Found problems: 5923
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
Let $(x_n)_{n\ge 0}$ a sequence of real numbers defined by $x_0>0$ and $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$. Compute $\lim_{n\to \infty}x_n$ and $\lim_{n\to \infty} \frac{x_n^3}{n^2}$.
Recall that an arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is the same. Suppose $x_1$, $x_2$, $x_3$ forms an arithmetic sequence. If $x_2 = 2023$, compute $x_1 + x_2 + x_3$.
Let $f(n)$ be a sequence of integers defined by $f(1)=1, f(2)=1,$ and $f(n)=f(n-1)+(-1)^nf(n-2)$ for all integers $n \geq 3.$ What is the value of $f(20)+f(21)?$
How many ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 10$ are there such that in the geometric sequence whose first term is $a$ and whose second term is $b$, the third term is an integer?
Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios
Proposed by [i]Morteza Saghafian[/i]
Find all positive integers $(r,s)$ such that there is a non-constant sequence $a_n$ os positive integers such that for all $n=1,2,\dots$
\[ a_{n+2}= \left(1+\frac{{a_2}^r}{{a_1}^s} \right ) \left(1+\frac{{a_3}^r}{{a_2}^s} \right ) \dots \left(1+\frac{{a_{n+1}}^r}{{a_n}^s} \right ).\]
Proposed by Navid Safaei, Iran
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
Let $x_0,x_1,x_2,\dots$ be a infinite sequence of real numbers, such that the following three equalities are true:
I- $x_{2k}=(4x_{2k-1}-x_{2k-2})^2$, for $k\geq 1$
II- $x_{2k+1}=|\frac{x_{2k}}{4}-k^2|$, for $k\geq 0$
III- $x_0=1$
a) Determine the value of $x_{2022}$
b) Prove that there are infinite many positive integers $k$, such that $2021|x_{2k+1}$
Let $n$ be a positive integer. Alice writes $n$ real numbers $a_1, a_2,\dots, a_n$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ($a_n$ is not a neighbor of $a_1$, nor vice versa). A number [i]changes sign[/i] if it changes from being nonnegative to negative or vice versa. In terms of $n$, determine the maximum number of times that $a_1$ can change sign, across all possible values of $a_1,a_2,\dots, a_n$ and all possible sequences of moves Alice may make.
Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.
For a fixed natural number $m \geq 2$, prove that
[b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\]
[b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.
Positive sequence $(a_n)_{n=0}^{\infty}$ satisfies that $\sqrt{a_na_{n-2}}-\sqrt{a_{n-1}a_{n-2}}=2a_{n-1}(n\geq2),a_0=a_1=1$. Find $a_n$.
Let the sequence $\{a_n\}_{n\ge 1}$ be defined by
$$a_n=\tan(n\theta)$$
where $\tan\theta =2$. Show that for all $n$, $a_n$ is a rational number which can be written with an odd denominator.
Does there exist a rational $x_1$, such that all members of the sequence $x_1, x_2, \ldots, x_{2024}$ defined by $x_{n+1}=x_n+\sqrt{x_n^2-1}$ for $n=1, 2, \ldots, 2023$ are greater than $1$ and rational?
Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.
[i]Proposed by Gabriel Carroll, USA[/i]
Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
The Fibonacci sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\in\mathbb N$.
(a) Show that $f_{1005}$ is divisible by $10$.
(b) Show that $f_{1005}$ is not divisible by $100$.
A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$. Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$?
The cells of an $n \times n$ chessboard are coloured in several colours so that no $2\times 2$ square contains four cells of the same colour. A [i]proper path [/i] of length $m$ is a sequence $a_1,a_2,..., a_m$ of distinct cells in which the cells $a_i$ and $a_{i+1}$ have a common side and are coloured in different colours for all $1 \le i < m$. Show that there exists a proper path of length $n$.
Let $n \ge k \ge 3$ be integers. Show that for every integer sequence $1 \le a_1 < a_2 < . . . <
a_k \le n$ one can choose non-negative integers $b_1, b_2, . . . , b_k$, satisfying the following conditions:
[list=i]
[*] $0 \le b_i \le n$ for each $1 \le i \le k$,
[*] all the positive $b_i$ are distinct,
[*] the sums $a_i + b_i$, $1 \le i \le k$, form a permutation of the first $k$ terms of a non-constant arithmetic
progression.
[/list]
Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$.
Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$.
Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.
Let $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$, and $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$ Determine $ a_{14^{14}}$.
We denote by $S(k)$ the sum of digits of a positive integer number $k$. We say that the positive integer $a$ is $n$-good, if there is a sequence of positive integers $a_0$, $a_1, \dots , a_n$, so that $a_n = a$ and $a_{i + 1} = a_i -S (a_i)$ for all $i = 0, 1,. . . , n-1$.
Is it true that for any positive integer $n$ there exists a positive integer $b$, which is $n$-good, but not $(n + 1)$-good?
A. Antropov
[hide=Intro]The answers to each of the ten questions in this section are integers containing only the digits $ 1$ through $ 8$, inclusive. Each answer can be written into the grid on the answer sheet, starting from the cell with the problem number, and continuing across or down until the entire answer has been written. Answers may cross dark lines. If the answers are correctly filled in, it will be uniquely possible to write an integer from $ 1$ to $ 8$ in every cell of the grid, so that each number will appear exactly once in every row, every column, and every marked $2$ by $4$ box. You will get $7$ points for every correctly filled answer, and a $15$ point bonus for filling in every gridcell. It will help to work back and forth between the grid and the problems, although every problem is uniquely solvable on its own.
Please write clearly within the boxes. No points will be given for a cell without a number, with multiple
numbers, or with illegible handwriting.[/hide]
[img]https://cdn.artofproblemsolving.com/attachments/9/b/f4db097a9e3c2602b8608be64f06498bd9d58c.png[/img]
[b]1 ACROSS:[/b] Jack puts $ 10$ red marbles, $ 8$ green marbles and 4 blue marbles in a bag. If he takes out $11$ marbles, what is the expected number of green marbles taken out?
[b]2 DOWN:[/b] What is the closest integer to $6\sqrt{35}$ ?
[b]3 ACROSS: [/b]Alan writes the numbers $ 1$ to $64$ in binary on a piece of paper without leading zeroes. How many more times will he have written the digit $ 1$ than the digit $0$?
[b]4 ACROSS:[/b] Integers a and b are chosen such that $-50 < a, b \le 50$. How many ordered pairs $(a, b)$ satisfy the below equation? $$(a + b + 2)(a + 2b + 1) = b$$
[b]5 DOWN: [/b]Zach writes the numbers $ 1$ through $64$ in binary on a piece of paper without leading zeroes. How many times will he have written the two-digit sequence “$10$”?
[b]6 ACROSS:[/b] If you are in a car that travels at $60$ miles per hour, $\$1$ is worth $121$ yen, there are $8$ pints in a gallon, your car gets $10$ miles per gallon, a cup of coffee is worth $\$2$, there are 2 cups in a pint, a gallon of gas costs $\$1.50$, 1 mile is about $1.6$ kilometers, and you are going to a coffee shop 32 kilometers away for a gallon of coffee, how much, in yen, will it cost?
[b]7 DOWN:[/b] Clive randomly orders the letters of “MIXING THE LETTERS, MAN”. If $\frac{p}{m^nq}$ is the probability that he gets “LMT IS AN EXTREME THING” where p and q are odd integers, and $m$ is a prime number, then what is $m + n$?
[b]8 ACROSS:[/b] Joe is playing darts. A dartboard has scores of $10, 7$, and $4$ on it. If Joe can throw $12$ darts, how many possible scores can he end up with?
[b]9 ACROSS:[/b] What is the maximum number of bounded regions that $6$ overlapping ellipses can cut the plane into?
[b]10 DOWN:[/b] Let $ABC$ be an equilateral triangle, such that $A$ and $B$ both lie on a unit circle with center $O$. What is the maximum distance between $O$ and $C$? Write your answer be in the form $\frac{a\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime, and $a$ and $c$ share no common factor. What is $abc$ ?
PS. You had better use hide for answers.