Found problems: 5923
Consider the sequence $ a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1$, where $ z\in \mathbb{C}^*$ is given.
i) Prove that if $ a_1>2$, then:
\[ a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^*\]
ii) Prove that if there is a $ k\in \mathbb{N}^*$ such that $ a_k\le 2$, then $ a_1\le 2$.
Find the number of binary sequences $S$ of length $2015$ such that for any two segments $I_1, I_2$ of $S$ of the same length, we have
• The sum of digits of $I_1$ differs from the sum of digits of $I_2$ by at most $1$,
• If $I_1$ begins on the left end of S then the sum of digits of $I_1$ is not greater than the sum of digits of $I_2$,
• If $I_2$ ends on the right end of S then the sum of digits of $I_2$ is not less than the sum of digits of $I_1$.
Lê Anh Vinh
Consider the sequence of natural numbers $a_n$ defined as $a_0=4$ and $a_{n+1}=\frac{a_n(a_n-1)}{2}$ for each $n\geq 0$.
Define a new sequence $b_n$ as follows: $b_n=0$ if $a_n$ is even, and $b_n=1$ if $a_n$ is odd. Prove that for each natural $m$, the sequence
\[b_m, b_{m+1}, b_{m+2},b_{m+3}, \dots\]
is not periodic.
The real numbers $x$, $y$, $z$, satisfy $0\leq x \leq y \leq z \leq 4$. If their squares form an arithmetic progression with common difference $2$, determine the minimum possible value of $|x-y|+|y-z|$.
Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$:
(a) $g_{n+1}\le g_n$
(b) $g_{10}= 000$.
In what case does the system of equations
$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$
have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
Let $a_1$, $a_2$, $ ...$ be a sequence of numbers, each either $1$ or $-1$. Show that if
$$\frac{a_1}{3}+\frac{a_2}{3^2} + ... =\frac{p}{q}$$ for integers $p$ and $q$ such that $3$ does not divide $q$, then the sequence $a_1$, $a_2$, $ ...$ is periodic; that is, there is some positive integer $n$ such that $a_i = a_{n+i}$ for $i = 1$, $2$,$...$.
A sequence $a_n$ is defined such that $a_0 =\frac{1 + \sqrt3}{2}$ and $a_{n+1} =\sqrt{a_n}$ for $n \ge 0$. Evaluate
$$\prod_{k=0}^{\infty} 1 - a_k + a^2_k$$
Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then
\[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\]
For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.
Let $1 = x_{1} < x_{2} < \dots < x_{k} = n$ denote the sequence of all divisors $x_{1}, x_{2} \dots x_{k}$ of $n$ in increasing order. Find the smallest possible value of $n$ such that
$$n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}.$$
[i]Proposed by Justin Lee[/i]
Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?
[i]Proposed by Andy Xu[/i]
Suppose that $(a_n)_{n\geq 1}$ is a sequence of real numbers satisfying $a_{n+1} = \frac{3a_n}{2+a_n}$.
(i) Suppose $0 < a_1 <1$, then prove that the sequence $a_n$ is increasing and hence show that $\lim_{n \to \infty} a_n =1$.
(ii) Suppose $ a_1 >1$, then prove that the sequence $a_n$ is decreasing and hence show that $\lim_{n \to \infty} a_n =1$.
Consider arithmetic sequences where all terms are natural numbers. If the first term of such a sequence is $1$, prove that that sequence contains infinitely many terms that are the cube of a natural number. Give an example of such a sequence in which no term is the cube of a natural number and show the correctness of this example.
Given a sequence of natural numbers $ (a_i) $, for each natural number $ n $ the sum of the terms of the sequence that are not greater than $ n $ is a number not less than $ n $. Prove that for every natural number $ k $ it is possible to choose from the sequence $ (a_i) $ a finite sequence with the sum of terms equal to $ k $.
Define a sequence $<x_n>$ of real numbers by specifying an initial $x_0$ and by the recurrence $x_{n+1}=\frac{1+x_n}{1-x_n}$. Find $x_n$ as a function of $x_0$ and $n$, in closed form. There may be multiple cases.
Consider all words constituted by eight letters from $\{C ,H,M, O\}$. We arrange the words in an alphabet sequence.
Precisely, the first word is $CCCCCCCC$, the second one is $CCCCCCCH$, the third is $CCCCCCCM$, the fourth one is $CCCCCCCO, ...,$ and the last word is $OOOOOOOO$.
a) Determine the $2017$th word of the sequence?
b) What is the position of the word $HOMCHOMC$ in the sequence?
The sequence ($a_n$) is defined by $a_1 = a_2 = 1, a_3 = 199$ and $a_{n+1} =\frac{1989+a_na_{n-1}}{a_{n-2}}$ for all $n \ge 3$. Prove that all terms of the sequence are positive integers
Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence
\[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \]
forms an arithmetic progression. Prove that the terms of the sequence are equal.
A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$
a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{} b$ and $ a_n \equal{} c.$
b.) Find the smallest natural number $ n$ for which there exists a quadratic sequence with $ a_0 \equal{} 0$ and $ a_n \equal{} 1996.$
Let $F_0,F_1,\dots$ be the sequence of Fibonacci numbers, with $F_0=0,F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n \ge 2$. For $m>2$, let $R_m$ be the remainder when the product $\prod_{k=1}^{F_m-1} k^k$ is divided by $F_m$. Prove that $R_m$ is also a Fibonacci number.
p1. Given $f(x)=\frac{1+x}{1-x}$ , for $x \ne 1$ . Defined $p @ q = \frac{p+q}{1+pq}$ for all positive rational numbers $p$ and $q$. Note the sequence with $a_1,a_2,a_3,...$ with $a_1=2 @3$, $a_{n}=a_{n-1}@ (n+2)$ for $n \ge 2$. Determine $f(a_{233})$ and $a_{233}$
p2. It is known that $ a$ and $ b$ are positive integers with $a > b > 2$. Is $\frac{2^a+1}{2^b-1}$ an integer? Write down your reasons.
p3. Given a cube $ABCD.EFGH$ with side length $ 1$ dm. There is a square $PQRS$ on the diagonal plane $ABGH$ with points $P$ on $HG$ and $Q$ on $AH$ as shown in the figure below. Point $T$ is the center point of the square $PQRS$. The line $HT$ is extended so that it intersects the diagonal line $BG$ at $N$. Point $M$ is the projection of $N$ on $BC$. Determine the volume of the truncated prism $DCM.HGN$.
[img]https://cdn.artofproblemsolving.com/attachments/f/6/22c26f2c7c66293ad7065a3c8ce3ac2ffd938b.png[/img]
4. Nine pairs of husband and wife want to take pictures in a three-line position with the background of the Palembang Ampera Bridge. There are $4$ people in the front row, $6$ people in the middle row, and $ 8$ people in the back row. They agreed that every married couple must be in the same row, and every two people next to each other must be a married couple or of the same sex. Specify the number of different possible arrangements of positions.
p5. p5. A hotel provides four types of rooms with capacity, rate, and number of rooms as presented in the following table.
[b] type of room, capacity of persons/ room, day / rate (Rp.), / number of rooms [/b][img]https://cdn.artofproblemsolving.com/attachments/3/c/e9e1ed86887e692f9d66349a82eaaffc730b46.jpg[/img]
A group of four families wanted to stay overnight at the hotel. Each family consists of husband and wife and their unmarried children. The number of family members by gender is presented in the following table.
[b]family / man / woman/ total[/b]
[img]https://cdn.artofproblemsolving.com/attachments/4/6/5961b130c13723dc9fa4e34b43be30c31ee635.jpg[/img]
The group leader enforces the following provisions.
I. Each husband and wife must share a room and may not share a room with other married couples.
II. Men and women may not share the same room unless they are from the same family.
III. At least one room is occupied by all family representatives (“representative room”)
IV. Each family occupies at most $3$ types of rooms.
V. No rooms are occupied by more than one family except representative rooms.
You are asked to arrange a room for the group so that the total cost of lodging is as low as possible. Provide two possible alternative room arrangements for each family and determine the total cost.
We call a polynomial $P(x)=a_dx^d+...+a_0$ of degree $d$ [i]nice[/i] if
$$\frac{2021(|a_d|+...+|a_0|)}{2022}<max_{0 \le i \le d}|a_i|$$
Initially Shayan has a sequence of $d$ distinct real numbers; $r_1,...,r_d \neq \pm 1$. At each step he choose a positive integer $N>1$ and raises the $d$ numbers he has to the exponent of $N$, then delete the previous $d$ numbers and constructs a monic polynomial of degree $d$ with these number as roots, then examine whether it is nice or not. Prove that after some steps, all the polynomials that shayan produces would be nice polynomials
Proposed by [i]Navid Safaei[/i]
The sequence $0, 1, 1, 1, 1, 1,....,1$ where have $1$ number zero and $1995$ numbers one.
If we choose two or more numbers in this sequence(but not the all $1996$ numbers) and substitute one number by arithmetic mean of the numbers selected, we obtain a new sequence with $1996$ numbers!!!
Show that, we can repeat this operation until we have all $1996$ numbers are equal
Note: It's not necessary to choose the same quantity of numbers in each operation!!!
A sequence of integers $a_1,a_2,a_3,\ldots$ is called [i]exact[/i] if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$.
An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=2, \; a_{n+1}=\left\lfloor \frac{3}{2}a_{n}\right\rfloor.\] Show that it has infinitely many even and infinitely many odd integers.