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(1) For $ a>0,\ b\geq 0$, Compare
$ \int_b^{b\plus{}1} \frac{dx}{\sqrt{x\plus{}a}},\ \frac{1}{\sqrt{a\plus{}b}},\ \frac{1}{\sqrt{a\plus{}b\plus{}1}}$.
(2) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{1}{\sqrt{n^2\plus{}k}}$. | ### Part (1)
We need to compare the following three expressions for \( a > 0 \) and \( b \geq 0 \):
1. \( \int_b^{b+1} \frac{dx}{\sqrt{x+a}} \)
2. \( \frac{1}{\sqrt{a+b}} \)
3. \( \frac{1}{\sqrt{a+b+1}} \)
#### Step 1: Evaluate the integral
First, we evaluate the integral \( \int_b^{b+1} \frac{dx}{\sqrt{x+a}} \).
Let... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ l_1,\ l_2$ be the tangent and nomal line respectively at the point $ (p,\ \ln (p \plus{} 1))$ on the curve $ C: y \equal{} \ln (x \plus{} 1)$.
Denote by $ T_i\ (i \equal{} 1,\ 2)$ the areas bounded by $ l_i\ (i \equal{} 1,\ 2), C$ and the $ y$ axis respectively. Find the limit $ \lim_{p\rightarrow 0} \frac {T_2}... | 1. **Find the equation of the tangent line \( l_1 \) at the point \( (p, \ln(p+1)) \) on the curve \( y = \ln(x+1) \):**
The slope of the tangent line at \( (p, \ln(p+1)) \) is given by the derivative of \( y = \ln(x+1) \) at \( x = p \):
\[
\frac{dy}{dx} = \frac{1}{x+1} \quad \text{so at} \quad x = p, \quad ... | -1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
There are many two kinds of black and white cards. When you have $ k$ cards in hand, consider the following procedure $ (\bf{A})$.
$ (\bf{A})$ You choose one card from $ k$ cards in hand with equal probability $ \frac {1}{k}$ and replace the card with different color one.
Answer the following questions.
(1) Wh... | ### Part (1): Probability of having 4 cards of the same color when starting with 2 white and 2 black cards
1. **Define the state variable**:
Let \( X_j \) be the number of white cards at step \( j \). Initially, \( X_0 = 2 \).
2. **Transition probabilities**:
- If \( X_j = m \), the probability of choosing a wh... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c$ be positive numbers such that $3a=b^3,\ 5a=c^2.$ Assume that a positive integer is limited to $d=1$ such that $a$ is divisible by $d^6.$
(1) Prove that $a$ is divisible by $3$ and $5.$
(2) Prove that the prime factor of $a$ are limited to $3$ and $5.$
(3) Find $a.$ | Given the conditions:
\[ 3a = b^3 \]
\[ 5a = c^2 \]
and the assumption that \( a \) is divisible by \( d^6 \) where \( d = 1 \).
### Part 1: Prove that \( a \) is divisible by \( 3 \) and \( 5 \).
1. From the equation \( 3a = b^3 \), we can see that \( a = \frac{b^3}{3} \). For \( a \) to be an integer, \( b^3 \) mus... | 1125 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $ | 1. **Define the problem and setup:**
- We are given a sphere with radius \(1\) and a point \(P\) on the sphere.
- Three mutually perpendicular rays emanate from \(P\) and intersect the sphere at points \(A\), \(B\), and \(C\).
- We need to prove that all such possible planes \(ABC\) pass through a fixed point ... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$. | 1. Given the sequence \( a_1, a_2, \dots, a_{1999} \) where \( a_n - a_{n-1} - a_{n-2} \) is divisible by \( 100 \) for \( 3 \leq n \leq 1999 \), we start by noting that this implies:
\[
a_n \equiv a_{n-1} + a_{n-2} \pmod{100}
\]
We are given \( a_1 = 19 \) and \( a_2 = 99 \).
2. To find the remainder of \... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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} $. | 1. We start with the initial condition \( a_1 = 1 \).
2. For \( n > 1 \), \( a_{n+1} \) is the smallest integer greater than \( a_n \) such that \( a_i + a_j \neq 3a_k \) for any \( i, j, k \) from \( \{1, 2, \dots, n+1\} \).
3. Let's find the first few terms of the sequence to identify a pattern:
- \( a_1 = 1 \)
... | 3006 | Number Theory | other | Yes | Yes | aops_forum | false |
Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions:
[b]1.[/b]Each plane contains at least $ 4$ of them
[b]2.[/b]No four points are collinear. | 1. **Understanding the problem**: We need to find the maximum number of planes in space such that there are 6 points, each plane contains at least 4 of these points, and no four points are collinear.
2. **Initial setup**: Let's denote the 6 points as \( P_1, P_2, P_3, P_4, P_5, P_6 \). We need to ensure that each plan... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]word[/i] is a sequence of 0 and 1 of length 8. Let $ x$ and $ y$ be two words differing in exactly three places.
Prove that the number of words differing from each of $ x$ and $ y$ in at least five places is 188. | 1. We start by assuming without loss of generality that \( x = 11111111 \) and \( y = 00011111 \). These two words differ in exactly three places (the first three positions).
2. Let \( z \) be a word that differs from \( x \) in \( a \) of its first 3 places and in \( b \) of its last 5 places. Then \( z \) differs fro... | 38 | Combinatorics | proof | Yes | Yes | aops_forum | false |
A permutation $a_1,a_2,\cdots ,a_6$ of numbers $1,2,\cdots ,6$ can be transformed to $1,2,\cdots,6$ by transposing two numbers exactly four times. Find the number of such permutations. | 1. **Understanding the Problem:**
We need to find the number of permutations of the numbers \(1, 2, \ldots, 6\) that can be transformed into the identity permutation \(1, 2, \ldots, 6\) by exactly four transpositions.
2. **Parity of Permutations:**
A permutation is even if it can be expressed as an even number ... | 360 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a $ 4\times 4$ squares table. How many ways that we can fill the table with $ \{0,1\}$ such that two neighbor squares (have one common side) have product which is equal to $ 0$? | To solve the problem of filling a \(4 \times 4\) table with \(\{0,1\}\) such that two neighboring squares (sharing a common side) have a product equal to \(0\), we need to ensure that no two adjacent squares both contain \(1\). This means that each \(1\) must be surrounded by \(0\)s.
1. **Define the Problem in Terms o... | 511 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For primes $a, b,c$ that satis fies the following, calculate $abc$.
$b + 8$ is a multiple of $a$, and $b^2 - 1$ is a multiple of $a$ and $c$. Also, $b + c = a^2 - 1$. | 1. **Given Conditions:**
- \( b + 8 \) is a multiple of \( a \), i.e., \( a \mid (b + 8) \).
- \( b^2 - 1 \) is a multiple of \( a \) and \( c \), i.e., \( a \mid (b^2 - 1) \) and \( c \mid (b^2 - 1) \).
- \( b + c = a^2 - 1 \).
2. **Analyzing \( a \mid (b + 8) \):**
- This implies \( b + 8 = ka \) for som... | 2009 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $3$ students from Korea, China, and Japan, so total of $9$ students are present. How many ways are there to make them sit down in a circular table, with equally spaced and equal chairs, such that the students from the same country do not sit next to each other? If array $A$ can become array $B$ by rotation, t... | 1. **Understanding the Problem:**
We need to arrange 9 students from 3 different countries (Korea, China, and Japan) around a circular table such that no two students from the same country sit next to each other. Additionally, arrangements that can be obtained by rotation are considered identical.
2. **Applying Bur... | 40320 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ be such that for every positive integer $n$, followings are satisfied.
i. $f(n+1) > f(n)$
ii. $f(f(n)) = 2n+2$
Find the value of $f(2013)$.
(Here, $\mathbb{N}$ is the set of all positive integers.) | To find the value of \( f(2013) \), we need to use the given properties of the function \( f \):
1. \( f(n+1) > f(n) \)
2. \( f(f(n)) = 2n + 2 \)
We will start by determining the initial values of \( f \) and then use induction and properties to find \( f(2013) \).
1. **Initial Values:**
- From \( f(f(1)) = 4 \), ... | 4026 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
for a positive integer $n$, there are positive integers $a_1, a_2, ... a_n$ that satisfy these two.
(1) $a_1=1, a_n=2020$
(2) for all integer $i$, $i$satisfies $2\leq i\leq n, a_i-a_{i-1}=-2$ or $3$.
find the greatest $n$ | 1. Let \( x \) be the number of values of \( i \) where \( a_i - a_{i-1} = 3 \) and \( y \) be the number of values of \( i \) where \( a_i - a_{i-1} = -2 \).
2. From the given conditions, we have:
\[
a_1 = 1 \quad \text{and} \quad a_n = 2020
\]
Therefore, we can write:
\[
1 + 3x - 2y = 2020
\]
... | 2019 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
[list]
[*] $a_1=2021^{2021}$
[*] $0 \le a_k < k$ for all integers $k \ge 2$
[*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
[/list]
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.... | 1. **Define the sequence and initial conditions:**
Let $\{a_n\}$ be a sequence of integers with the following properties:
- $a_1 = 2021^{2021}$
- $0 \le a_k < k$ for all integers $k \ge 2$
- The alternating sum $a_1 - a_2 + a_3 - a_4 + \cdots + (-1)^{k+1}a_k$ is a multiple of $k$ for all positive integers $... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player".
[b](Condition)[/b] For each player $B$ who won $A$, there is a playe... | To solve this problem, we need to determine the minimum number of skilled players in a tournament with 2023 players, where each player plays exactly one match against every other player, and there are no draws. A player \( A \) is defined as skilled if for every player \( B \) who defeats \( A \), there exists a player... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions.
(i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$
(ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$
(iii) For all $i=1,2,\cdots, k, a_i\le a_{... | 1. **Define the Set \( S \)**:
The set \( S \) is given by \( S = \{1996^n \mid n = 0, 1, 2, \cdots\} \). This means \( S \) contains powers of 1996, i.e., \( S = \{1, 1996, 1996^2, 1996^3, \cdots\} \).
2. **Conditions to Satisfy**:
- \( a_i, b_i \in S \) for all \( i = 1, 2, \cdots, k \).
- \( a_i \neq b_i \... | 1997 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle:
Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.
Supposing there ... | 1. **Label the students and define the groups:**
Let the students be labeled as \(1, 2, 3, 4, 5, 6\). We need to form study groups and service activity groups such that each group has exactly 3 members, and for any pair of students, there are the same number of study groups and service activity groups that both stud... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$ | 1. **Using Vieta's Formulas:**
Given the polynomial \( P(x) = x^{2006} - 2006x^{2005} + a_{2004}x^{2004} + \cdots + a_2x^2 + a_1x + 1 \), we know from Vieta's formulas that the sum of the roots \( x_1, x_2, \ldots, x_{2006} \) is given by:
\[
\sum_{k=1}^{2006} x_k = 2006
\]
and the sum of the reciprocals... | -2006 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S$ be the set of all positive integers whose all digits are $ 1$ or $ 2$. Denote $ T_{n}$ as the set of all integers which is divisible by $ n$, then find all positive integers $ n$ such that $ S\cap T_{n}$ is an infinite set. | 1. **Define the sets and the problem:**
- Let \( S \) be the set of all positive integers whose digits are either \( 1 \) or \( 2 \).
- Let \( T_n \) be the set of all integers divisible by \( n \).
- We need to find all positive integers \( n \) such that \( S \cap T_n \) is an infinite set.
2. **Initial obs... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality?
$ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ . | 1. We start with the given inequality:
\[
\frac{a}{c + kb} + \frac{b}{a + kc} + \frac{c}{b + ka} \geq \frac{1}{2007}
\]
We need to find the value of the positive constant \( k \) that satisfies this inequality for all positive reals \( a \), \( b \), and \( c \).
2. Let's test the inequality with \( a = b ... | 6020 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $V=[(x,y,z)|0\le x,y,z\le 2008]$ be a set of points in a 3-D space.
If the distance between two points is either $1, \sqrt{2}, 2$, we color the two points differently.
How many colors are needed to color all points in $V$? | 1. **Claim**: We need 7 colors to color all points in \( V \).
2. **Example of 7 Points**: Consider the seven points \((2,2,2)\), \((3,2,2)\), \((1,2,2)\), \((2,3,2)\), \((2,1,2)\), \((2,2,3)\), and \((2,2,1)\). These points are such that the distance between any two of them is either \(1\), \(\sqrt{2}\), or \(2\). Th... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest value of $M$ for which for any choice of positive integer $n$ and positive real numbers $x_1<x_2<\ldots<x_n \le 2023$ the inequality
$$\sum_{1\le i < j \le n , x_j-x_i \ge 1} 2^{i-j}\le M$$
holds. | To determine the smallest value of \( M \) for which the inequality
\[ \sum_{1 \le i < j \le n, x_j - x_i \ge 1} 2^{i-j} \le M \]
holds for any choice of positive integer \( n \) and positive real numbers \( x_1 < x_2 < \ldots < x_n \le 2023 \), we proceed as follows:
1. **Define Sets \( S \) and \( T \)**:
Let \(... | 4044 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operati... | ### Part (1)
To determine if there exists a series of operations such that all 9 numbers turn out to be equal from the given initial arrangements, we need to analyze the operations and their effects on the numbers.
#### Initial Arrangement a)
\[ \begin{array}{ccc}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array} \... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Given vector $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$ and recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n\geq 0}$
$$\mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1}$$
Evaluate the value of infinite series $\sum_{n=1}^\infty (3,2,1)\cdot \ma... | 1. **Initial Setup and Definitions:**
Given the vector $\mathbf{u} = \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \in \mathbb{R}^3$ and the recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n \geq 0}$:
\[
\mathbf{v}_0 = (1, 2, 3), \quad \mathbf{v}_n = \mathbf{u} \times \mathbf{v}_{n-1}
\]
... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$.
$$\mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0)$$
Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}... | To solve the given integral, we will use the vector calculus identity known as the curl of a cross product. Specifically, we will use the following identity:
\[
\mathbf{G} \cdot \text{curl}(\mathbf{F}) - \mathbf{F} \cdot \text{curl}(\mathbf{G}) = \nabla \cdot (\mathbf{F} \times \mathbf{G})
\]
This identity simplifies t... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
For a real number $a$ and an integer $n(\geq 2)$, define
$$S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}$$
Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real. | To find the value of \( a \) such that the sequence \( \{S_n(a)\}_{n \geq 2} \) converges to a positive real number, we start by analyzing the given expression:
\[ S_n(a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}. \]
We need to determine the behavior of \( S_n(a) \) as \( n \to \infty \). To do this, we w... | 2019 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than $11$.
P.s. the current year in the problem is $2010$. | 1. Let the year of birth be represented as \(1000a + 100b + 10c + d\), where \(a, b, c, d\) are the digits of the year.
2. Let the current year be \(2010\).
3. The person's age is the sum of the digits of their year of birth, i.e., \(a + b + c + d\).
4. Given that the person is older than \(11\), we have \(a + b + c + ... | 24 | Logic and Puzzles | other | Yes | Yes | aops_forum | false |
A student read the book with $480$ pages two times. If he in the second time for every day read $16$ pages more than in the first time and he finished it $5$ days earlier than in the first time. For how many days did he read the book in the first time? | 1. Let \( x \) be the number of pages the student reads per day the first time.
2. Let \( y \) be the number of days the student reads the book the first time. Then, we have:
\[
y = \frac{480}{x}
\]
3. The student reads \( x + 16 \) pages per day the second time and finishes the book in \( y - 5 \) days. Thus:... | 15 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Ana has $22$ coins. She can take from her friends either $6$ coins or $18$ coins, or she can give $12$ coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have. | 1. **Identify the operations and their effects on the number of coins:**
- Ana can take 6 coins: \( n \rightarrow n + 6 \)
- Ana can take 18 coins: \( n \rightarrow n + 18 \)
- Ana can give 12 coins: \( n \rightarrow n - 12 \)
2. **Analyze the problem using modular arithmetic:**
- Since \( 6 \equiv 0 \pmod... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest possible value of the expression $$gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )$$ for naturals $n$?
[hide]original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n? [/hide] | To find the largest possible value of the expression $\gcd(n^2 + 3, (n + 1)^2 + 3)$ for natural numbers $n$, we start by simplifying the given expressions.
1. **Simplify the expressions:**
\[
(n + 1)^2 + 3 = n^2 + 2n + 1 + 3 = n^2 + 2n + 4
\]
Therefore, we need to find:
\[
\gcd(n^2 + 3, n^2 + 2n + 4)... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$2017$ chess players participated in the chess tournament, each of them played exactly one chess game with each other. Let's call a trio of chess players $A, B, C$ a [i]principled [/i]one, if $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. What is the largest possible number of threes of principled chess play... | To solve the problem, we need to determine the largest possible number of principled trios among 2017 chess players. A principled trio is defined as a set of three players \(A, B, C\) such that \(A\) defeats \(B\), \(B\) defeats \(C\), and \(C\) defeats \(A\).
Given the general formula for the maximum number of princi... | 341606288 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$22$ football players took part in the football training. They were divided into teams of equal size for each game ($11:11$). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training. | 1. **Define the problem and notation:**
Let \( f(n) \) be the minimum number of games required for \( n \) players such that each player has played against every other player at least once. In this problem, we have \( n = 22 \) players, and they are divided into two teams of 11 players each for each game.
2. **Init... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive constant $k$ such that no matter what $3$ lattice points we choose the following inequality holds:
$$ L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}} $$
where $L_{\max}$, $L_{\min}$ is the maximal and minimal distance between chosen points. | 1. **Observations:**
- We observe that \( L_{\max} \) is strictly greater than \( L_{\min} \), as the lattice points create a triangle with rational area. However, the area of an equilateral triangle is irrational.
- Both \( L_{\max} \) and \( L_{\min} \) are square roots of some positive integers. This follows d... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A kingdom has $2021$ towns. All of the towns lie on a circle, and there is a one-way road going from every town to the next $101$ towns in a clockwise order. Each road is colored in one color. Additionally, it is known that for any ordered pair of towns $A$ and $B$ it is possible to find a path from $A$ to $B$ so that ... | 1. **Identify the towns with elements of $\mathbb{Z}/2021\mathbb{Z}$**:
- Each town can be represented as an element of the set $\{0, 1, 2, \ldots, 2020\}$.
- There is a one-way road from each town $n$ to the next $101$ towns in a clockwise order, i.e., from town $n$ to towns $n+1, n+2, \ldots, n+101$ (mod 2021).... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
At a strange party, each person knew exactly $22$ others.
For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew.
For any pair of people $X$ and $Y$ who did not know one another, there were exactly 6 other people that they both knew.
How many people were at th... | 1. **Define the problem and variables:**
Let \( n \) be the number of people at the party. Each person knows exactly 22 others. For any pair of people \( X \) and \( Y \) who know each other, there is no other person at the party that they both know. For any pair of people \( X \) and \( Y \) who do not know each ot... | 100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd$(4, 6, 8)=2$ and gcd $(12, 15)=3$.) Suppose that positive integers $a, b, c$ satisfy the following four conditions:
$\bullet$ gcd $(a, b, c)=1$,
$\bullet$ gcd $(a, b + c)>1$,
$\bullet$ gcd $(b, c + a)>1$,
$\bullet$ gcd $(c, a... | Let's break down the problem and solution step by step.
### Part (a)
We need to determine if it is possible for \(a + b + c = 2015\) given the conditions:
1. \(\gcd(a, b, c) = 1\)
2. \(\gcd(a, b + c) > 1\)
3. \(\gcd(b, c + a) > 1\)
4. \(\gcd(c, a + b) > 1\)
To show that it is possible, we need to find specific values... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest integer $n$ such that an $n\times n$ square can be partitioned into $40\times 40$ and $49\times 49$ squares, with both types of squares present in the partition, if
a) $40|n$; b) $49|n$; c) $n\in \mathbb N$. | To solve the problem, we need to find the smallest integer \( n \) such that an \( n \times n \) square can be partitioned into \( 40 \times 40 \) and \( 49 \times 49 \) squares, with both types of squares present in the partition. We will consider the conditions \( 40|n \), \( 49|n \), and \( n \in \mathbb{N} \).
1. ... | 1960 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA =
17, PB = 15,$ and $PC = 6.$
What is the length of $PD$? | 1. Let the distances from point \( P \) to the vertices \( A, B, C, \) and \( D \) be denoted as \( PA = 17, PB = 15, \) and \( PC = 6 \). We need to find the distance \( PD \).
2. We can use the British flag theorem, which states that for any point \( P \) inside a rectangle \( ABCD \), the sum of the squares of the ... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest positive multiple of $225$ that can be written using
digits $0$ and $1$ only? | 1. **Understanding the Problem:**
We need to find the smallest positive multiple of \(225\) that can be written using only the digits \(0\) and \(1\).
2. **Factorization of 225:**
\[
225 = 15^2 = (3 \times 5)^2 = 3^2 \times 5^2
\]
Therefore, any multiple of \(225\) must be divisible by both \(9\) and \(... | 11111111100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given positive integers $a, b,$ and $c$ with $a + b + c = 20$.
Determine the number of possible integer values for $\frac{a + b}{c}.$ | 1. Given the equation \(a + b + c = 20\), we can let \(d = a + b\). Therefore, we have \(d + c = 20\).
2. We need to determine the number of possible integer values for \(\frac{a + b}{c} = \frac{d}{c}\).
3. Since \(d + c = 20\), we can express \(d\) as \(d = 20 - c\).
4. We need to find the integer values of \(\frac{d}... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest
possible value of $p$. | 1. We start with the given condition: when $2020$ is divided by a prime $p$, the remainder is $6$. This can be written in modular arithmetic as:
\[
2020 \equiv 6 \pmod{p}
\]
This implies:
\[
2020 - 6 \equiv 0 \pmod{p} \implies 2014 \equiv 0 \pmod{p}
\]
Therefore, $p$ must be a divisor of $2014$.... | 53 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a right-angled triangle with perimeter $18$. The sum of the squares
of the three side lengths is $128$. What is the area of the triangle?
| 1. Let the side lengths of the right-angled triangle be \(a\), \(b\), and \(c\), where \(c\) is the hypotenuse. We are given the following conditions:
\[
a + b + c = 18
\]
\[
a^2 + b^2 + c^2 = 128
\]
\[
a^2 + b^2 = c^2
\]
2. From the Pythagorean theorem, we know:
\[
a^2 + b^2 = c^2
... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integer solutions $(a,b,c,d)$ to the equation \[(a^2+b^2)(c^2-d^2)=2020.\]
Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different. | To find the number of positive integer solutions \((a,b,c,d)\) to the equation \((a^2+b^2)(c^2-d^2)=2020\), we start by factoring 2020:
\[ 2020 = 2^2 \cdot 5 \cdot 101 \]
We need to find pairs \((a^2 + b^2, c^2 - d^2)\) such that their product is 2020. The possible pairs are:
\[ (a^2 + b^2, c^2 - d^2) = (404, 5), (2... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A set $S$ has $7$ elements. Several $3$-elements subsets of $S$ are listed, such
that any $2$ listed subsets have exactly $1$ common element. What is the maximum number of subsets that can be listed?
| To solve this problem, we need to determine the maximum number of 3-element subsets of a 7-element set \( S \) such that any two subsets have exactly one common element.
1. **Understanding the Problem:**
- Let \( S = \{a_1, a_2, a_3, a_4, a_5, a_6, a_7\} \).
- We need to find the maximum number of 3-element sub... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest
side has length $12$. What is the length of the altitude (height) of the triangle that
is perpendicular to the longest side? | 1. **Determine the angles of the triangle:**
Given the ratio of the interior angles is \(1 : 5 : 6\), let the angles be \(x\), \(5x\), and \(6x\). Since the sum of the interior angles in a triangle is \(180^\circ\), we have:
\[
x + 5x + 6x = 180^\circ
\]
\[
12x = 180^\circ
\]
\[
x = 15^\circ
... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Geetha wants to cut a cube of size $4 \times 4\times 4$ into $64$ unit cubes (of size
$1\times 1\times 1$). Every cut must be straight, and parallel to a face of the big cube. What
is the minimum number of cuts that Geetha needs?
Note: After every cut, she can rearrange the pieces before cutting again. At every
cut, sh... | To solve this problem, we need to determine the minimum number of straight cuts required to divide a $4 \times 4 \times 4$ cube into $64$ unit cubes of size $1 \times 1 \times 1$.
1. **Initial Setup**:
- The large cube has dimensions $4 \times 4 \times 4$.
- We need to cut it into $64$ smaller cubes of dimensio... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
International Mathematical Olympiad National Selection Test
Malaysia 2020 Round 1 Primary
Time: 2.5 hours [hide=Rules]
$\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer.
$\bullet$ No mark is deducted for a wrong answer.
$\bullet$ The maximum number of p... | ### Problem 6
The number \( N \) is the smallest positive integer with the sum of its digits equal to \( 2020 \). What is the first (leftmost) digit of \( N \)?
1. To minimize \( N \), we need to maximize the leftmost digit.
2. The largest possible digit is \( 9 \).
3. We can use as many \( 9 \)'s as possible to get c... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider the following one-person game: A player starts with score $0$ and writes the number $20$ on an
empty whiteboard. At each step, she may erase any one integer (call it a) and writes two positive integers
(call them $b$ and $c$) such that $b + c = a$. The player then adds $b\times c$ to her score. She repeats the... | 1. **Initial Setup and Problem Understanding:**
- The player starts with a score of $0$ and writes the number $20$ on the whiteboard.
- At each step, the player can erase any integer $a$ and replace it with two positive integers $b$ and $c$ such that $b + c = a$. The score is then increased by $b \times c$.
- ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue? | 1. **Understanding the Problem:**
We need to form a queue of 10 girls such that no girl stands directly between two girls shorter than her. This implies that for any girl in the queue, all girls to her left must be either all shorter or all taller, and similarly for the girls to her right.
2. **Placement of the Sho... | 512 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The two diagonals of a rhombus have lengths with ratio $3 : 4$ and sum $56$. What is the perimeter of the rhombus? | 1. **Determine the lengths of the diagonals:**
- Let the lengths of the diagonals be \(d_1\) and \(d_2\).
- Given the ratio of the diagonals is \(3:4\), we can write \(d_1 = 3x\) and \(d_2 = 4x\) for some positive number \(x\).
- The sum of the diagonals is given as \(56\). Therefore, we have:
\[
d_1... | 80 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many integers $n$ (with $1 \le n \le 2021$) have the property that $8n + 1$ is a perfect square? | 1. We start by noting that we need \(8n + 1\) to be a perfect square. Let \(8n + 1 = k^2\) for some integer \(k\). Then we have:
\[
k^2 = 8n + 1
\]
Rearranging this equation, we get:
\[
8n = k^2 - 1 \implies n = \frac{k^2 - 1}{8}
\]
2. For \(n\) to be an integer, \(\frac{k^2 - 1}{8}\) must also be... | 63 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions? | To solve the problem, we need to find the number of four-digit passcodes where the product of the digits is 18. Let's denote the digits by \(a, b, c,\) and \(d\). Therefore, we need to find the number of solutions to the equation:
\[ a \cdot b \cdot c \cdot d = 18 \]
where \(a, b, c,\) and \(d\) are digits from 1 to ... | 36 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so?
[img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg[/img] | 1. **Understanding the Problem:**
Dinesh wants to arrange squares and regular pentagons alternately to form a closed loop. Each shape has a side length of 1. We need to determine how many pentagons are required to complete this loop.
2. **Analyzing the Geometry:**
- A regular pentagon has internal angles of \(10... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the ar... | 1. **Define the vertices and midpoints:**
Given the square \(ABCD\) with side length \(6\), we can place the vertices at:
\[
A(0,0), \quad B(6,0), \quad C(6,6), \quad D(0,6)
\]
The midpoints of each side are:
\[
E(3,0), \quad F(6,3), \quad G(3,6), \quad H(0,3)
\]
2. **Find the intersection poin... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$, contains the number $0$. For every other cell $C$, we consider a route from $(1, 1)$ to $C$, where at each step we can only go one cell to the right or one cell up (not... | 1. **Understanding the Problem:**
- We have a $2021 \times 2021$ table.
- The bottom-left cell $(1, 1)$ contains the number $0$.
- For any other cell $(x, y)$, the number in the cell is determined by the sum of the steps taken to reach that cell from $(1, 1)$ plus the sum of the numbers in the cells along the ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$.
What is the area of triangle $A... | 1. Let the area of the equilateral triangle \(ABC\) be \(n\) square centimeters.
2. The triangle is divided into 10 bands of equal width by 9 lines parallel to \(BC\). These bands form smaller similar triangles within the larger triangle.
3. The side lengths of these smaller triangles are proportional to the original t... | 200 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$. | 1. We need to determine the smallest prime \( p \) such that \( 2018! \) is divisible by \( p^3 \) but not by \( p^4 \). This implies that the \( p \)-adic valuation \( v_p(2018!) = 3 \).
2. Using Legendre's formula, we can express \( v_p(2018!) \) as:
\[
v_p(2018!) = \left\lfloor \frac{2018}{p} \right\rfloor + ... | 509 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a regular polygon with $n$ sides. It is known that there are $1200$ ways to choose three of the vertices of the polygon such that they form the vertices of a [b]right triangle[/b]. What is the value of $n$? | 1. **Understanding the problem**: We need to find the number of sides \( n \) of a regular polygon such that there are 1200 ways to choose three vertices that form a right triangle.
2. **Key observation**: In a regular polygon, a right triangle can be formed if and only if one of its sides is the diameter of the circu... | 50 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$, and $L$. What is the smallest possible integer value for $L$? | 1. Let the sides of the cuboid be \(a\), \(b\), and \(c\). The volume of the cuboid is given by \(abc\), and the areas of the three different faces are \(ab\), \(ac\), and \(bc\).
2. We are given that the areas of three faces are \(7\), \(27\), and \(L\). Therefore, we have:
\[
ab = 7, \quad ac = 27, \quad bc = L... | 21 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers? | 1. We start by noting that the product of 10 integers is 1024. We need to find the greatest possible sum of these 10 integers.
2. Since 1024 is a power of 2, we can express it as \(1024 = 2^{10}\).
3. To maximize the sum of the integers, we should use the smallest possible integers that multiply to 1024. The smallest i... | 1033 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM ,each mango costs $4$ RM , and each papaya costs $5$ RM . He bought at least one of each fruit, and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought? | 1. **Define the variables:**
Let \( a \) be the number of apples, \( m \) be the number of mangoes, and \( p \) be the number of papayas.
2. **Set up the cost equation:**
Each apple costs 3 RM, each mango costs 4 RM, and each papaya costs 5 RM. The total cost is 50 RM.
\[
3a + 4m + 5p = 50
\]
3. **Acco... | 15 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Daud want to paint some faces of a cube with green paint. At least one face must be painted. How many ways are there for him to paint the cube?
Note: Two colorings are considered the same if one can be obtained from the other by rotation. | To solve this problem, we need to consider the symmetry of the cube and the different ways to paint its faces. We will use Burnside's Lemma to count the distinct colorings under the cube's rotational symmetries.
1. **Paint one face only:**
- There are 6 faces on a cube. Painting one face can be done in 6 ways.
-... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{S}$ be a set of $2023$ points in a plane, and it is known that the distances of any two different points in $S$ are all distinct. Ivan colors the points with $k$ colors such that for every point $P \in \mathcal{S}$, the closest and the furthest point from $P$ in $\mathcal{S}$ also have the same color as $... | 1. **Graph Representation**:
- Consider a directed graph \( G \) where each vertex represents a point in the set \( \mathcal{S} \).
- Draw a directed edge \( a \rightarrow b \) if \( b \) is the closest point to \( a \).
- Since the distances between any two points are distinct, each point has a unique closest... | 506 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
It is known that a polynomial $P$ with integer coefficients has degree $2022$. What is the maximum $n$ such that there exist integers $a_1, a_2, \cdots a_n$ with $P(a_i)=i$ for all $1\le i\le n$?
[Extra: What happens if $P \in \mathbb{Q}[X]$ and $a_i\in \mathbb{Q}$ instead?] | 1. **Understanding the problem**: We need to find the maximum number \( n \) such that there exist integers \( a_1, a_2, \ldots, a_n \) with \( P(a_i) = i \) for all \( 1 \le i \le n \), given that \( P \) is a polynomial of degree 2022 with integer coefficients.
2. **Analyzing the polynomial**: Since \( P \) has inte... | 2022 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A biologist found a pond with frogs. When classifying them by their mass, he noticed the following:
[i]The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass.
[/i]As fate would have it, the frogs escaped and the bio... | 1. Let \( T \) be the total mass of all the frogs in the pond.
2. Let \( N \) be the total number of frogs in the pond.
3. The mass of the 50 lightest frogs is \( 0.3T \).
4. The mass of the 44 heaviest frogs is \( 0.27T \).
5. The mass of the remaining \( N - 50 - 44 = N - 94 \) frogs is \( 0.43T \).
We need to find ... | 165 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors? | To solve this problem, we need to find the pages that have both a blue mark and a red mark. A page has a blue mark if it is formed only by even digits, and it has a red mark if it is congruent to $2 \pmod{3}$.
1. **Identify pages with only even digits:**
- The even digits are $0, 2, 4, 6, 8$.
- We need to list a... | 44 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square. | 1. **Identify numbers with exactly three positive divisors:**
- A number \( n \) has exactly three positive divisors if and only if it is of the form \( p^2 \), where \( p \) is a prime number. This is because the divisors of \( p^2 \) are \( 1, p, \) and \( p^2 \).
2. **List the squares of primes less than 100:**
... | 44100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can one arrange seven white and five black balls in a line in such a way that there are no two neighboring black balls? | 1. **Initial Placement of Black Balls:**
- We need to place the five black balls such that no two black balls are adjacent. To achieve this, we can place the black balls with at least one white ball between each pair.
- Consider the arrangement of black balls as follows: B _ B _ B _ B _ B. Here, the underscores r... | 56 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest possible natural number $n = \overline{a_m ...a_2a_1a_0} $ (in decimal system) such that the number $r = \overline{a_1a_0a_m ..._20} $ equals $2n$. | 1. **Understanding the Problem:**
We need to find the smallest natural number \( n = \overline{a_m \ldots a_2a_1a_0} \) such that when the digits are rearranged to form \( r = \overline{a_1a_0a_m \ldots a_2} \), the number \( r \) equals \( 2n \).
2. **Analyzing the Last Digit:**
Since \( r = 2n \), the last dig... | 263157894736842105 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of paths from $A$ to $B$ on the picture that go along gridlines only, do not pass through any point twice, and never go upwards?
[img]https://cdn.artofproblemsolving.com/attachments/0/2/87868e24a48a2e130fb5039daeb85af42f4b9d.png[/img] | To determine the number of paths from point \( A \) to point \( B \) on the given grid, we need to consider the constraints: the paths must go along gridlines only, not pass through any point twice, and never go upwards.
1. **Understanding the Grid and Constraints**:
- The grid is a 6x6 grid.
- Movement is restr... | 924 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A company of $n$ soldiers is such that
(i) $n$ is a palindrome number (read equally in both directions);
(ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively.
Find the smallest $n$ satisfying these conditions and prove that there are infinitely... | 1. **Identify the conditions:**
- \( n \) is a palindrome.
- \( n \equiv 2 \pmod{3} \)
- \( n \equiv 3 \pmod{4} \)
- \( n \equiv 0 \pmod{5} \)
2. **Use the Chinese Remainder Theorem (CRT):**
- We need to solve the system of congruences:
\[
\begin{cases}
n \equiv 2 \pmod{3} \\
n \equi... | 515 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... $ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$. | 1. **Understanding the sequence pattern**:
The sequence alternates between odd and even numbers, increasing the count of each type in each subsequent block. Specifically:
- 1 odd number
- 2 even numbers
- 3 odd numbers
- 4 even numbers
- and so on...
2. **Identifying the pattern in differences**:
... | 1993 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A capricious mathematician writes a book with pages numbered from $2$ to $400$. The pages are to be read in the following order. Take the last unread page ($400$), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the ... | To determine the last page to be read, we need to follow the given reading order and identify the pages that remain unread until the end. Let's break down the steps:
1. **Identify the first page to be read:**
The first page to be read is the last page, which is \(400\).
2. **Identify pages not relatively prime to ... | 397 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $handshakes, what is $N$? | 1. **Categorize the students based on their positions:**
- **Corner students:** There are 4 corner students in an \(m \times n\) array. Each corner student shakes hands with 3 other students.
- **Edge students (excluding corners):** There are \(2(m-2) + 2(n-2) = 2(m+n-4)\) edge students. Each edge student shakes ... | 280 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numb... | 1. **Determine the position of each chip at minute \( m \):**
- Each chip \( x \) moves \( x \) booths every minute. Therefore, at minute \( m \), chip \( x \) will be at booth \( mx \mod 64 \).
- Specifically, chip 1 will be at booth \( m \mod 64 \).
2. **Condition for lighting bulbs:**
- A bulb is lit if ch... | 64 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear? | 1. **Initial Setup**: We start with 6 points in space, which are not all coplanar, and no three of which are collinear. We need to determine the minimum number of planes these points can determine.
2. **Tetrahedron Formation**: Select 4 points to form a non-degenerate tetrahedron. Since the points are not all coplanar... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$ | 1. **Understanding the Problem:**
We need to find the maximum number of positive integers such that for any two distinct integers \(a\) and \(b\), the inequality \(|a - b| \ge \frac{ab}{100}\) holds.
2. **Rewriting the Inequality:**
The given inequality is \(|a - b| \ge \frac{ab}{100}\). This can be rewritten in... | 10 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The fraction $\frac1{10}$ can be expressed as the sum of two unit fraction in many ways, for example, $\frac1{30}+\frac1{15}$ and $\frac1{60}+\frac1{12}$.
Find the number of ways that $\frac1{2007}$ can be expressed as the sum of two distinct positive unit fractions. | 1. We start with the given equation:
\[
\frac{1}{2007} = \frac{1}{m} + \frac{1}{n}
\]
where \(m\) and \(n\) are distinct positive integers.
2. Rewrite the equation with a common denominator:
\[
\frac{1}{2007} = \frac{m + n}{mn}
\]
3. Cross-multiplying to clear the fractions, we get:
\[
mn =... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$. Find all values of $n$ such that $n=d_2^2+d_3^3$. | 1. **Assume \( n \) is odd.**
- We know that \( d_2 \) is the smallest divisor of \( n \) other than 1, so it must be a prime number.
- If \( n \) is odd, then all its divisors \( d_1, d_2, \ldots, d_k \) must also be odd.
- However, \( d_2^2 + d_3^3 \) would be even if both \( d_2 \) and \( d_3 \) are odd, wh... | 68 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.
\[\begin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
\hline
17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\
\hline
25 & 26 & 27 & 28 &... | 1. **Understanding the Knight's Movement**:
- A knight in chess moves in an "L" shape: two squares in one direction and then one square perpendicular, or one square in one direction and then two squares perpendicular.
- This means a knight placed on a square can attack up to 8 different squares, depending on its ... | 1056 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all four digit positive integers such that the sum of the squares of the digits equals twice the sum of the digits. | Let \( N = \overline{a_3a_2a_1a_0} \) be a four-digit number satisfying the equation:
\[
\sum_{i=0}^3 a_i^2 = 2 \sum_{i=0}^3 a_i
\]
We can rewrite this equation as:
\[
\sum_{i=0}^3 (a_i^2 - 2a_i) = 0
\]
This can be further simplified by completing the square:
\[
\sum_{i=0}^3 (a_i^2 - 2a_i + 1 - 1) = 0
\]
\[
\sum_{i=0... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Certain tickets are numbered as follows: $1, 2, 3, \dots, N $. Exactly half of the tickets have the digit $ 1$ on them. If $N$ is a three-digit number, determine all possible values of $N $. | To solve this problem, we need to determine the number \( N \) such that exactly half of the tickets from 1 to \( N \) contain the digit 1. We will break down the problem into smaller steps and analyze the occurrences of the digit 1 in different ranges.
1. **Count the occurrences of the digit 1 in the range 1 to 99:**... | 598 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_ {32} = 6$, $p_ {203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$. Find the largest prime that divides $S $. | To solve the problem, we need to find the sum \( S = p_1 + p_2 + p_3 + \dots + p_{999} \) and determine the largest prime that divides \( S \).
1. **Define \( p_n \)**:
- \( p_n \) is the product of all non-zero digits of \( n \).
- For example, \( p_6 = 6 \), \( p_{32} = 6 \), \( p_{203} = 6 \).
2. **Classify ... | 103 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $. | 1. **Claim**: The maximum value of \( n \) is \( 25 \).
2. **Constructing a Set of 25 Elements**:
- Consider the set \( S = \{p^2 : p \text{ is a prime number and } p < 100\} \).
- There are 25 prime numbers less than 100. These primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ... | 25 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two types of pieces, bishops and rooks, are to be placed on a $10\times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to [i]attack[/i] a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on ... | To solve this problem, we need to determine the maximum number of pieces on a $10 \times 10$ chessboard such that no piece attacks another. We will use the definitions of how bishops and rooks attack other pieces to guide our placement strategy.
1. **Understanding the Attack Mechanism:**
- A bishop attacks any piec... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Found the smaller multiple of $2019$ of the form $abcabc\dots abc$, where $a,b$ and $c$ are digits. | To find the smallest multiple of $2019$ of the form $abcabc\ldots abc$, where $a$, $b$, and $c$ are digits, we can follow these steps:
1. **Factorize 2019**:
\[
2019 = 673 \times 3
\]
This tells us that any multiple of $2019$ must be divisible by both $673$ and $3$.
2. **Divisibility by 3**:
For a numb... | 673673673 | Number Theory | other | Yes | Yes | aops_forum | false |
Let us have $6050$ points in the plane, no three collinear. Find the maximum number $k$ of non-overlapping triangles without common vertices in this plane. | 1. **Determine the maximum number of non-overlapping triangles:**
Given 6050 points in the plane, no three of which are collinear, we need to find the maximum number \( k \) of non-overlapping triangles without common vertices.
Since each triangle requires 3 vertices, the maximum number of triangles we can for... | 2016 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ x,y\in\mathbb{R}$ , and $ x,y \in $ $ \left(0,\frac{\pi}{2}\right) $, and $ m \in \left(2,+\infty\right) $ such that $ \tan x * \tan y = m $ . Find the minimum value of the expression $ E(x,y) = \cos x + \cos y $. | Given \( x, y \in \left(0, \frac{\pi}{2}\right) \) and \( m \in \left(2, +\infty\right) \) such that \( \tan x \cdot \tan y = m \), we need to find the minimum value of the expression \( E(x, y) = \cos x + \cos y \).
1. **Expressing the given condition in terms of trigonometric identities:**
\[
\tan x \cdot \tan... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $(a_n)_{n\geq1}$ is defined as:
$$a_1=2, a_2=20, a_3=56, a_{n+3}=7a_{n+2}-11a_{n+1}+5a_n-3\cdot2^n.$$ Prove that $a_n$ is positive for every positive integer $n{}$. Find the remainder of the divison of $a_{673}$ to $673$. | 1. **Define the sequence transformation:**
Let \( a_n = b_n + 2^n \). We need to find the recurrence relation for \( b_n \).
2. **Derive the recurrence relation for \( b_n \):**
Given the original recurrence relation:
\[
a_{n+3} = 7a_{n+2} - 11a_{n+1} + 5a_n - 3 \cdot 2^n,
\]
substitute \( a_n = b_n ... | 663 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Show that the expression $(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1)$, where $a =\sqrt{1 + x^2}$, $b =\sqrt{1 + y^2}$ and $x + y = 1$ is constant ¸and be calculated that constant value.
| 1. Given the expression \((a + b + 1)(a + b - 1)(a - b + 1)(-a + b + 1)\), where \(a = \sqrt{1 + x^2}\), \(b = \sqrt{1 + y^2}\), and \(x + y = 1\), we need to show that this expression is constant and calculate its value.
2. First, let's express \(a\) and \(b\) in terms of \(x\) and \(y\):
\[
a = \sqrt{1 + x^2},... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions:
1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$
2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$
3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$ | To find the smallest natural number \( n \) such that there exist real numbers \( x_1, x_2, \ldots, x_n \) satisfying the given conditions, we will analyze the inequalities step by step.
1. **Given Conditions:**
- \( x_i \in \left[\frac{1}{2}, 2\right] \) for \( i = 1, 2, \ldots, n \)
- \( x_1 + x_2 + \cdots + x... | 9 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest natural number $n =>2$ with the property:
For every positive integers $a_1, a_2,. . . , a_n$ the product of all differences $a_j-a_i$,
$1 <=i <j <=n$, is divisible by 2001. | To determine the smallest natural number \( n \geq 2 \) such that for every set of positive integers \( a_1, a_2, \ldots, a_n \), the product of all differences \( a_j - a_i \) (for \( 1 \leq i < j \leq n \)) is divisible by 2001, we need to consider the prime factorization of 2001 and the properties of differences.
1... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$. | 1. The sequence is constructed from powers of 5 and sums of distinct powers of 5, written in ascending order. Let's denote the sequence as \( a_n \).
2. We start by listing the first few terms:
\[
\begin{aligned}
&5^0 = 1, \\
&5^1 = 5, \\
&5^1 + 5^0 = 6, \\
&5^2 = 25, \\
&5^2 + 5^0 = 26, \\
&5^... | 81281 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let the number $x$. Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example, $x\cdot x=x^{2}$, $x^{2}\cdot x^{2}=x^{4}$, $x^{4}: x=x^{3}$, etc). Determine the minimal number of operations needed for calculating $x^{2006}$. | To determine the minimal number of operations needed to calculate \( x^{2006} \), we can use the binary representation of the exponent 2006. The binary representation of 2006 is \( (11111010110)_2 \).
1. **Binary Representation**:
\[
2006_{10} = 11111010110_2
\]
This means:
\[
2006 = 2^{10} + 2^9 + 2... | 17 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$. | To determine the smallest natural number written in the decimal system with the product of the digits equal to \(10! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot 9 \cdot 10\), we need to factorize \(10!\) and then find the smallest number that can be formed using these factors as digits.
1. **Calculate \(10!\):**
\[
1... | 45578899 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
a) Calculate the product $$\left(1+\frac{1}{2}\right) \left(1+\frac{1}{3}\right) \left(1+\frac{1}{4}\right)... \left(1+\frac{1}{2006}\right) \left(1+\frac{1}{2007}\right)$$
b) Let the set $$A =\left\{\frac{1}{2}, \frac{1}{3},\frac{1}{4}, ...,\frac{1}{2006}, \frac{1}{2007}\right\}$$
Determine the sum of all produ... | ### Part (a)
1. Consider the product:
\[
\left(1 + \frac{1}{2}\right) \left(1 + \frac{1}{3}\right) \left(1 + \frac{1}{4}\right) \cdots \left(1 + \frac{1}{2006}\right) \left(1 + \frac{1}{2007}\right)
\]
2. Each term in the product can be rewritten as:
\[
1 + \frac{1}{k} = \frac{k+1}{k}
\]
Therefore... | 1003 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The positive real numbers $x$ and $y$ satisfy the relation $x + y = 3 \sqrt{xy}$. Find the value of the numerical expression $$E=\left| \frac{x-y}{x+y}+\frac{x^2-y^2}{x^2+y^2}+\frac{x^3-y^3}{x^3+y^3}\right|.$$
| 1. Given the relation \( x + y = 3 \sqrt{xy} \), we start by squaring both sides to eliminate the square root:
\[
(x + y)^2 = (3 \sqrt{xy})^2
\]
\[
x^2 + 2xy + y^2 = 9xy
\]
\[
x^2 + y^2 + 2xy = 9xy
\]
\[
x^2 + y^2 = 7xy
\]
2. Next, we need to express \( E \) in terms of \( x \) and ... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The natural numbers $m$ and $k$ satisfy the equality $$1001 \cdot 1002 \cdot ... \cdot 2010 \cdot 2011 = 2^m (2k + 1)$$. Find the number $m$. | 1. We start with the given product:
\[
1001 \cdot 1002 \cdot \ldots \cdot 2010 \cdot 2011
\]
This product can be expressed as:
\[
\frac{2011!}{1000!}
\]
2. To find the power of 2 in the prime factorization of this product, we use Legendre's formula. Legendre's formula for the exponent of a prime \... | 1008 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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