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Let $P(x) = x^2 + ax + b$ be a quadratic polynomial. For how many pairs $(a, b)$ of positive integers where $a, b < 1000$ do the quadratics $P(x+1)$ and $P(x) + 1$ have at least one root in common? | 1. We start with the given quadratic polynomial \( P(x) = x^2 + ax + b \). We need to find the pairs \((a, b)\) of positive integers where \(a, b < 1000\) such that the quadratics \(P(x+1)\) and \(P(x) + 1\) have at least one root in common.
2. First, we express \(P(x+1)\) and \(P(x) + 1\):
\[
P(x+1) = (x+1)^2 +... | 30 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers $(1, 1)$ and $(4, 5)$ and radii $r_1 < r_2$, respectively, are drawn on the coordinate plane. The product of the slopes of the two common external tangents of $\mathcal{C}_1$ and $\mathcal{C}_2$ is $3$. If the value of $(r_2 - r_1)^2$ can be expressed as a co... | 1. Let the centers of the circles $\mathcal{C}_1$ and $\mathcal{C}_2$ be $O_1$ and $O_2$ respectively, with coordinates $(1, 1)$ and $(4, 5)$. The radii of the circles are $r_1$ and $r_2$ respectively, with $r_1 < r_2$.
2. The distance between the centers $O_1$ and $O_2$ is calculated as:
\[
d = \sqrt{(4 - 1)^2 +... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integers $n$ between $1$ and $1000$, inclusive, satisfying
\[ \lfloor \sqrt{n - 1} \rfloor + 1 = \left\lfloor \sqrt{n + \sqrt{n} + 1}\right\rfloor\]
where $\lfloor n \rfloor$ denotes the greatest integer not exceeding $n$. | 1. We start by analyzing the given equation:
\[
\lfloor \sqrt{n - 1} \rfloor + 1 = \left\lfloor \sqrt{n + \sqrt{n} + 1}\right\rfloor
\]
Let \( k = \lfloor \sqrt{n - 1} \rfloor \). Then, \( k \leq \sqrt{n - 1} < k + 1 \), which implies:
\[
k^2 \leq n - 1 < (k + 1)^2 \implies k^2 + 1 \leq n < (k + 1)^2 ... | 503 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider all $2^{20}$ paths of length $20$ units on the coordinate plane starting from point $(0, 0)$ going only up or right, each one unit at a time. Each such path has a unique [i]bubble space[/i], which is the region of points on the coordinate plane at most one unit away from some point on the path. The average are... | 1. **Understanding the Problem:**
We need to find the average area enclosed by the bubble space of all paths of length 20 units starting from the point \((0,0)\) and moving only up or right. Each path has a unique bubble space, which is the region of points on the coordinate plane at most one unit away from some poi... | 261 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$. | To solve the problem, we need to find the number of ordered pairs \((i, j)\) such that \(d_i - d_j\) is divisible by \(11\), where \(d_1, d_2, \ldots, d_k\) are the distinct positive integer divisors of \(6^8\).
1. **Determine the Divisors of \(6^8\)**:
\[
6^8 = (2 \cdot 3)^8 = 2^8 \cdot 3^8
\]
The divisor... | 665 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x) = x^3 - 3x^2 + 3$. For how many positive integers $n < 1000$ does there not exist a pair $(a, b)$ of positive integers such that the equation
\[ \underbrace{P(P(\dots P}_{a \text{ times}}(x)\dots))=\underbrace{P(P(\dots P}_{b \text{ times}}(x)\dots))\]
has exactly $n$ distinct real solutions? | 1. **Substitution and Simplification**:
We start by setting \( x = t + \frac{1}{t} + 1 \). This substitution helps us simplify the polynomial \( P(x) \). Given \( P(x) = x^3 - 3x^2 + 3 \), we substitute \( x \) to find \( P(x) \):
\[
P(x) = \left( t + \frac{1}{t} + 1 \right)^3 - 3 \left( t + \frac{1}{t} + 1 \r... | 984 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Adam has a single stack of $3 \cdot 2^n$ rocks, where $n$ is a nonnegative integer. Each move, Adam can either split an existing stack into two new stacks whose sizes differ by $0$ or $1$, or he can combine two existing stacks into one new stack.
Adam keeps performing such moves until he eventually gets at least ... | To solve this problem, we need to determine the minimum number of times Adam could have combined two stacks to eventually get at least one stack with \(2^n\) rocks. Let's break down the problem step-by-step.
1. **Initial Setup**:
Adam starts with a single stack of \(3 \cdot 2^n\) rocks. Our goal is to reach a confi... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a polynomial $f(x)=x^{2020}+\sum_{i=0}^{2019} c_ix^i$, where $c_i \in \{ -1,0,1 \}$. Denote $N$ the number of positive integer roots of $f(x)=0$ (counting multiplicity). If $f(x)=0$ has no negative integer roots, find the maximum of $N$. | To find the maximum number of positive integer roots \( N \) of the polynomial \( f(x) = x^{2020} + \sum_{i=0}^{2019} c_i x^i \) where \( c_i \in \{-1, 0, 1\} \), and given that \( f(x) = 0 \) has no negative integer roots, we proceed as follows:
1. **Construct a Polynomial with Positive Integer Roots:**
Consider t... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=2^n$ . Find the maximum possible value of positive integer $n$ . | 1. **Understanding the Problem:**
We are given a permutation \(a_1, a_2, \ldots, a_{17}\) of the numbers \(1, 2, \ldots, 17\) such that the product \((a_1 - a_2)(a_2 - a_3) \cdots (a_{17} - a_1) = 2^n\). We need to find the maximum possible value of the positive integer \(n\).
2. **Considering the Parity:**
Reve... | 40 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ . | 1. **Understanding the problem**: We need to find the maximum possible value of \( n \) such that the product \((a_1 - a_2)(a_2 - a_3) \cdots (a_{17} - a_1) = n^{17}\), where \(a_1, a_2, \ldots, a_{17}\) is a permutation of \(1, 2, \ldots, 17\).
2. **Initial observation**: Since \(a_1, a_2, \ldots, a_{17}\) is a permu... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ is the set of permutation of {1,2,3,4,5,6,7,8}
(1)For all $\sigma=\sigma_1\sigma_2...\sigma_8\in S$
Evaluate the sum of S=$\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8$. Then for all elements in $S$,what is the arithmetic mean of S?
(Notice $S$ and S are different.)
(2)In $S$, how many p... | ### Part 1: Arithmetic Mean of \( S \)
1. **Define the set \( S \)**:
Let \( S \) be the set of all permutations of \(\{1, 2, 3, 4, 5, 6, 7, 8\}\). Each permutation \(\sigma \in S\) can be written as \(\sigma = \sigma_1 \sigma_2 \sigma_3 \sigma_4 \sigma_5 \sigma_6 \sigma_7 \sigma_8\).
2. **Define the sum \( S \)**... | 41787 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every... | To solve this problem, we need to determine the smallest number of evaluations of the polynomial \( f(x) \) that Alice needs to determine the parity of \( f(0) \).
1. **Understanding the Polynomial Properties**:
- \( f(x) \) is a polynomial of degree less than 187.
- For any integer \( n \), \( f(n) \) is an in... | 187 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$ What~ is~ the~ maximum~ number~ of~ distinct~ integers~ in~ a~ row~ such~ that~ the~sum~ of~ any~ 11~ consequent~ integers~ is~ either~ 100~ or~ 101~?$
I'm posting this problem for people to discuss | To solve this problem, we need to find the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101. Let's break down the solution step-by-step:
1. **Understanding the Problem:**
- We need to find a sequence of integers where the sum of any 11 consecutive in... | 22 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice has a deck of $36$ cards, $4$ suits of $9$ cards each. She picks any $18$ cards and gives the rest to Bob. Now each turn Alice picks any of her cards and lays it face-up onto the table, then Bob similarly picks any of his cards and lays it face-up onto the table. If this pair of cards has the same suit or the sam... | 1. **Restate the Problem in Grid Terms**:
- Alice and Bob are playing a game on a \(4 \times 9\) grid.
- Alice selects 18 cells to color black.
- Bob attempts to pair as many pairs of squares which are differently colored and lie in the same row or column as possible.
- We need to find the maximum number of... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0... | 1. **Define the period modulo \( n \)**:
The period of a polynomial \( P(x) \) modulo \( n \) is the smallest positive integer \( p \) such that \( P^p(0) \equiv 0 \pmod{n} \). We denote the set of possible periods modulo \( n \) as \( P_n \).
2. **Determine the largest element of \( P_{2020} \)**:
We need to fi... | 1980 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Elmo bakes cookies at a rate of one per 5 minutes. Big Bird bakes cookies at a rate of one per 6 minutes. Cookie Monster [i]consumes[/i] cookies at a rate of one per 4 minutes. Together Elmo, Big Bird, Cookie Monster, and Oscar the Grouch produce cookies at a net rate of one per 8 minutes. How many minutes does it take... | 1. Let's denote the rates of baking and consuming cookies as follows:
- Elmo's rate: $\frac{1}{5}$ cookies per minute
- Big Bird's rate: $\frac{1}{6}$ cookies per minute
- Cookie Monster's rate: $-\frac{1}{4}$ cookies per minute (since he consumes cookies)
- Oscar the Grouch's rate: $\frac{1}{o}$ cookies pe... | 120 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(n)$ denote the largest odd factor of $n$, including possibly $n$. Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer. | To solve the problem, we need to determine the value of the sum
\[
\sum_{n=1}^{2048} \frac{f(n)}{n},
\]
where \( f(n) \) denotes the largest odd factor of \( n \).
1. **Understanding \( f(n) \)**:
- If \( n \) is odd, then \( f(n) = n \).
- If \( n \) is even, we can write \( n = 2^k \cdot m \) where \( m \) is ... | 1365 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be the product of all linear polynomials $ax+b$, where $a,b\in \{0,\ldots,2016\}$ and $(a,b)\neq (0,0)$. Let $R(x)$ be the remainder when $P(x)$ is divided by $x^5-1$. Determine the remainder when $R(5)$ is divided by $2017$. | 1. We start by considering the polynomial \( P(x) \) which is the product of all linear polynomials \( ax + b \) where \( a, b \in \{0, \ldots, 2016\} \) and \((a, b) \neq (0, 0)\). We need to find the remainder \( R(x) \) when \( P(x) \) is divided by \( x^5 - 1 \).
2. Since we are working modulo \( 2017 \) (a prime ... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the value of
\[\sum_{i=0}^{2026} \frac{i^2}{9+i^4} \pmod{2027},\]
where $\frac{1}{a}$ denotes the multiplicative inverse of $a$ modulo $2027$. | To solve the problem, we need to compute the sum
\[
\sum_{i=0}^{2026} \frac{i^2}{9+i^4} \pmod{2027},
\]
where $\frac{1}{a}$ denotes the multiplicative inverse of $a$ modulo $2027$.
1. **Finite Field Extension**:
We work within the finite field extension $\mathbb{Z}_{2027}[\sqrt{-1}]$. Here, $i = \sqrt{-1}$.
2. **S... | 1689 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Alex is thinking of a number that is divisible by all of the positive integers 1 through 200 inclusive except for two consecutive numbers. What is the smaller of these numbers? | 1. **Identify the smallest number divisible by all integers from 1 to 200 except for two consecutive numbers:**
- The smallest number that is divisible by all integers from 1 to 200 is the least common multiple (LCM) of these numbers.
- However, we need to exclude two consecutive numbers from this set.
2. **Cons... | 128 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Yu Semo and Yu Sejmo have created sequences of symbols $\mathcal{U} = (\text{U}_1, \ldots, \text{U}_6)$ and $\mathcal{J} = (\text{J}_1, \ldots, \text{J}_6)$. These sequences satisfy the following properties.
[list]
[*] Each of the twelve symbols must be $\Sigma$, $\#$, $\triangle$, or $\mathbb{Z}$.
[*] In each of the s... | 1. **Define the symbols and sets:**
- Each symbol in the sequences $\mathcal{U} = (\text{U}_1, \text{U}_2, \text{U}_3, \text{U}_4, \text{U}_5, \text{U}_6)$ and $\mathcal{J} = (\text{J}_1, \text{J}_2, \text{J}_3, \text{J}_4, \text{J}_5, \text{J}_6)$ must be one of $\Sigma$, $\#$, $\triangle$, or $\mathbb{Z}$.
- Th... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Farmer John has a $47 \times 53$ rectangular square grid. He labels the first row $1, 2, \cdots, 47$, the second row $48, 49, \cdots, 94$, and so on. He plants corn on any square of the form $47x + 53y$, for non-negative integers $x, y$. Given that the unplanted squares form a contiguous region $R$, find the perimeter ... | 1. **Understanding the Problem:**
Farmer John has a $47 \times 53$ rectangular grid. He plants corn on any square of the form $47x + 53y$ for non-negative integers $x$ and $y$. We need to find the perimeter of the contiguous region $R$ formed by the unplanted squares.
2. **Expressibility:**
A number is expressib... | 794 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = \{1, \cdots, 6\}$ and $\mathcal{P}$ be the set of all nonempty subsets of $S$. Let $N$ equal the number of functions $f:\mathcal P \to S$ such that if $A,B\in \mathcal P$ are disjoint, then $f(A)\neq f(B)$.
Determine the number of positive integer divisors of $N$. | 1. **Understanding the Problem:**
We need to find the number of functions \( f: \mathcal{P} \to S \) such that if \( A, B \in \mathcal{P} \) are disjoint, then \( f(A) \neq f(B) \). Here, \( S = \{1, 2, 3, 4, 5, 6\} \) and \( \mathcal{P} \) is the set of all nonempty subsets of \( S \).
2. **Initial Observations:**... | 13872 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of positive integers $k$ such that the two parabolas$$y=x^2-k~~\text{and}~~x=2(y-20)^2-k$$intersect in four distinct points, and these four points lie on a circle with radius at most $21$. Find the sum of the least element of $S$ and the greatest element of $S$. | 1. **Combine the given equations:**
We start with the given parabolas:
\[
y = x^2 - k \quad \text{and} \quad x = 2(y - 20)^2 - k
\]
To find a relationship between \(x\) and \(y\), we add the first equation to half of the second equation:
\[
y + \frac{1}{2}x = x^2 + (y - 20)^2 - \frac{3}{2}k
\]
... | 285 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, th... | 1. **Define Moves and Constraints**:
- Call a move \( I \) if it moves from the bottom face to the bottom face or the top face to the top face.
- Call a move \( O \) if it moves from the top face to the bottom face or the bottom face to the top face.
- Note that there are no two consecutive \( O \)'s and there... | 121 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are real numbers $a, b, c, $ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i, $ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$ | 1. **Identify the roots and use Vieta's formulas:**
- For the polynomial \(x^3 + ax + b\), we know that \(-20\) is a root. Let the other roots be \(m + \sqrt{n}i\) and \(m - \sqrt{n}i\) (since complex roots come in conjugate pairs).
- The sum of the roots of \(x^3 + ax + b = 0\) is zero (since the coefficient of ... | 330 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products
$$x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2$$
is divisible by $3$. | 1. **Identify the key observation**: One of \( x_1, x_2, x_3, x_4, x_5 \) must be equal to 3. This is because the sum of the products \( x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2 \) must be divisible by 3, and having one of the numbers as 3 simplifies the problem.
2. **Assume \( x_1 = 3 \)**: Without l... | 80 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a circle there're $1000$ marked points, each colored in one of $k$ colors. It's known that among any $5$ pairwise intersecting segments, endpoints of which are $10$ distinct marked points, there're at least $3$ segments, each of which has its endpoints colored in different colors. Determine the smallest possible val... | 1. **Understanding the problem**: We have a circle with 1000 marked points, each colored in one of \( k \) colors. Among any 5 pairwise intersecting segments, endpoints of which are 10 distinct marked points, there are at least 3 segments with endpoints of different colors. We need to determine the smallest possible va... | 143 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For some positive integer $n>m$, it turns out that it is representable as sum of $2021$ non-negative integer powers of $m$, and that it is representable as sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$. | To solve the problem, we need to find the maximal value of the positive integer \( m \) such that \( n \) can be represented as the sum of 2021 non-negative integer powers of \( m \) and also as the sum of 2021 non-negative integer powers of \( m+1 \).
1. **Representation in terms of powers of \( m \):**
We need to... | 43 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$g(x):\mathbb{Z}\rightarrow\mathbb{Z}$ is a function that satisfies $$g(x)+g(y)=g(x+y)-xy.$$ If $g(23)=0$, what is the sum of all possible values of $g(35)$? | 1. Given the function \( g(x): \mathbb{Z} \rightarrow \mathbb{Z} \) that satisfies the functional equation:
\[
g(x) + g(y) = g(x+y) - xy
\]
and the condition \( g(23) = 0 \).
2. Define a new function \( f(x) \) such that:
\[
f(x) = g(x) - \frac{x^2}{2}
\]
This transformation is chosen to simpli... | 210 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Ga... | 1. **Calculate the total number of unrestricted paths:**
- Gamakichi needs to move from \((0,0)\) to \((5,5)\). This requires 5 steps in the \(x\)-direction and 5 steps in the \(y\)-direction, for a total of 10 steps. The number of ways to arrange these steps is given by the binomial coefficient:
\[
\binom... | 19152 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$ABCD$ is an isosceles trapezium such that $AD=BC$, $AB=5$ and $CD=10$. A point $E$ on the plane is such that $AE\perp{EC}$ and $BC=EC$. The length of $AE$ can be expressed as $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is not divisible by any square number other than $1$. Find the value of $(a+b)$. | 1. Given that \(ABCD\) is an isosceles trapezium with \(AD = BC\), \(AB = 5\), and \(CD = 10\). We need to find the length of \(AE\) where \(AE \perp EC\) and \(BC = EC\).
2. Let \(X\) and \(Y\) be the feet of the perpendiculars from \(A\) and \(B\) onto \(CD\), respectively. Since \(ABCD\) is an isosceles trapezium, ... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
On a table near the sea, there are $N$ glass boxes where $N<2021$, each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he... | 1. **Determine the winning strategy for each player:**
- If $N$ is even, Rafi wins by copying Sowdha's moves. This is because after each of Sowdha's moves, Rafi can always make a move that leaves an even number of boxes, maintaining the symmetry.
- If $N$ is odd, Sowdha wins by removing all balls from one pile. T... | 101 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$. | 1. **Prime Factorization and Divisors**:
For a positive integer \( n \) with prime factorization \( n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k} \), the number of divisors of \( n \) that are perfect squares is given by:
\[
s(n) = \prod_{i=1}^k \left( \left\lfloor \frac{\alpha_i}{2} \right\rfloor + ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of $80$ Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations:
1. Ever... | 1. **Understanding the Problem:**
- Cynthia wants to catch as many Pokemon as possible from a total of 80 Pokemon.
- Each Pokemon is enemies with exactly two other Pokemon.
- Cynthia cannot catch any two Pokemon that are enemies with each other.
- We need to find the sum of all possible values of \( n \), w... | 469 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square such that $A=(0,0)$ and $B=(1,1)$. $P(\frac{2}{7},\frac{1}{4})$ is a point inside the square. An ant starts walking from $P$, touches $3$ sides of the square and comes back to the point $P$. The least possible distance traveled by the ant can be expressed as $\frac{\sqrt{a}}{b}$, where $a$ and $b... | To solve this problem, we need to find the shortest path that the ant can take to touch three sides of the square and return to the point \( P \left( \frac{2}{7}, \frac{1}{4} \right) \). We will use the method of reflecting the square and the point \( P \) to find the shortest path.
1. **Reflect the Square and Point \... | 19 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$? | To find the number of ordered pairs of integers \((m, n)\) such that \(m\) and \(n\) are the legs of a right triangle with an area equal to a prime number not exceeding \(80\), we start by analyzing the given condition.
1. **Area of the Right Triangle**:
The area of a right triangle with legs \(m\) and \(n\) is giv... | 87 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\}+[r]=125$? | 1. Let \( r \) be a positive real number such that \( 25\{r\} + [r] = 125 \). We denote the integer part of \( r \) by \( [r] \) and the fractional part of \( r \) by \( \{r\} \).
2. Since \( [r] \) is a positive integer, let \( [r] = k \) where \( k \) is a positive integer. Then, \( 25\{r\} \) must also be an intege... | 2837 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut.
What is the smallest possible volume of the largest of the three cuboids?
| 1. **Initial Setup**: We start with a cube of side length \(10\). The volume of the cube is:
\[
V = 10^3 = 1000
\]
2. **First Cut**: We need to divide the cube into two cuboids with integral side lengths. Let the dimensions of the first cuboid be \(a \times 10 \times 10\) and the second cuboid be \((10-a) \ti... | 500 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For two sets $A, B$, define the operation $$A \otimes B = \{x \mid x=ab+a+b, a \in A, b \in B\}.$$ Set $A=\{0, 2, 4, \cdots, 18\}$ and $B=\{98, 99, 100\}$. Compute the sum of all the elements in $A \otimes B$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 7)[/i] | 1. First, we start by understanding the operation defined for sets \( A \) and \( B \):
\[
A \otimes B = \{x \mid x = ab + a + b, \, a \in A, \, b \in B\}
\]
We can rewrite \( x = ab + a + b \) as:
\[
x = (a+1)(b+1) - 1
\]
2. Given sets \( A = \{0, 2, 4, \ldots, 18\} \) and \( B = \{98, 99, 100\} ... | 29970 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ have its vertices at $A(0, 0), B(7, 0), C(3, 4)$ in the Cartesian plane. Construct a line through the point $(6-2\sqrt 2, 3-\sqrt 2)$ that intersects segments $AC, BC$ at $P, Q$ respectively. If $[PQC] = \frac{14}3$, what is $|CP|+|CQ|$?
[i](Source: China National High School Mathematics League 202... | 1. **Determine the coordinates of the vertices of the triangle:**
- \( A(0, 0) \)
- \( B(7, 0) \)
- \( C(3, 4) \)
2. **Find the equation of the line through the point \((6-2\sqrt{2}, 3-\sqrt{2})\):**
- Let the equation of the line be \( y = mx + c \).
- Since the line passes through \((6-2\sqrt{2}, 3-\s... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all natural numbers $n$ for which equality holds $n + d (n) + d (d (n)) +... = 2021$, where $d (0) = d (1) = 0$ and for $k> 1$, $ d (k)$ is the [i]superdivisor [/i] of the number $k$ (i.e. its largest divisor of $d$ with property $d <k$).
(Tomáš Bárta) | 1. **Define the function \( f(n) \)**:
We are given the function \( f(n) = n + d(n) + d(d(n)) + \cdots \) where \( d(k) \) is the largest divisor of \( k \) that is less than \( k \). For \( k = 0 \) or \( k = 1 \), \( d(k) = 0 \).
2. **Identify the target value**:
We need to find all natural numbers \( n \) suc... | 1919 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the remainder when the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024})$$ is divided by $1000$.
[i]Proposed by pog[/i] | 1. Let \( x = 3^{2020} \). Then the given expression can be rewritten as:
\[
(3^{2020} + 3^{2021})(3^{2021} + 3^{2022})(3^{2022} + 3^{2023})(3^{2023} + 3^{2024})
\]
Substituting \( x \) into the expression, we get:
\[
(x + 3x)(3x + 9x)(9x + 27x)(27x + 81x)
\]
Simplifying each term inside the par... | 783 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $x$ is a real number satisfying the equation $$9\log_3 x - 10\log_9 x =18 \log_{27} 45,$$ then the value of $x$ is equal to $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
[i]Proposed by pog[/i] | 1. We start with the given equation:
\[
9\log_3 x - 10\log_9 x = 18 \log_{27} 45
\]
2. We use the change of base formula for logarithms. Recall that:
\[
\log_9 x = \frac{\log_3 x}{\log_3 9} = \frac{\log_3 x}{2}
\]
and
\[
\log_{27} x = \frac{\log_3 x}{\log_3 27} = \frac{\log_3 x}{3}
\]
3.... | 140 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There are $7$ balls in a jar, numbered from $1$ to $7$, inclusive. First, Richard takes $a$ balls from the jar at once, where $a$ is an integer between $1$ and $6$, inclusive. Next, Janelle takes $b$ of the remaining balls from the jar at once, where $b$ is an integer between $1$ and the number of balls left, inclusive... | 1. **Determine the number of ways Tai can take the balls:**
- Tai can take $0, 1, 2, 3, 4, 5,$ or $6$ balls. We need to calculate the number of ways for each scenario.
2. **Calculate the number of ways Richard and Janelle can take the remaining balls:**
- If Tai takes $k$ balls, there are $7 - k$ balls left for ... | 932 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{S}$ be the set of all positive integers which are both a multiple of $3$ and have at least one digit that is a $1$. For example, $123$ is in $\mathcal{S}$ and $450$ is not. The probability that a randomly chosen $3$-digit positive integer is in $\mathcal{S}$ can be written as $\tfrac{m}{n}$, where $m$ and... | To solve this problem, we need to determine the probability that a randomly chosen 3-digit positive integer is in the set $\mathcal{S}$, which consists of numbers that are multiples of 3 and have at least one digit that is 1. We will use the Principle of Inclusion-Exclusion (PIE) to count the numbers in $\mathcal{S}$.
... | 491 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games before Jacob wins $3$ games can be written as... | 1. **Determine the probability that Jimmy wins a single game:**
- The product of two numbers is even if at least one of the numbers is even.
- The only way for the product to be odd is if both numbers are odd.
- The probability that a die roll is odd:
- If a player rolls a fair six-sided die, the probabi... | 360 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There exist complex numbers $z_1,z_2,\dots,z_{10}$ which satisfy$$|z_ki^k+ z_{k+1}i^{k+1}| = |z_{k+1}i^k+ z_ki^{k+1}|$$for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. If $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1|+|z_2|+\cdots+... | 1. **Analyzing the given condition:**
The given condition is:
\[
|z_k i^k + z_{k+1} i^{k+1}| = |z_{k+1} i^k + z_k i^{k+1}|
\]
for all integers \(1 \leq k \leq 9\). We need to understand what this implies about the relationship between \(z_k\) and \(z_{k+1}\).
2. **Simplifying the condition:**
Notice ... | 183 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Call a positive integer $k$ $\textit{pretty}$ if for every positive integer $a$, there exists an integer $n$ such that $n^2+n+k$ is divisible by $2^a$ but not $2^{a+1}$. Find the remainder when the $2021$st pretty number is divided by $1000$.
[i]Proposed by i3435[/i] | To solve the problem, we need to determine the conditions under which a positive integer \( k \) is considered "pretty." Specifically, for every positive integer \( a \), there must exist an integer \( n \) such that \( n^2 + n + k \) is divisible by \( 2^a \) but not \( 2^{a+1} \).
1. **Evenness of \( k \)**:
- Fi... | 42 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\omega_1, \omega_2, \omega_3, \ldots, \omega_{2020!}$ be the distinct roots of $x^{2020!} - 1$. Suppose that $n$ is the largest integer such that $2^n$ divides the value $$\sum_{k=1}^{2020!} \frac{2^{2019!}-1}{\omega_{k}^{2020}+2}.$$ Then $n$ can be written as $a! + b$, where $a$ and $b$ are positive integers, and... | 1. **Identify the roots of the polynomial**: The roots of \( x^{2020!} - 1 \) are the \( 2020! \)-th roots of unity, denoted as \( \omega_1, \omega_2, \ldots, \omega_{2020!} \). These roots can be expressed as \( \omega_k = e^{2\pi i k / 2020!} \) for \( k = 0, 1, 2, \ldots, 2020! - 1 \).
2. **Simplify the given sum**... | 31 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $$\sum_{n=1}^{210} nf(n)$$ is divided by $211$.
[i]Proposed by ApraTrip[/i] | 1. **Understanding \( f(n) \)**:
- \( f(n) \) is the smallest positive integer \( k \) such that \( n^k \equiv 1 \pmod{211} \).
- This means \( f(n) \) is the order of \( n \) modulo 211.
- Since 211 is a prime, the order of any integer \( n \) modulo 211 must divide \( \phi(211) = 210 \).
2. **Rewriting the ... | 48 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We consider sports tournaments with $n \ge 4$ participating teams and where every pair of teams plays against one another at most one time. We call such a tournament [i]balanced [/i] if any four participating teams play exactly three matches between themselves. So, not all teams play against one another.
Determine the ... | 1. **Lemma Statement and Proof:**
We start by proving the lemma: If \( n \geq 5 \), there can't exist any triangle (three teams having met each other) or any antitriangle (three teams none of which faced any other).
- **Proof of Lemma:**
Assume, for contradiction, that there exists a triangle involving team... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$2021$ people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up $1000$ positions to the left of their original position. | 1. **Understanding the Problem:**
We have 2021 people sitting around a circular table. We need to determine the minimum number of adjacent swaps required to move each person 1000 positions to the left of their original position.
2. **Initial Setup:**
Let's denote the positions of the people as \( P_0, P_1, P_2, ... | 1021000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a set of $2020$ distinct points in the plane. Let
\[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\]
Find the least possible value of the number of points in $M$. | 1. **Constructing the Set \( S \)**:
Let \( S \) be a set of 2020 distinct points in the plane. For simplicity, consider the points to be on the x-axis:
\[
S = \{(1,0), (2,0), \ldots, (2020,0)\}
\]
2. **Defining the Set \( M \)**:
The set \( M \) consists of midpoints of all pairs of distinct points in ... | 4037 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $r \ge 2$ is an integer, and let $m_1, n_1, m_2, n_2, \dots, m_r, n_r$ be $2r$ integers such that $$\left|m_in_j-m_jn_i\right|=1$$ for any two integers $i$ and $j$ satisfying $1 \le i<j \le r$. Determine the maximum possible value of $r$.
[i]Proposed by B Sury[/i] | 1. We start by noting the given condition: for any two integers \(i\) and \(j\) such that \(1 \le i < j \le r\), we have \(\left|m_i n_j - m_j n_i\right| = 1\). This implies that the determinant of the matrix formed by \((m_i, n_i)\) and \((m_j, n_j)\) is \(\pm 1\).
2. We need to determine the maximum possible value o... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$.
Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$. | To determine the largest possible \( n \) such that there exists a sequence of prime powers \( a_1, a_2, \dots, a_n \) satisfying \( a_i = a_{i-1} + a_{i-2} \) for all \( 3 \le i \le n \), we will analyze the given cases and prove by contradiction that \( n \geq 8 \) is not possible.
1. **Case 1: \( a_1 \) and \( a_2 ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$. Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process. | 1. **Initial Sum Calculation:**
The sum of the numbers from 1 to 9 is:
\[
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
\]
2. **Effect of Adding and Subtracting Signs:**
If we place a minus sign between any two numbers, it effectively subtracts twice the value of the number following the minus sign from the to... | 43 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have.
(Two cells of the grid are adjacent if they have a common vertex) | 1. **Understanding the Problem:**
- We are given an infinite grid where each cell contains a natural number.
- The number in each cell is equal to the number of its adjacent cells (including diagonals) that have the same number.
- We need to find the maximum number of distinct numbers that can be placed in thi... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps.
What is the re... | 1. **Define the sequences:**
Let \( t_{2k+1} \) represent the number of ways to cover a \( 3 \times (2k+1) \) chessboard with one corner square removed, and let \( s_{2k} \) represent the number of ways to cover a \( 3 \times 2k \) chessboard with no corner squares removed.
2. **Establish recurrence relations:**
... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each square of an $n \times n$ grid is coloured either blue or red, where $n$ is a positive integer. There are $k$ blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form $S_B$. He then performs the sa... | 1. Let \( b_i \) be the number of blue cells in the \( i \)-th row. Then, the total number of blue cells in the grid is given by:
\[
b_1 + b_2 + \dots + b_n = k
\]
The number of red cells in the \( i \)-th row is \( n - b_i \).
2. The sum of the squares of the number of blue cells in each row is:
\[
... | 313 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,\dots,a_{2021}$ be $2021$ integers which satisfy
\[ a_{n+5}+a_n>a_{n+2}+a_{n+3}\]
for all integers $n=1,2,\dots,2016$. Find the minimum possible value of the difference between the maximum value and the minimum value among $a_1,a_2,\dots,a_{2021}$. | 1. Given the inequality \( a_{n+5} + a_n > a_{n+2} + a_{n+3} \) for all integers \( n = 1, 2, \dots, 2016 \), we can rewrite it as:
\[
a_{n+5} - a_{n+2} > a_{n+3} - a_n
\]
This suggests that the difference between terms spaced 3 apart is increasing.
2. By induction, we can generalize this inequality. Suppo... | 85008 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
In a country, there are $28$ cities and between some cities there are two-way flights. In every city there is exactly one airport and this airport is either small or medium or big. For every route which contains more than two cities, doesn't contain a city twice and ends where it begins; has all types of airports. What... | 1. **Define the sets and constraints:**
- Let \( A \) be the set of cities with big airports.
- Let \( B \) be the set of cities with small airports.
- Let \( C \) be the set of cities with medium airports.
- We know that \( |A| + |B| + |C| = 28 \).
2. **Analyze the structure of the graph:**
- For any c... | 286 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any non-empty subset $X$ of $M=\{1,2,3,...,2021\}$, let $a_X$ be the sum of the greatest and smallest elements of $X$. Determine the arithmetic mean of all the values of $a_X$, as $X$ covers all the non-empty subsets of $M$. | 1. **Define the problem and notation:**
Let \( M = \{1, 2, 3, \ldots, 2021\} \). For any non-empty subset \( X \subseteq M \), let \( a_X \) be the sum of the greatest and smallest elements of \( X \). We need to determine the arithmetic mean of all the values of \( a_X \) as \( X \) covers all the non-empty subsets... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.
Find the largest possible value of $T_A$. | 1. **Define the set and its properties:**
Let \( A = \{x_1, x_2, x_3, x_4, x_5\} \) be a set of five distinct positive integers. Denote by \( S_A \) the sum of its elements:
\[
S_A = x_1 + x_2 + x_3 + x_4 + x_5
\]
Denote by \( T_A \) the number of triples \((i, j, k)\) with \(1 \le i < j < k \le 5\) for ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, ..., 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a + b - c | > 10$.
Determine the largest possible number of elements of $M$. | 1. **Claim**: The largest possible number of elements in \( M \) is \( 1006 \).
2. **Construction**: Consider the set \( M = \{1016, 1017, 1018, \ldots, 2021\} \). This set contains \( 2021 - 1016 + 1 = 1006 \) elements.
3. **Verification**: We need to verify that for any three elements \( a, b, c \) in \( M \), the ... | 1006 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Nine weights are placed in a scale with the respective values $1kg,2kg,...,9kg$. In how many ways can we place six weights in the left side and three weights in the right side such that the right side is heavier than the left one? | 1. **Calculate the total sum of all weights:**
\[
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \text{ kg}
\]
This is the sum of the first nine natural numbers.
2. **Determine the sum of weights on each side:**
Let the sum of the weights on the left side be \( S_L \) and on the right side be \( S_R \). Since t... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each of the spots in a $8\times 8$ chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse?
[b]Remark:[/b] A black horse could attack another black horse. | 1. **Understanding the Problem:**
We need to place as many black horses as possible on an $8 \times 8$ chessboard such that no black horse can attack more than one other black horse. In chess, a horse (knight) moves in an "L" shape: two squares in one direction and then one square perpendicular, or one square in one... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $O$ be the circumcenter of triangle $ABC$. Suppose the perpendicular bisectors of $\overline{OB}$ and $\overline{OC}$ intersect lines $AB$ and $AC$ at $D\neq A$ and $E\neq A$, respectively. Determine the maximum possible number of distinct intersection points between line $BC$ and the circumcircle of $\triangle AD... | 1. **Identify the circumcenter and perpendicular bisectors:**
Let \( O \) be the circumcenter of triangle \( ABC \). The perpendicular bisectors of \( \overline{OB} \) and \( \overline{OC} \) intersect lines \( AB \) and \( AC \) at points \( D \neq A \) and \( E \neq A \), respectively.
2. **Establish angle relati... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For each prime $p$, let $\mathbb S_p = \{1, 2, \dots, p-1\}$. Find all primes $p$ for which there exists a function $f\colon \mathbb S_p \to \mathbb S_p$ such that
\[ n \cdot f(n) \cdot f(f(n)) - 1 \; \text{is a multiple of} \; p \]
for all $n \in \mathbb S_p$.
[i]Andrew Wen[/i] | 1. **Primitive Root and Function Definition:**
- For any prime \( p \), there exists a primitive root \( g \) modulo \( p \).
- Let \( n = g^k \) for some integer \( k \).
- Define a function \( c \) such that \( g^{c(k)} = f(n) \). This implies \( g^{c(c(k))} = f(f(n)) \).
2. **Rewriting the Problem:**
... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A certain country wishes to interconnect $2021$ cities with flight routes, which are always two-way, in the following manner:
• There is a way to travel between any two cities either via a direct flight or via a sequence of connecting flights.
• For every pair $(A, B)$ of cities that are connected by a direct flight... | To solve this problem, we need to show that at least 3030 flight routes are needed to interconnect 2021 cities under the given conditions. We will use graph theory to model the cities and flight routes.
1. **Model the Problem as a Graph:**
- Let \( G = (V, E) \) be a graph where \( V \) represents the set of cities... | 3030 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ with the following property: there are rectangles $A_1, ... , A_n$ and $B_1,... , B_n,$ on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles $A_i$ and $B_i$ are disjoint for all $i \in \{1,..., n\}$, but the rectangles $A_i$ and $B_... | 1. **Claim**: The largest positive integer \( n \) with the given property is at most \( 4 \).
2. **Construction for \( n = 4 \)**:
- Consider four rectangles \( A_1, A_2, A_3, A_4 \) arranged in a cyclic manner such that:
- \( A_1 \) is connected to \( A_2 \),
- \( A_2 \) is connected to \( A_3 \)... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The figure below, composed of four regular pentagons with a side length of $1$, was glued in space as follows. First, it was folded along the broken sections, by combining the bold sections, and then formed in such a way that colored sections formed a square. Find the length of the segment $AB$ created in this way.
[im... | 1. **Define the problem and setup the coordinate system:**
We are given a figure composed of four regular pentagons with a side length of 1. The figure is folded such that the colored sections form a square. We need to find the length of the segment \( AB \).
2. **Define the vertices and vectors:**
Denote the ve... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$ | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( k \) such that for any subset \( A \) of \( M = \{1, 2, 3, \ldots, 2020\} \) with \( k \) elements, there exist three distinct numbers \( a, b, c \) from \( M \) such that \( a+b \), \( b+c \), and \( c+a \) are all in \( A \).
2. **... | 1011 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The cells of a $100 \times 100$ table are colored white. In one move, it is allowed to select some $99$ cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?
[i]S. Berlov[/i] | 1. **Proving the minimality:**
- To achieve a chessboard coloring, we need to ensure that the cells alternate in color such that no two adjacent cells (horizontally or vertically) share the same color.
- Consider the set \( S \) of cells on the main diagonal, i.e., cells \((i, i)\) for \( i = 1, 2, \ldots, 100 \)... | 100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $2021$ points on a circle. Kostya marks a point, then marks the adjacent point to the right, then he marks the point two to its right, then three to the next point's right, and so on. Which move will be the first time a point is marked twice?
[i]K. Kokhas[/i] | 1. **Understanding the Problem:**
Kostya marks points on a circle with 2021 points. He starts at a point, marks it, then marks the next point to the right, then the point two to the right of the last marked point, and so on. We need to find the first time a point is marked twice.
2. **Formulating the Problem Mathem... | 66 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the country of Graphia there are $100$ towns, each numbered from $1$ to $100$. Some pairs of towns may be connected by a (direct) road and we call such pairs of towns [i]adjacent[/i]. No two roads connect the same pair of towns.
Peter, a foreign tourist, plans to visit Graphia $100$ times. For each $i$, $i=1,2,\do... | To solve this problem, we need to determine the largest possible number of roads in Graphia such that each town is visited the same number of times by Peter after all 100 trips. Let's break down the solution step by step.
1. **Understanding the Problem:**
- There are 100 towns numbered from 1 to 100.
- Peter sta... | 4851 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We have $2021$ colors and $2021$ chips of each color. We place the $2021^2$ chips in a row. We say that a chip $F$ is [i]bad[/i] if there is an odd number of chips that have a different color to $F$ both to the left and to the right of $F$.
(a) Determine the minimum possible number of bad chips.
(b) If we impose the ... | ### Part (a)
1. **Define the problem in terms of parity:**
- We have \(2021\) colors and \(2021\) chips of each color, making a total of \(2021^2\) chips.
- A chip is considered *bad* if there is an odd number of chips of different colors on both sides of it.
2. **Classify chips and positions by parity:**
- ... | 1010 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. All numbers $m$ which are coprime to $n$ all satisfy $m^6\equiv 1\pmod n$. Find the maximum possible value of $n$. | 1. **Determine the constraints on \( n \) being even:**
- Since \( n \) is even, we consider the highest power of 2 dividing \( n \). Let \( v_2(n) \) denote the highest power of 2 dividing \( n \).
- Suppose \( v_2(n) \geq 4 \). Then \( n \) is divisible by 16. We need \( m^6 \equiv 1 \pmod{16} \) for all \( m \... | 504 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $110$ guinea pigs for each of the $110$ species, arranging as a $110\times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs align, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species. | 1. **Claim**: The maximum integer \( n \) such that no matter how the guinea pigs align, we can always find a column or a row of \( 110 \) guinea pigs containing at least \( n \) different species is \( 11 \).
2. **Upper Bound Construction**: To show that \( n \leq 11 \), we construct an arrangement where every row or... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There is a fence that consists of $n$ planks arranged in a line. Each plank is painted with one of the available $100$ colors. Suppose that for any two distinct colors $i$ and $j$, there is a plank with color $i$ located to the left of a (not necessarily adjacent) plank with color $j$. Determine the minimum possible va... | 1. **Understanding the Problem:**
We need to determine the minimum number of planks, \( n \), such that for any two distinct colors \( i \) and \( j \) (where \( i \neq j \)), there is a plank with color \( i \) located to the left of a plank with color \( j \). There are 100 colors available.
2. **Initial Observat... | 199 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We are numbering the rows and columns of a $29 \text{x} 29$ chess table with numbers $1, 2, ..., 29$ in order (Top row is numbered with $1$ and first columns is numbered with $1$ as well). We choose some of the squares in this chess table and for every selected square, we know that there exist at most one square having... | 1. **Understanding the Problem:**
We are given a $29 \times 29$ chessboard and need to select squares such that for every selected square, there is at most one other selected square with both a row number and a column number greater than or equal to the selected square's row and column numbers, respectively.
2. **G... | 43 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If a polynomial with real coefficients of degree $d$ has at least $d$ coefficients equal to $1$ and has $d$ real roots, what is the maximum possible value of $d$?
(Note: The roots of the polynomial do not have to be different from each other.) | 1. **Define the polynomial and its properties:**
Given a polynomial \( P(x) \) of degree \( d \) with real coefficients, it has at least \( d \) coefficients equal to 1 and \( d \) real roots. We can write:
\[
P(x) = x^d + x^{d-1} + \cdots + x + 1 + a \cdot x^k
\]
where \( a \) is a real number and \( k ... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A set of 1990 persons is divided into non-intersecting subsets in such a way that
1. No one in a subset knows all the others in the subset,
2. Among any three persons in a subset, there are always at least two who do not know each other, and
3. For any two persons in a subset who do not know each other, there ... | ### Part (a): Prove that within each subset, every person has the same number of acquaintances.
1. **Graph Representation**:
- Represent the problem using a graph \( G \) where each person is a vertex and an edge between two vertices indicates that the two people know each other.
- The conditions given translate... | 398 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral? | 1. **Identify the given elements and properties:**
- Triangle \(ABC\) with midpoints \(D\), \(E\), and \(F\) of sides \(BC\), \(AC\), and \(AB\) respectively.
- \(G\) is the centroid of the triangle.
- \(AEGF\) is a cyclic quadrilateral.
2. **Use properties of midpoints and centroids:**
- Since \(D\), \(E\... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose there are $997$ points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least $1991$ red points in the plane. Can you find a special case with exactly $1991$ red points? | To show that there are at least $1991$ red points in the plane, we will use a combinatorial argument.
1. **Total Number of Line Segments:**
Given $997$ points, the total number of line segments formed by joining every pair of points is given by the combination formula:
\[
\binom{997}{2} = \frac{997 \times 99... | 1991 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ with the following property: there does not exist an arithmetic progression of $1999$ real numbers containing exactly $n$ integers. | 1. **Define the problem and the arithmetic progression:**
We need to find the smallest positive integer \( n \) such that there does not exist an arithmetic progression of 1999 real numbers containing exactly \( n \) integers.
2. **Consider the general form of an arithmetic progression:**
An arithmetic progressi... | 70 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both). | 1. We need to find the largest positive integer \( N \) such that the number of integers in the set \(\{1, 2, \dots, N\}\) which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).
2. Let \( a = \left\lfloor \frac{N}{3} \right\rfloor \) be the number of integers divisible by ... | 65 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a circus, there are $n$ clowns who dress and paint themselves up using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set of colours and no more than 20 clowns may use any one par... | To solve this problem, we need to ensure that no two clowns have the same set of colors, each clown uses at least five different colors, and no more than 20 clowns use any one particular color. We will use combinatorial methods to find the largest number \( n \) of clowns.
1. **Determine the total number of possible s... | 240 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\] | 1. **Prime Factorization of 1980**:
First, we need to find the prime factorization of 1980.
\[
1980 = 2^2 \times 3^2 \times 5 \times 11
\]
This shows that 1980 can be expressed as a product of six prime factors (counting multiplicities).
2. **Understanding the Equations**:
We are given the equations:... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Numbers $\frac{49}{1}, \frac{49}{2}, ... , \frac{49}{97}$ are writen on a blackboard. Each time, we can replace two numbers (like $a, b$) with $2ab-a-b+1$. After $96$ times doing that prenominate action, one number will be left on the board. Find all the possible values fot that number. | 1. **Define the initial set of numbers:**
The numbers on the blackboard are \(\frac{49}{1}, \frac{49}{2}, \ldots, \frac{49}{97}\).
2. **Define the operation:**
Each time, we replace two numbers \(a\) and \(b\) with \(2ab - a - b + 1\).
3. **Identify the invariant:**
We need to find an invariant that remains ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all real functions $f$ definited on positive integers and satisying:
(a) $f(x+22)=f(x)$,
(b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$
for all positive integers $x$ and $y$. | 1. **Periodicity of \( f \)**:
From condition (a), we know that \( f \) is periodic with period 22. This means:
\[
f(x + 22) = f(x) \quad \text{for all positive integers } x.
\]
Therefore, \( f \) is completely determined by its values on the set \(\{1, 2, \ldots, 22\}\).
2. **Functional Equation**:
... | 13 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $S(n)$ be the sum of digits for any positive integer n (in decimal notation).
Let $N=\displaystyle\sum_{k=10^{2003}}^{10{^{2004}-1}} S(k)$. Determine $S(N)$. | 1. **Define the sum of digits function and the problem setup:**
Let \( S(n) \) be the sum of digits of a positive integer \( n \). We need to determine \( S(N) \) where \( N = \sum_{k=10^{2003}}^{10^{2004}-1} S(k) \).
2. **Establish the sum of digits for a range:**
Let \( P_n \) denote the sum of digits from \( ... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that
\[\frac{a^p - a}{p}=b^2.\] | To determine the smallest natural number \( a \geq 2 \) for which there exists a prime number \( p \) and a natural number \( b \geq 2 \) such that
\[
\frac{a^p - a}{p} = b^2,
\]
we start by rewriting the given equation as:
\[
a(a^{p-1} - 1) = pb^2.
\]
Since \( a \) and \( a^{p-1} - 1 \) are coprime (i.e., \(\gcd(a, a^... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given the sets $R_{mn} = \{ (x,y) \mid x=0,1,\dots,m; y=0,1,\dots,n \}$, consider functions $f:R_{mn}\to \{-1,0,1\}$ with the following property: for each quadruple of points $A_1,A_2,A_3,A_4\in R_{mn}$ which form a square with side length $0<s<3$, we have
$$f(A_1)+f(A_2)+f(A_3)+f(A_4)=0.$$
For each pair $(m,n)$ of pos... | 1. **Lemma:**
Consider a square of size $2 \times 2$ within $R_{mn}$. Suppose there exists a $1 \times 1$ square $A_1A_2A_3A_4$ within it such that the values of $f(A_1), f(A_2), f(A_3), f(A_4)$ are determined. Then, all points in this $2 \times 2$ square are uniquely determined.
2. **Proof of Lemma:**
Assume th... | 18 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let the sequence $(a_n)_{n \in \mathbb{N}}$, where $\mathbb{N}$ denote the set of natural numbers, is given with $a_1=2$ and $a_{n+1}$ $=$ $a_n^2$ $-$ $a_n+1$. Find the minimum real number $L$, such that for every $k$ $\in$ $\mathbb{N}$
\begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*} | 1. **Claim 1:**
We need to prove that \(\prod_{i=1}^j a_i = a_{j+1} - 1\).
**Proof:**
By the given recurrence relation, we have:
\[
a_{k+1} - 1 = a_k^2 - a_k = a_k(a_k - 1)
\]
We can continue this process:
\[
a_k(a_k - 1) = a_k(a_{k-1}^2 - a_{k-1}) = a_k a_{k-1}(a_{k-1} - 1)
\]
Repeati... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x),...,f_n(x)$ (not necessarily distinct) and $g(x)$ such that $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$ | To solve the problem, we need to find all positive integers \( n \) such that there exist non-constant polynomials with integer coefficients \( f_1(x), \ldots, f_n(x) \) and \( g(x) \) satisfying the equation:
\[ 1 + \prod_{k=1}^{n}\left(f_k^2(x) - 1\right) = (x^2 + 2013)^2 g^2(x). \]
1. **Analyzing the given equation... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$. | To prove that the sum \(1^{p+2} + 2^{p+2} + \cdots + (p-1)^{p+2}\) is divisible by \(p^2\) for a prime number \(p > 3\), we will use properties of modular arithmetic and Fermat's Little Theorem.
1. **Using Fermat's Little Theorem:**
Fermat's Little Theorem states that for any integer \(a\) and a prime \(p\), we hav... | 0 | Number Theory | proof | Yes | Yes | aops_forum | false |
Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$.
[i]Demetres Christofides, Cyprus[/i] | To determine the maximum possible value of \( r + s \) where \( P(x) \) and \( Q(x) \) are distinct polynomials of degree 2020 with non-zero coefficients, and they have \( r \) common real roots (counting multiplicity) and \( s \) common coefficients, we proceed as follows:
1. **Construction**:
- Let \( P(x) = (x^2... | 3029 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks int... | 1. **Initial Setup and Definitions:**
- Angel starts with 100 piles, each containing 100 pieces of rubbish.
- Each morning, Angel can either:
- Clear every piece of rubbish from a single pile.
- Clear one piece of rubbish from each pile.
- Each evening, the demon can either:
- Add one piece of r... | 199 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. Determine, in terms of $n$, the greatest integer which divides
every number of the form $p + 1$, where $p \equiv 2$ mod $3$ is a prime number which does not divide $n$.
| 1. **Identify the form of the prime number \( p \):**
Given that \( p \equiv 2 \pmod{3} \), we can write:
\[
p = 3k + 2 \quad \text{for some integer } k.
\]
2. **Express \( p + 1 \) in terms of \( k \):**
Adding 1 to both sides of the equation \( p = 3k + 2 \), we get:
\[
p + 1 = 3k + 3 = 3(k + 1)... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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