\[(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3\]
Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.
The sequence $\{a_n\}$ satisfies $a_0=1, a_1=2011,$ and $a_n=2a_{n-1}+a_{n-2}$ for all $n \geq 2$. Let
\[ S = \sum_{i=1}^{\infty} \frac{a_{i-1}}{a_i^2-a_{i-1}^2} \]
What is $\frac{1}{S}$?
[i]Author: Ray Li[/i]
An integer $m\ge 3$ and an infinite sequence of positive integers $(a_n)_{n\ge 1}$ satisfies the equation
\[a_{n+2} = 2\sqrt[m]{a_{n+1}^{m-1} + a_n^{m-1}} - a_{n+1}. \]
for all $n\ge 1$. Prove that $a_1 < 2^m$.
[b]p1.[/b] Determine the number of ways to place $4$ rooks on a $4 \times 4$ chessboard such that:
(a) no two rooks attack one another, and
(b) the main diagonal (the set of squares marked $X$ below) does not contain any rooks.
[img]https://cdn.artofproblemsolving.com/attachments/e/e/e3aa96de6c8ed468c6ef3837e66a0bce360d36.png[/img]
The rooks are indistinguishable and the board cannot be rotated. (Two rooks attack each other if they are in the same row or column.)
[b]p2.[/b] Seven students, numbered $1$ to $7$ in counter-clockwise order, are seated in a circle. Fresh Mann has 100 erasers, and he wants to distribute them to the students, albeit unfairly. Starting with person $ 1$ and proceeding counter-clockwise, Fresh Mann gives $i$ erasers to student $i$; for example, he gives $ 1$ eraser to student $ 1$, then $2$ erasers to student $2$, et cetera. He continues around the circle until he does not have enough erasers to give to the next person. At this point, determine the number of erasers that Fresh Mann has.
[b]p3.[/b] Let $ABC$ be a triangle with $AB = AC = 17$ and $BC = 24$. Approximate $\angle ABC$ to the nearest multiple of $10$ degrees.
[b]p4.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 =\frac35$ and for $n \ge 1$, $x_{n+1} = 2 - \frac{1}{x_n}$ . Compute the product $x_1x_2x_3... x_{2013}$.
[b]p5.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PR = 2$, $IM = 3$, and $ES = 5$, compute the area of hexagon $PRIMES$.
[b]p6.[/b] Let $f(a, b) = \frac{a^2}{a+b}$ . Let $A$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le i < j \le 10$; that is,
$$A = (f(1, 2) + f(1, 3) + ...+ f(1, 10)) + (f(2, 3) + f(2, 4) +... + f(2, 10)) +... + f(9, 10).$$
Similarly, let $B$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le j < i \le 10$, that is, $$B = (f(2, 1) + f(3, 1) + ... + f(10, 1)) + (f(3, 2) + f(4, 2) +... + f(10, 2)) +... + f(10, 9).$$ Compute $B - A$.
[b]p7.[/b] Fresh Mann has a pile of seven rocks with weights $1, 1, 2, 4, 8, 16$, and $32$ pounds and some integer X between $1$ and $64$, inclusive. He would like to choose a set of the rocks whose total weight is exactly $X$ pounds. Given that he can do so in more than one way, determine the sum of all possible values of $X$. (The two $1$-pound rocks are indistinguishable.)
[b]p8.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = CA$. Suppose that point $P$ lies inside the quadrilateral with $AP = PD = DA$ and $\angle PCD = 30^o$. Given that $CP = 2$ and $CD = 3$, compute $CA$.
[b]p9.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 = 2$, $x_2 = \frac{13}{2}$ , and for $n \ge 1$, $x_{n+2} = 3 -\frac{3}{x_{n+1}}+\frac{1}{x_nx_{n+1}}$. Compute $x_{100}$.
[b]p10.[/b] Ten prisoners are standing in a line. A prison guard wants to place a hat on each prisoner. He has two colors of hats, red and blue, and he has $10$ hats of each color. Determine the number of ways in which the prison guard can place hats such that among any set of consecutive prisoners, the number of prisoners with red hats and the number of prisoners with blue hats differ by at most $2$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.
Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying
\[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\]
Prove that $P(1983) = F_{1983} - 1.$
Call a triple $(a, b, c)$ of positive integers a [b]nice[/b] triple if $a, b, c$ forms a non-decreasing arithmetic progression, $gcd(b, a) = gcd(b, c) = 1$ and the product $abc$ is a perfect square. Prove that given a nice triple, there exists some other nice triple having at least one element common with the given triple.
Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let $\{a_n\}, n\ge 1$ be such a sequence. We call it [i]consistent [/i] if, for an arbitrary natural $k$ and every natural $n ,m$ such that $a_n <a_m$, the inequality $a_{kn} <a _{km}$ also holds. For example, the sequence $a_n = n$ is consistent .
a) Prove that there are consistent sequences other than $a_n = n$.
b) Are there consistent sequences for which $a_n \ne n, n\ge 2$ ?
c) Are there consistent sequences for which $a n \ne n, n\ge 1$ ?
[b]p19.[/b] Sequences $a_n$ and $b_n$ of positive integers satisfy the following properties:
(1) $a_1 = b_1 = 1$
(2) $a_5 = 6, b_5 \ge 7$
(3) Both sequences are strictly increasing
(4) In each sequence, the difference between consecutive terms is either $1$ or $2$
(5) $\sum^5_{n=1}na_n =\sum^5_{n=1}nb_n = S$
Compute $S$.
[b]p20.[/b] Let $A$, $B$, and $C$ be points lying on a line in that order such that $AB = 4$ and $BC = 2$. Let $I$ be the circle centered at B passing through $C$, and let $D$ and $E$ be distinct points on $I$ such that $AD$ and $AE$ are tangent to $I$. Let $J$ be the circle centered at $C$ passing through $D$, and let $F$ and $G$ be distinct points on $J$ such that $AF$ and $AG$ are tangent to $J$ and $DG < DF$. Compute the area of quadrilateral $DEFG$.
[b]p21.[/b] Twain is walking randomly on a number line. They start at $0$, and flip a fair coin $10$ times. Every time the coin lands heads, they increase their position by 1, and every time the coin lands tails, they decrease their position by $1$. What is the probability that at some point the absolute value of their position is at least $3$?
PS. You should use hide for answers.
A sequence $(x_{i})$ is said to have a [i]Cesaro limit[/i] exactly if $\lim_{n\to\infty} \frac{x_{1}+\ldots+x_{n}}{n}$ exists.
Find all real-valued functions $f$ on the closed interval $[0, 1]$ such that $(f(x_i))$ has a Cesaro limit if and only if $(x_i)$ has a Cesaro limit.
Let $a_1, a_2, ..., a_n$ be real numbers such that $a_1 = a_n = a$ and $a_{k+1} \le \frac{a_k + a_{k+2}}{2} $, for all $k = 1, 2, ..., n - 2$. Prove that $a_k \le a,$ for all $k = 1, 2, ..., n.$
Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled:
1) For every prime number $p$ and every natural number $n$, the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour.
2) There does not exist an infinite geometric sequence of natural numbers of the same colour.
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
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 $\{x_n\}$ be a real sequence defined by:
\[x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n\]
For all $n=1,2,3...$ and a is a real number.
Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases.
Let $A > 1$ and $B > 1$ be real numbers and (xn) be a sequence of numbers in the interval $[1,AB]$. Prove that there exists a sequence $(y_n)$ of numbers in the interval $[1,A]$ such that
$$\frac{x_m}{x_n}\le B\frac{y_m}{y_n} \,\,\, for \,\,\, all \,\,\, m,n = 1,2,...$$
Let $\alpha > 1$ be an irrational number and $L$ be a integer such that $L > \frac{\alpha^2}{\alpha - 1}$. A sequence $x_1, x_2, \cdots$ satisfies that $x_1 > L$ and for all positive integers $n$, \[ x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \textup{if} \; x_n \leqslant L \\\left \lfloor \frac{x_n}{\alpha} \right \rfloor & \textup{if} \; x_n > L \end{cases}. \]
Prove that
(i) $\left\{x_n\right\}$ is eventually periodic.
(ii) The eventual fundamental period of $\left\{x_n\right\}$ is an odd integer which doesn't depend on the choice of $x_1$.