Found problems: 5923
Consider the sequence $1, \frac12, \frac13, \frac14 ,...$
Does there exist an arithmetic progression composed of terms of this sequence
(a) of length $5$,
(b) of length greater than $5$ (if so, what possible length)?
(G Galperin, Moscow)
Each term in a sequence $1,0,1,0,1,0...$starting with the seventh is the sum of the last 6 terms mod 10 .Prove that the sequence $...,0,1,0,1,0,1...$ never occurs
Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k \minus{} n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either [i]on[/i] or [i]off[/i]. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).
Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off.
Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off, but where none of the lamps $ n \plus{} 1$ through $ 2n$ is ever switched on.
Determine $ \frac {N}{M}$.
[i]Author: Bruno Le Floch and Ilia Smilga, France[/i]
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_ib_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$ for all $i \ge 1$. If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$, then what is $b_{6}$?
[i]Proposed by Andrew Wu[/i]
Let $a$ be an odd natural number and $b$ be a positive integer. We define a sequence of reals $(u_n)$ as follows: $u_0=b$ and, for all $n\in\mathbb N_0$, $u_{n+1}$ is $\frac{u_n}2$ if $u_n$ is even and $a+u_n$ otherwise.
(a) Prove that one can find an element of $u_n$ smaller than $a$.
(b) Prove that the sequence is eventually periodic.
For how many pairs of sequences of nonnegative integers $(b_1,b_2,\ldots, b_{2018})$ and $(c_1,c_2,\ldots, c_{2018})$ does there exist a sequence of nonnegative integers $(a_0,\ldots, a_{2018})$ with the following properties:
[list]
[*] For $0\leq i\leq 2018,$ $a_i<2^{2018}.$
[*] For $1\leq i \leq 2018, b_i=a_{i-1}+a_i$ and $c_i=a_{i-1}|a_i$;
[/list]
where $|$ denotes the bitwise or operation?
We define the sequences $a_n =\frac{n (n + 1)}{2}$ and $b_n = a_1 + a_2 +… + a_n$.
Prove that there is no integer $n$ such that $b_n = 2017$.
Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$.
[b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \]
[b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]
We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions:
[b]a)[/b] every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares.
[b]b)[/b] the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division.
[color=#008000]Moderator says: see [url]https://artofproblemsolving.com/community/c6h58591[/url][/color]
A sequence $ (x_n)$ is given as follows: $ x_0,x_1$ are arbitrary positive real numbers, and $ x_{n\plus{}2}\equal{}\frac{1\plus{}x_{n\plus{}1}}{x_n}$ for $ n \ge 0$. Find $ x_{1998}$.
Given a pair $(a_0, b_0)$ of real numbers, we define two sequences $a_0, a_1, a_2,...$ and $b_0, b_1, b_2, ...$ of real numbers by $a_{n+1}= a_n + b_n$ and $b_{n+1}=a_nb_n$ for all $n = 0, 1, 2,...$. Find all pairs $(a_0, b_0)$ of real numbers such that $a_{2022}= a_0$ and $b_{2022}= b_0$.
The sequence (a_n) is defined as follows:
$ a_0\equal{}1, a_1\equal{}3$
For $ n\ge 2$, $ a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}9a_n$ if n is even, $ a_{n\plus{}2}\equal{}9a_{n\plus{}1}\plus{}5a_n$ if n is odd.
Prove that
1) $ (a_{1995})^2\plus{}(a_{1996})^2\plus{}...\plus{}(a_{2000})^2$ is divisible by 20
2) $ a_{2n\plus{}1}$ is not a perfect square for every natural numbers $ n$.
Determine all increasing sequences $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for each two natural numbers $i$ and $j$ (not necessarily different), the numbers $i+j$ and $a_i+a_j$ have an equal number of distinct natural divisors.
Given positive integers $a,c$ and integer $b$, prove that there exists a positive integer $x$ such that
\[ a^x + x \equiv b \pmod c, \]
that is, there exists a positive integer $x$ such that $c$ is a divisor of $a^x + x - b$.
[b]p1.[/b] Velociraptor $A$ is located at $x = 10$ on the number line and runs at $4$ units per second. Velociraptor $B$ is located at $x = -10$ on the number line and runs at $3$ units per second. If the velociraptors run towards each other, at what point do they meet?
[b]p2.[/b] Let $n$ be a positive integer. There are $n$ non-overlapping circles in a plane with radii $1, 2, ... , n$. The total area that they enclose is at least $100$. Find the minimum possible value of $n$.
[b]p3.[/b] How many integers between $1$ and $50$, inclusive, are divisible by $4$ but not $6$?
[b]p4.[/b] Let $a \star b = 1 + \frac{b}{a}$. Evaluate $((((((1 \star 1) \star 1) \star 1) \star 1) \star 1) \star 1) \star 1$.
[b]p5.[/b] In acute triangle $ABC$, $D$ and $E$ are points inside triangle $ABC$ such that $DE \parallel BC$, $B$ is closer to $D$ than it is to $E$, $\angle AED = 80^o$ , $\angle ABD = 10^o$ , and $\angle CBD = 40^o$. Find the measure of $\angle BAE$, in degrees.
[b]p6. [/b]Al is at $(0, 0)$. He wants to get to $(4, 4)$, but there is a building in the shape of a square with vertices at $(1, 1)$, $(1, 2)$, $(2, 2)$, and $(2, 1)$. Al cannot walk inside the building. If Al is not restricted to staying on grid lines, what is the shortest distance he can walk to get to his destination?
[b]p7. [/b]Point $A = (1, 211)$ and point $B = (b, 2011)$ for some integer $b$. For how many values of $b$ is the slope of $AB$ an integer?
[b]p8.[/b] A palindrome is a number that reads the same forwards and backwards. For example, $1$, $11$ and $141$ are all palindromes. How many palindromes between $1$ and 1000 are divisible by $11$?
[b]p9.[/b] Suppose $x, y, z$ are real numbers that satisfy: $$x + y - z = 5$$
$$y + z - x = 7$$
$$z + x - y = 9$$ Find $x^2 + y^2 + z^2$.
[b]p10.[/b] In triangle $ABC$, $AB = 3$ and $AC = 4$. The bisector of angle $A$ meets $BC$ at $D$. The line through $D$ perpendicular to $AD$ intersects lines $AB$ and $AC$ at $F$ and $E$, respectively. Compute $EC - FB$. (See the following diagram.)
[img]https://cdn.artofproblemsolving.com/attachments/2/7/e26fbaeb7d1f39cb8d5611c6a466add881ba0d.png[/img]
[b]p11.[/b] Bob has a six-sided die with a number written on each face such that the sums of the numbers written on each pair of opposite faces are equal to each other. Suppose that the numbers $109$, $131$, and $135$ are written on three faces which share a corner. Determine the maximum possible sum of the numbers on the three remaining faces, given that all three are positive primes less than $200$.
[b]p12.[/b] Let $d$ be a number chosen at random from the set $\{142, 143, ..., 198\}$. What is the probability that the area of a rectangle with perimeter $400$ and diagonal length $d$ is an integer?
[b]p13.[/b] There are $3$ congruent circles such that each circle passes through the centers of the other two. Suppose that $A, B$, and $C$ are points on the circles such that each circle has exactly one of $A, B$, or $C$ on it and triangle $ABC$ is equilateral. Find the ratio of the maximum possible area of $ABC$ to the minimum possible area of $ABC$. (See the following diagram.)
[img]https://cdn.artofproblemsolving.com/attachments/4/c/162554fcc6aa21ce3df3ce6a446357f0516f5d.png[/img]
[b]p14.[/b] Let $k$ and $m$ be constants such that for all triples $(a, b, c)$ of positive real numbers,
$$\sqrt{ \frac{4}{a^2}+\frac{36}{b^2}+\frac{9}{c^2}+\frac{k}{ab} }=\left| \frac{2}{a}+\frac{6}{b}+\frac{3}{c}\right|$$
if and only if $am^2 + bm + c = 0$. Find $k$.
[b]p15.[/b] A bored student named Abraham is writing $n$ numbers $a_1, a_2, ..., a_n$. The value of each number is either $1, 2$, or $3$; that is, $a_i$ is $1, 2$ or $3$ for $1 \le i \le n$. Abraham notices that the ordered triples $$(a_1, a_2, a_3), (a_2, a_3, a_4), ..., (a_{n-2}, a_{n-1}, a_n), (a_{n-1}, a_n, a_1), (a_n, a_1, a_2)$$ are distinct from each other. What is the maximum possible value of $n$? Give the answer n, along with an example of such a sequence. Write your answer as an ordered pair. (For example, if the answer were $5$, you might write $(5, 12311)$.)
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
The numbers $a^2, b^2, c^2$ form an arithmetic progression. Show that the numbers $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ also form arithmetic progression.
A finite number of coins are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one coin is chosen. Two coins are taken from this square; one of them is placed on the square immediately to the left while the other is placed on the square immediately to the right of the chosen square. The sequence terminates if at some point there is at most one coin on each square. Given some initial configuration, show that any legal sequence of moves will terminate after the same number of steps and with the same final configuration.
Let $P(x)$ be a non-constant polynomial with integer coefficients and let $n$ be a positive integer. The sequence $a_0,a_1,\ldots$ is defined as follows: $a_0=n$ and $a_k=P(a_{k-1})$ for all positive integers $k.$ Assume that for every positive integer $b$ the sequence contains a $b$th power of an integer greater than $1.$ Show that $P(x)$ is linear.
a) There is an infinite sequence of $0,1$, like $\dots,a_{-1},a_{0},a_{1},\dots$ (i.e. an element of $\{0,1\}^{\mathbb Z}$). At each step we make a new sequence. There is a function $f$ such that for each $i$, $\mbox{new }a_{i}=f(a_{i-100},a_{i-99},\dots,a_{i+100})$. This operation is mapping $F: \{0,1\}^{\mathbb Z}\longrightarrow\{0,1\}^{\mathbb Z}$. Prove that if $F$ is 1-1, then it is surjective.
b) Is the statement correct if we have an $f_{i}$ for each $i$?
There are $N$ monsters, each with a positive weight. On each step, two of the monsters are merged into one, whose weight is the sum of weights for the two original monsters. At the end, all monsters will be merged into one giant monster. During this process, if at any mergence, one of the two monsters has a weight greater than $2.020$ times the other monster's weight, we will call this mergence [b]dangerous[/b]. The dangerous level of a sequence of mergences is the number of dangerous mergence throughout its process.
Prove that, no matter how the weights being distributed among the monsters, "for every step, merge the lightest two monsters" is always one of the merging sequences that obtain the minimum possible dangerous level.
[i]Proposed by houkai[/i]
The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $F_{mn}-F_{n+1}^{m}+F_{n-1}^{m}$ is divisible by $F_{n}^{3}$ for all $m \ge 1$ and $n>1$.
An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a [i]binary sequence of length [/i]$n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.
Let $a_1, . . . , a_n$ be positive integers satisfying the inequality
$\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}$.
Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each $i = 1, . . . , n$,the possible values of the $i-th$ indicator are $1, 2, . . . , a_i$. The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.
A sequence $(a_n)$ is defined by means of the recursion
\[a_1 = 1, a_{n+1} = \frac{1 + 4a_n +\sqrt{1+ 24a_n}}{16}.\]
Find an explicit formula for $a_n.$
Does there exist a sequence of natural numbers $a_1,a_2,\ldots$ such that the number $a_i+a_j$ has an even number of different prime divisors for any two different natural indices $i{}$ and $j{}$?
[i]From the folklore[/i]