Found problems: 5923
For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.
Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order. What is the $10$th term of $\{a_n+g_n\}$?
The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition.
$a_1=1, a_{n+1}=2019a_{n}+1$
Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.)
Prove that $\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{2020-k}-1)^2$
Suppose that $ n \geq 2$ and $ x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $ i$ between $ 1$ and $ n \minus{} 1$ the
inequality
\[ x_i (1 \minus{} x_{i\plus{}1}) \geq \frac{1}{4} x_1 (1 \minus{} x_{n})\]
The sequence of integers $ a_1 $, $ a_2 $, $ \dots $ is defined as follows:
$ a_1 = 1 $ and $ n> 1 $, $ a_ {n + 1} $ is the smallest integer greater than $ a_n $ and such, that $ a_i + a_j \neq 3a_k $ for any $ i, j $ and $ k $ from $ \{1, 2, \dots, n + 1 \} $ are not necessarily different.
Define $ a_ {2004} $.
Given a positive integer whose base-$10$ representation is $\overline{d_k\ldots d_0}$ for some integer $k \geq 0$, where $d_k \neq 0$, a move consists of selecting some integers $0 \leq i \leq j \leq k$, such that the digits $d_j,\ldots,d_i$ are not all $0$, erasing them from $n$, and replacing them with a divisor of $\overline{d_j\ldots d_i}$ (this divisor need not have the same number of digits as $\overline{d_j\ldots d_i}$).
Prove that for all sufficiently large even integers $n$, we may apply some sequence of moves to $n$ to transform it into $2024$.
[i]Allen Wang[/i]
Santa Clause had $n$ sorts of candies, $k$ candies of each sort. He distributed them at random between $k$ gift bags, $n$ candies per a bag and gave a bag to everyone of $k$ children at Christmas party. The children learned what they had in their bags and decided to trade. Two children trade one candy for one candy in case if each of them gets the candy of the sort which was absent in his/her bag. Prove that they can organize a sequence of trades so that finally every child would have candies of each sort.
Let $X{}$ be a set of integers which can be partitioned into $N{}$ disjoint increasing arithmetic progressions (infinite in both directions), and cannot be partitioned into a smaller number of such progressions. Is such partition into $N{}$ progressions unique for every such $X{}$ if a) $N = 2{}$ and b) $N = 3$?
[i]Viktor Kleptsyn[/i]
Consider the sequence \[ 1, \minus{} 2,3, \minus{} 4,5, \minus{} 6,\ldots,\] whose $ n$th term is $ ( \minus{} 1)^{n \plus{} 1}\cdot n$. What is the average of the first $ 200$ terms of the sequence?
$ \textbf{(A)}\minus{}\!1\qquad \textbf{(B)}\minus{}\!0.5\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 0.5\qquad \textbf{(E)}\ 1$
For a positive integer $n$, let $y_n$ be the number of $n$-digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$
Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.
Let $a_0$ be an arbitrary positive integer. Let $(a_n)$ be infinite sequence of positive integers such that for every positive integer $n$, the term $a_n$ is the smallest positive integer such that $a_0 + a_1 +... + a_n$ is divisible by $n$. Prove that there exist $N$ such that $a_{n+1} = a_n$ for all $n \ge N$
Let $x_0, x_1, x_2 \dots$ be a sequence of positive real numbers such that for all $n \geq 0$, $$x_{n+1} = \dfrac{(n^2+1)x_n^2}{x_n^3+n^2}$$ For which values of $x_0$ is this sequence bounded?
Given a sequence $\{x_k\}$ such that $x_1 = 1$, $x_{n+1} = n \sin x_n+ 1$. Prove that the sequence is non-periodic.
Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties:
(a). $x_1 < x_2 < \cdots < x_{n-1}$ ;
(b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$;
(c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.
Given a rearrangement of the numbers from $1$ to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a < b$) or decreasing (if $b < a$). How many rearrangements of the numbers from $1$ to $n$ have exactly two increasing pairs of consecutive elements? Express your answer in terms of $n$.
Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions.
[i]Vasile Pop[/i]
[i]25 problems for 30 minutes.[/i]
[b]p1.[/b] Chad, Ravi, Kevin, and Meena are four of the $551$ residents of Chadwick, Illinois. Expressing your answer to the nearest percent, how much of the population do they represent?
[b]p2.[/b] Points $A$, $B$, and $C$ are on a line for which $AB = 625$ and $BC = 256$. What is the sum of all possible values of the length $AC$?
[b]p3.[/b] An increasing arithmetic sequence has first term $2014$ and common difference $1337$. What is the least odd term of this sequence?
[b]p4.[/b] How many non-congruent scalene triangles with integer side lengths have two sides with lengths $3$ and $4$?
[b]p5.[/b] Let $a$ and $b$ be real numbers for which the function $f(x) = ax^2+bx+3$ satisfies $f(0)+2^0 = f(1)+2^1 = f(2) + 2^2$. What is $f(0)$?
[b]p6.[/b] A pentomino is a set of five planar unit squares that are joined edge to edge. Two pentominoes are considered the same if and only if one can be rotated and translated to be identical to the other. We say that a pentomino is compact if it can fit within a $2$ by $3$ rectangle. How many distinct compact pentominoes exist?
[b]p7.[/b] Consider a hexagon with interior angle measurements of $91$, $101$, $107$, $116$, $152$, and $153$ degrees. What is the average of the interior angles of this hexagon, in degrees?
[b]p8.[/b] What is the smallest positive number that is either one larger than a perfect cube and one less than a perfect square, or vice versa?
[b]p9.[/b] What is the first time after $4:56$ (a.m.) when the $24$-hour expression for the time has three consecutive digits that form an increasing arithmetic sequence with difference $1$? (For example, $23:41$ is one of those moments, while $23:12$ is not.)
[b]p10.[/b] Chad has trouble counting. He wants to count from $1$ to $100$, but cannot pronounce the word "three," so he skips every number containing the digit three. If he tries to count up to $100$ anyway, how many numbers will he count?
[b]p11.[/b] In square $ABCD$, point $E$ lies on side $BC$ and point $F$ lies on side $CD$ so that triangle $AEF$ is equilateral and inside the square. Point $M$ is the midpoint of segment $EF$, and $P$ is the point other than $E$ on $AE$ for which $PM = FM$. The extension of segment $PM$ meets segment $CD$ at $Q$. What is the measure of $\angle CQP$, in degrees?
[b]p12.[/b] One apple is five cents cheaper than two bananas, and one banana is seven cents cheaper than three peaches. How much cheaper is one apple than six peaches, in cents?
[b]p13.[/b] How many ordered pairs of integers $(a, b)$ exist for which |a| and |b| are at most $3$, and $a^3-a = b^3-b$?
[b]p14.[/b] Five distinct boys and four distinct girls are going to have lunch together around a table. They decide to sit down one by one under the following conditions: no boy will sit down when more boys than girls are already seated, and no girl will sit down when more girls than boys are already seated. How many possible sequences of taking seats exist?
[b]p15.[/b] Jordan is swimming laps in a pool. For each lap after the first, the time it takes her to complete is five seconds more than that of the previous lap. Given that she spends 10 minutes on the first six laps, how long does she spend on the next six laps, in minutes?
[b]p16.[/b] Chad decides to go to trade school to ascertain his potential in carpentry. Chad is assigned to cut away all the vertices of a wooden regular tetrahedron with sides measuring four inches. Each vertex is cut away by a plane which passes through the three midpoints of the edges adjacent to that vertex. What is the surface area of the resultant solid, in square inches?
Note: A tetrahedron is a solid with four triangular faces. In a regular tetrahedron, these faces are all equilateral triangles.
[b]p17.[/b] Chad and Jordan independently choose two-digit positive integers. The two numbers are then multiplied together. What is the probability that the result has a units digit of zero?
[b]p18.[/b] For art class, Jordan needs to cut a circle out of the coordinate grid. She would like to find a circle passing through at least $16$ lattice points so that her cut is accurate. What is the smallest possible radius of her circle?
Note: A lattice point is defined as one whose coordinates are both integers. For example, $(5, 8)$ is a lattice point whereas $(3.5, 5)$ is not.
[b]p19.[/b] Chad's ant Arctica is on one of the eight corners of Chad's toolbox, which measures two decimeters in width, three decimeters in length, and four decimeters in height. One day, Arctica wanted to go to the opposite corner of this box. Assuming she can only crawl on the surface of the toolbox, what is the shortest distance she has to crawl to accomplish this task, in decimeters? (You may assume that the toolbox is oating in the Exeter Space Station, so that Arctica can crawl on all six faces.)
[b]p20.[/b] Jordan is counting numbers for fun. She starts with the number $1$, and then counts onward, skipping any number that is a divisor of the product of all previous numbers she has said. For example, she starts by counting $1$, $2$, $3$, $4$, $5$, but skips 6, a divisor of $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$. What is the $20^{th}$ number she counts?
[b]p21.[/b] Chad and Jordan are having a race in the lake shown below. The lake has a diameter of four kilometers and there is a circular island in the middle of the lake with a diameter of two kilometers. They start at one point on the edge of the lake and finish at the diametrically opposite point. Jordan makes the trip only by swimming in the water, while Chad swims to the island, runs across it, and then continues swimming. They both take the fastest possible route and, amazingly, they tie! Chad swims at two kilometers an hour and runs at five kilometers an hour. At what speed does Jordan swim?
[img]https://cdn.artofproblemsolving.com/attachments/f/6/22b3b0bba97d25ab7aabc67d30821d0b12efc0.png[/img]
[b]p22.[/b] Cameron has stolen Chad's barrel of oil and is driving it around on a truck on the coordinate grid on his truck. Cameron is a bad truck driver, so he can only move the truck forward one kilometer at a $4$ $EMC^2$ $2014$ Problems time along one of the gridlines. In fact, Cameron is so bad at driving the truck that between every two one-kilometer movements, he has to turn exactly $90$ degrees. After $50$ one-kilometer movements, given that Cameron's first one-kilometer movement was westward, how many points he could be on?
[b]p23.[/b] Let $a$, $b$, and $c$ be distinct nonzero base ten digits. Assume there exist integers $x$ and $y$ for which $\overline{abc} \cdot \overline{cb} = 100x^2 + 1$ and $\overline{acb} \cdot \overline{bc} = 100y^2 + 1$. What is the minimum value of the number $\overline{abbc}$?
Note: The notation $\overline{pqr}$ designates the number whose hundreds digit is $p$, tens digit is $q$, and units digit is $r$, not the product $p \cdot q \cdot r$.
[b]p24.[/b] Let $r_1, r_2, r_3, r_4$ and $r_5$ be the five roots of the equation $x^5-4x^4+3x^2-2x+1 = 0$. What is the product of $(r_1 +r_2 +r_3 +r_4)$, $(r_1 +r_2 +r_3 +r_5)$, $(r_1 +r_2 +r_4 +r_5)$, $(r_1 +r_3 +r_4 +r_5)$, and $(r_2 +r_3 +r_4 +r_5)$?
[b]p25.[/b] Chad needs seven apples to make an apple strudel for Jordan. He is currently at 0 on the metric number line. Every minute, he randomly moves one meter in either the positive or the negative direction with equal probability. Arctica's parents are located at $+4$ and $-4$ on the number line. They will bite Chad for kidnapping Arctica if he walks onto those numbers. Also, there is one apple located at each integer between $-3$ and $3$, inclusive. Whenever Chad lands on an integer with an unpicked apple, he picks it. What is the probability that Chad picks all the apples without getting bitten by Arctica's parents?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
There are two letter sequences $A$ and $B$, both with length $100$ letters. In one move you can insert in any place of sequence ( possibly to start or to end) any number of same letters or remove any number of consecutive same letters.
Prove that it is possible to make second sequence from first sequence using not more than $100$ moves.
Let $x_n$ the sequence defined by any nonnegatine integer $x_0$ and $x_{n+1}=1+\prod_{0 \leq i \leq n}{x_i}$
Show that there exists prime $p$ such that $p\not|x_n$ for any $n$.
Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that:
(i) the first cell in the sequence is a valley,
(ii) each subsequent cell in the sequence is adjacent to the previous cell, and
(iii) the numbers written in the cells in the sequence are in increasing order.
Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square.
Author: Nikola Petrović
Consider the arithmetic sequence $8, 21,34,47,....$
a) Prove that this sequence contains infinitely many integers written only with digit $9$.
b) How many such integers less than $2010^{2010}$ are in the sequence?
Given a real number $q$, $1 < q < 2$ define a sequence $ \{x_n\}$ as follows:
for any positive integer $n$, let
\[x_n=a_0+a_1 \cdot 2+ a_2 \cdot 2^2 + \cdots + a_k \cdot 2^k \qquad (a_i \in \{0,1\}, i = 0,1, \cdots m k)\]
be its binary representation, define
\[x_k= a_0 +a_1 \cdot q + a_2 \cdot q^2 + \cdots +a_k \cdot q^k.\]
Prove that for any positive integer $n$, there exists a positive integer $m$ such that $x_n < x_m \leq x_n+1$.
In a mathematical competition $n=10\,000$ contestants participate.
During the final party, in sequence, the first one takes $1/n$ of the cake, the second one takes $2/n$ of the remaining cake, the third one takes $3/n$ of the cake that remains after the first and the second contestant, and so on until the last one, who takes all of the remaining cake.
Determine which competitor takes the largest piece of cake.
Find the number of sequences $a_1,a_2,\dots,a_{100}$ such that
$\text{(i)}$ There exists $i\in\{1,2,\dots,100\}$ such that $a_i=3$, and
$\text{(ii)}$ $|a_i-a_{i+1}|\leq 1$ for all $1\leq i<100$.