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Let $A\subset \{1,2,\ldots,4014\}$, $|A|=2007$, such that $a$ does not divide $b$ for all distinct elements $a,b\in A$. For a set $X$ as above let us denote with $m_{X}$ the smallest element in $X$. Find $\min m_{A}$ (for all $A$ with the above properties). | 1. **Partitioning the Set**:
We start by partitioning the set \(\{1, 2, \ldots, 4014\}\) into 2007 parts \(P_1, P_2, \ldots, P_{2007}\) such that \(P_a\) contains all numbers of the form \(2^b(2a-1)\) where \(b\) is a nonnegative integer. This ensures that no two elements in the same part can divide each other, as o... | 128 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \] | To find the greatest positive integer \( k \) such that the inequality holds for all \( a, b, c \in \mathbb{R}^+ \) satisfying \( abc = 1 \):
\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{k}{a+b+c+1} \geq 3 + \frac{k}{4}
\]
we start by substituting \( a = t, b = t, c = \frac{1}{t^2} \) for \( t \neq 1 \). This s... | 13 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
In the Cartesian plane is given a set of points with integer coordinate \[ T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\} \] We colour some points of $ T $ such that for each point $ (x;y)\in T $ then either $ (x;y) $ or $ (-x;-y) $ is coloured. Denote $ N $ to be the number of couples $ {(x_1;... | 1. **Understanding the Set \( T \)**:
The set \( T \) consists of all points \((x, y)\) where \(x\) and \(y\) are integers, \(|x| \leq 20\), \(|y| \leq 20\), and \((x, y) \neq (0, 0)\). This gives us a total of \(41 \times 41 - 1 = 1680\) points.
2. **Coloring Condition**:
For each point \((x, y) \in T\), either... | 420 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ be the positive root of the equation $x^2+x=5$. Let $n$ be a positive integer number, and let $c_0,c_1,\ldots,c_n\in \mathbb{N}$ be such that $ c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n=2015. $
a. Prove that $c_0+c_1+c_2+\cdots+c_n\equiv 2 \pmod{3}$.
b. Find the minimum value of the sum $c_0+c_1+c_2+\cd... | ### Part (a)
1. **Identify the positive root of the equation \(x^2 + x = 5\):**
\[
x^2 + x - 5 = 0
\]
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 1\), and \(c = -5\):
\[
x = \frac{-1 \pm \sqrt{1 + 20}}{2} = \frac{-1 \pm \sqrt{21}}{2}
\]
The posi... | 29 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive interger number $n$ such that there exists $n$ real numbers $a_1,a_2,\ldots,a_n$ satisfied three conditions as follow:
a. $a_1+a_2+\cdots+a_n>0$;
b. $a_1^3+a_2^3+\cdots+a_n^3<0$;
c. $a_1^5+a_2^5+\cdots+a_n^5>0$. | To find the smallest positive integer \( n \) such that there exist \( n \) real numbers \( a_1, a_2, \ldots, a_n \) satisfying the conditions:
a. \( a_1 + a_2 + \cdots + a_n > 0 \)
b. \( a_1^3 + a_2^3 + \cdots + a_n^3 < 0 \)
c. \( a_1^5 + a_2^5 + \cdots + a_n^5 > 0 \)
we will analyze the cases for \( n = 1, 2, 3, 4 \... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be a set contains $2000$ distinct integers and $B$ be a set contains $2016$ distinct integers. $K$ is the numbers of pairs $(m,n)$ satisfying \[ \begin{cases} m\in A, n\in B\\ |m-n|\leq 1000 \end{cases} \] Find the maximum value of $K$ | 1. Let \( A = \{A_1, A_2, \ldots, A_{2000}\} \) and \( B = \{B_1, B_2, \ldots, B_{2016}\} \) be the sets of distinct integers, where \( A_1 < A_2 < \cdots < A_{2000} \) and \( B_1 < B_2 < \cdots < B_{2016} \).
2. We need to find the maximum number of pairs \((m, n)\) such that \( m \in A \), \( n \in B \), and \(|m - ... | 3015636 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the movie ”Prison break $4$”. Michael Scofield has to break into The Company. There, he encountered a kind of code to protect Scylla from being taken away. This code require picking out every number in a $2015\times 2015$ grid satisfying:
i) The number right above of this number is $\equiv 1 \mod 2$
ii) The number ... | To solve this problem, we need to identify the numbers in a \(2015 \times 2015\) grid that satisfy the given modular conditions. Let's break down the problem step by step.
1. **Understanding the Grid Layout:**
- The grid is filled with numbers from \(1\) to \(2015^2\).
- The first row is filled with numbers in a... | 270681 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Students in a school are arranged in an order that when you count from left to right, there will be $n$ students in the first row, $n-1$ students in the second row, $n - 2$ students in the third row,... until there is one student in the $n$th row. All the students face to the first row. For example, here is an arrang... | 1. **Understanding the Problem:**
- We have \( n \) rows of students.
- The first row has \( n \) students, the second row has \( n-1 \) students, and so on until the \( n \)-th row which has 1 student.
- Each student (except the first in each row) can make one of two statements:
1. The student in front o... | 2031120 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd n... | 1. **Initial Analysis**:
- Anna and Berta take turns removing marbles.
- Anna starts first.
- The player can remove \( k \) marbles where \( k \) is either:
- An even number with \( k \leq \frac{n}{2} \), or
- An odd number with \( \frac{n}{2} \leq k \leq n \).
- The player who removes the last ma... | 131070 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $x,y,z$ are chosen such that $$\frac{1}{|x^2+2yz|} ,\frac{1}{|y^2+2zx|} ,\frac{1}{|x^2+2xy|} $$
are lengths of a non-degenerate triangle .
Find all possible values of $xy+yz+zx$ .
[i]Proposed by Michael Rolínek[/i] | To solve the problem, we need to find all possible values of \( xy + yz + zx \) such that the given expressions form the sides of a non-degenerate triangle. The expressions are:
\[
\frac{1}{|x^2 + 2yz|}, \quad \frac{1}{|y^2 + 2zx|}, \quad \frac{1}{|z^2 + 2xy|}
\]
For these to be the sides of a non-degenerate triangle... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
There are $2022$ numbers arranged in a circle $a_1, a_2, . . ,a_{2022}$. It turned out that for any three consecutive $a_i$, $a_{i+1}$, $a_{i+2}$ the equality $a_i =\sqrt2 a_{i+2} - \sqrt3 a_{i+1}$. Prove that $\sum^{2022}_{i=1} a_ia_{i+2} = 0$, if we know that $a_{2023} = a_1$, $a_{2024} = a_2$. | 1. Given the equation for any three consecutive terms \(a_i, a_{i+1}, a_{i+2}\):
\[
a_i = \sqrt{2} a_{i+2} - \sqrt{3} a_{i+1}
\]
We can rewrite this equation as:
\[
a_i + \sqrt{3} a_{i+1} = \sqrt{2} a_{i+2}
\]
Squaring both sides, we get:
\[
(a_i + \sqrt{3} a_{i+1})^2 = (\sqrt{2} a_{i+2})^... | 0 | Algebra | proof | Yes | Yes | aops_forum | false |
In equality
$$1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023$$
Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used? | 1. **Initial Consideration**:
- We need to find the smallest number of multiplication signs (denoted as "•") to make the equation \(1 * 2 * 3 * 4 * 5 * \ldots * 60 * 61 * 62 = 2023\) true.
- If we use only addition and subtraction, the maximum sum is \(1 + 2 + 3 + \ldots + 62\), which is less than 2023.
2. **Sum... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many of the fractions $ \frac{1}{2023}, \frac{2}{2023}, \frac{3}{2023}, \cdots, \frac{2022}{2023} $ simplify to a fraction whose denominator is prime?
$\textbf{(A) } 20 \qquad \textbf{(B) } 22 \qquad \textbf{(C) } 23 \qquad \textbf{(D) } 24 \qquad \textbf{(E) } \text{none of the above}$ | To determine how many of the fractions $\frac{1}{2023}, \frac{2}{2023}, \frac{3}{2023}, \cdots, \frac{2022}{2023}$ simplify to a fraction whose denominator is prime, we need to analyze the prime factorization of 2023.
1. **Prime Factorization of 2023:**
\[
2023 = 7 \times 17^2
\]
This means that the prime ... | 22 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Hexagon $ ABCDEF, $ as pictured [in the attachment], is such that $ ABDE $ is a square with side length 20, $ \overline{AB}, \overline{CF} $, and $ \overline{DE} $ are all parallel, and $ BC = CD = EF = FA = 23 $. What is the largest integer not exceeding the length of $ \overline{CF} $?
$\textbf{(A) } 41 \qquad \text... | 1. **Identify the given information and the structure of the hexagon:**
- Hexagon \( ABCDEF \) has \( ABDE \) as a square with side length 20.
- \( \overline{AB}, \overline{CF}, \) and \( \overline{DE} \) are all parallel.
- \( BC = CD = EF = FA = 23 \).
2. **Determine the coordinates of the vertices:**
- ... | 28 | Geometry | MCQ | Yes | Yes | aops_forum | false |
We have $ 23^2 = 529 $ ordered pairs $ (x, y) $ with $ x $ and $ y $ positive integers from 1 to 23, inclusive. How many of them have the property that $ x^2 + y^2 + x + y $ is a multiple of 6?
$\textbf{(A) } 225 \qquad \textbf{(B) } 272 \qquad \textbf{(C) } 324 \qquad \textbf{(D) } 425 \qquad \textbf{(E) } \text{none... | To solve the problem, we need to determine how many ordered pairs \((x, y)\) with \(x\) and \(y\) being positive integers from 1 to 23 satisfy the condition that \(x^2 + y^2 + x + y\) is a multiple of 6.
1. **Rewrite the expression**:
\[
P = x^2 + y^2 + x + y = x(x+1) + y(y+1)
\]
Notice that \(x(x+1)\) and... | 225 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A positive real number $ A $ rounds to 20, and another positive real number $ B $ rounds to 23. What is the largest possible value of the largest integer not exceeding the value of $ \frac{100A}{B}? $
$\textbf{(A) } 91 \qquad \textbf{(B) } 89 \qquad \textbf{(C) } 88 \qquad \textbf{(D) } 87 \qquad \textbf{(E) } \text{n... | 1. **Identify the range of values for \( A \) and \( B \):**
- Since \( A \) rounds to 20, \( A \) must be in the interval \( [19.5, 20.5) \).
- Since \( B \) rounds to 23, \( B \) must be in the interval \( [22.5, 23.5) \).
2. **Determine the maximum value of \( \frac{100A}{B} \):**
- To maximize \( \frac{10... | 91 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
For any positive integer $a$, let $\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest possible value of $4\tau(n)-n$ over all positive integers $n$. | 1. **Understanding the function $\tau(n)$**:
- The function $\tau(n)$ represents the number of positive divisors of $n$. For example, if $n = 12$, then the divisors are $1, 2, 3, 4, 6, 12$, so $\tau(12) = 6$.
2. **Establishing an upper bound for $\tau(n)$**:
- We need to find an upper bound for $\tau(n)$. Note t... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 13, BC = 14, $and$ CA = 15$. Suppose $PQRS$ is a square such that $P$ and $R$ lie on line $BC, Q$ lies on line $CA$, and $S$ lies on line $AB$. Compute the side length of this square. | 1. **Identify the coordinates and projections:**
- Let $A = (0, 0)$, $B = (13, 0)$, and $C = (x, y)$.
- Given $AB = 13$, $BC = 14$, and $CA = 15$, we can use the distance formula to find $x$ and $y$.
- The projection $D$ of $A$ onto $BC$ lies on $BC$ and is perpendicular to $BC$.
2. **Use the distance formula... | 42 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $ABCD$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $ABE$ is $10\pi$ and the perimeter of triangle $ADE$ is $n$. Compute the number of possible integer values of $n$. | 1. **Understanding the Problem:**
- We are given a rectangle \(ABCD\) with diagonals intersecting at \(E\).
- The perimeter of triangle \(ABE\) is \(10\pi\).
- We need to find the number of possible integer values for the perimeter of triangle \(ADE\), denoted as \(n\).
2. **Setting Up the Equations:**
- L... | 47 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Richard starts with the string $HHMMMMTT$. A move consists of replacing an instance of $HM$ with $MH$, replacing an instance of $MT$ with $TM$, or replacing an instance of $TH$ with $HT$. Compute the number of possible strings he can end up with after performing zero or more moves. | 1. **Initial String and Moves**: Richard starts with the string \( HHMMMMTT \). The allowed moves are:
- Replace \( HM \) with \( MH \)
- Replace \( MT \) with \( TM \)
- Replace \( TH \) with \( HT \)
2. **Observing the Moves**: Notice that the moves do not change the relative order of \( H \) and \( T \). S... | 70 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Elbert and Yaiza each draw $10$ cards from a $20$-card deck with cards numbered $1,2,3,\dots,20$. Then, starting with the player with the card numbered $1$, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, ... | 1. **Identify the cards placed and their order:**
Let the cards placed be \(1, a, b, c, d\) in that order. This means that Elbert starts with the card numbered \(1\), and then Yaiza places \(a\), Elbert places \(b\), Yaiza places \(c\), and Elbert places \(d\).
2. **Determine the cards in Yaiza's hand:**
Since Y... | 240 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $E$, $I$, $L$, $V$ are (not necessarily distinct) nonzero digits in base ten for which
[list]
[*] the four-digit number $\underline{E}\ \underline{V}\ \underline{I}\ \underline{L}$ is divisible by $73$, and
[*] the four-digit number $\underline{V}\ \underline{I}\ \underline{L}\ \underline{E}$ is divisible by $... | To solve the problem, we need to find the four-digit number \(\underline{L}\ \underline{I}\ \underline{V}\ \underline{E}\) given the conditions:
1. The four-digit number \(\underline{E}\ \underline{V}\ \underline{I}\ \underline{L}\) is divisible by 73.
2. The four-digit number \(\underline{V}\ \underline{I}\ \underlin... | 5499 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define a common chord between two intersecting circles to be the line segment connecting their two intersection points. Let $\omega_1,\omega_2,\omega_3$ be three circles of radii $3, 5,$ and $7$, respectively. Suppose they are arranged in such a way that the common chord of $\omega_1$ and $\omega_2$ is a diameter of $\... | 1. **Identify the centers and radii of the circles:**
- Let the centers of the circles $\omega_1, \omega_2, \omega_3$ be $O_1, O_2, O_3$ respectively.
- The radii of the circles are $r_1 = 3$, $r_2 = 5$, and $r_3 = 7$.
2. **Determine the distances between the centers:**
- The common chord of $\omega_1$ and $\... | 275 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
5. Let $S$ denote the set of all positive integers whose prime factors are elements of $\{2,3,5,7,11\}$. (We include 1 in the set $S$.) If
$$
\sum_{q \in S} \frac{\varphi(q)}{q^{2}}
$$
can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, find $a+b$. (Here $\varphi$ denotes Euler's totie... | 1. **Understanding the Problem:**
We need to find the sum \(\sum_{q \in S} \frac{\varphi(q)}{q^2}\), where \(S\) is the set of all positive integers whose prime factors are elements of \(\{2, 3, 5, 7, 11\}\), and \(\varphi\) is Euler's totient function. We need to express this sum as a fraction \(\frac{a}{b}\) in si... | 1537 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positiv... | 1. **Identify the polynomial and its coefficients:**
Given the polynomial \( f(x) = x^3 + px^2 + qx + r \) with roots \( a, b, c \), we can express the coefficients in terms of the roots using Vieta's formulas:
\[
p = -(a + b + c), \quad q = ab + bc + ca, \quad r = -abc
\]
2. **Sum of the squares of the co... | 551 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are. | 1. Given the quartic polynomial \( P x^{4}+U x^{3}+M x^{2}+A x+C \) with four different positive real roots, we need to find the square of the smallest real number \( z \) for which the expression \( M^{2}-2 U A+z P C \) is always positive.
2. Since \( P \neq 0 \) (as the polynomial has four roots), we can assume with... | 16 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
10. The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form
$$
\frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)}
$$
where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$. Determin... | 1. We start with the sum \(\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)\). To simplify this, we use the complex exponential form of the cosine function: \(\cos x = \frac{e^{ix} + e^{-ix}}{2}\).
2. Let \(z = e^{i \frac{4 \pi}{4041}}\). Then, \(\cos \left(\frac{4 k \pi}{4041}\right) = \frac{z^k + z^{-k}}{2... | 4049 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
11. Let $f(z)=\frac{a z+b}{c z+d}$ for $a, b, c, d \in \mathbb{C}$. Suppose that $f(1)=i, f(2)=i^{2}$, and $f(3)=i^{3}$. If the real part of $f(4)$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m^{2}+n^{2}$. | 1. Given the function \( f(z) = \frac{az + b}{cz + d} \) and the conditions \( f(1) = i \), \( f(2) = i^2 \), and \( f(3) = i^3 \), we need to find the values of \( a, b, c, \) and \( d \).
2. Substitute \( z = 1 \) into \( f(z) \):
\[
f(1) = \frac{a \cdot 1 + b}{c \cdot 1 + d} = i \implies \frac{a + b}{c + d} =... | 34 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
12. What is the sum of all possible $\left(\begin{array}{l}i \\ j\end{array}\right)$ subject to the restrictions that $i \geq 10, j \geq 0$, and $i+j \leq 20$ ? Count different $i, j$ that yield the same value separately - for example, count both $\left(\begin{array}{c}10 \\ 1\end{array}\right)$ and $\left(\begin{array... | 1. **Identify the constraints and the sum to be calculated:**
We need to find the sum of all possible pairs \((i, j)\) such that \(i \geq 10\), \(j \geq 0\), and \(i + j \leq 20\). Each pair \((i, j)\) is counted separately.
2. **Set up the double summation:**
The sum can be expressed as:
\[
\sum_{j=0}^{10... | 27633 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \mid a_{2023}$ ? | To solve the problem, we need to find the largest integer \( m \) such that \( 3^m \) divides \( a_{2023} \). We start by understanding the recurrence relation for \( a_n \), the number of ternary strings of length \( n \) that do not have any \( k < n \) such that the first \( k \) digits equal the last \( k \) digits... | 2022 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square, and let $l$ be a line passing through the midpoint of segment $AB$ that intersects segment $BC$. Given that the distances from $A$ and $C$ to $l$ are $4$ and $7$, respectively, compute the area of $ABCD$. | 1. **Identify the midpoint and distances:**
Let $M$ be the midpoint of segment $AB$. Since $M$ is the midpoint, the distance from $B$ to $l$ is also $4$ by symmetry. Let $l \cap BC = K$. Given the distances from $A$ and $C$ to $l$ are $4$ and $7$ respectively, we can use these distances to find the area of the squar... | 256 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Omega$ and $\omega$ be circles with radii $123$ and $61$, respectively, such that the center of $\Omega$ lies on $\omega$. A chord of $\Omega$ is cut by $\omega$ into three segments, whose lengths are in the ratio $1 : 2 : 3$ in that order. Given that this chord is not a diameter of $\Omega$, compute the length o... | 1. **Define the centers and radii:**
Let the center of the larger circle \(\Omega\) be \(O\) with radius \(R = 123\), and the center of the smaller circle \(\omega\) be \(O'\) with radius \(r = 61\). Since \(O\) lies on \(\omega\), the distance \(OO'\) is equal to \(r = 61\).
2. **Define the chord and segment lengt... | 42 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A two-digit integer $\underline{a}\,\, \underline{b}$ is multiplied by $9$. The resulting three-digit integer is of the form $\underline{a} \,\,\underline{c} \,\,\underline{b}$ for some digit $c$. Evaluate the sum of all possible $\underline{a} \,\, \underline{b}$. | 1. Let the two-digit integer be represented as \( \underline{a}\, \underline{b} \), where \( a \) and \( b \) are the tens and units digits, respectively. This can be expressed as \( 10a + b \).
2. When this number is multiplied by 9, the resulting number is \( 9(10a + b) \).
3. According to the problem, the resultin... | 120 | Number Theory | other | Yes | Yes | aops_forum | false |
Five boys and six girls are to be seated in a row of eleven chairs so that they sit one at a time from one end to the other. The probability that there are no more boys than girls seated at any point during the process is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Evaluate $m + n$. | 1. **Understanding the Problem:**
We need to find the probability that there are no more boys than girls seated at any point during the process of seating 5 boys and 6 girls in a row of 11 chairs. This can be visualized as a path on a grid from \((0,0)\) to \((6,5)\) without crossing the line \(y = x\).
2. **Catala... | 9 | Combinatorics | other | Yes | Yes | aops_forum | false |
Point $P$ is situated inside regular hexagon $ABCDEF$ such that the feet from $P$ to $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$ respectively are $G$, $H$, $I$, $J$, $K$, and $L$. Given that $PG = \frac92$ , $PI = 6$, and $PK =\frac{15}{2}$ , the area of hexagon $GHIJKL$ can be written as $\frac{a\sqrt{b}}{c}$ for positive ... | 1. **Understanding the Problem:**
We are given a regular hexagon \(ABCDEF\) with a point \(P\) inside it. The perpendicular distances from \(P\) to the sides \(AB\), \(BC\), \(CD\), \(DE\), \(EF\), and \(FA\) are given as \(PG = \frac{9}{2}\), \(PI = 6\), and \(PK = \frac{15}{2}\). We need to find the area of the he... | 736 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
One face of a tetrahedron has sides of length $3$, $4$, and $5$. The tetrahedron’s volume is $24$ and surface area is $n$. When $n$ is minimized, it can be expressed in the form $n = a\sqrt{b} + c$, where $a$, $b$, and $c$ are positive integers and b is not divisible by the square of any prime. Evaluate $a + b + c$. | 1. **Identify the given information and set up the problem:**
- One face of the tetrahedron is a triangle with sides \(AB = 3\), \(BC = 4\), and \(AC = 5\).
- The volume of the tetrahedron is \(24\).
- We need to find the minimum surface area \(n\) of the tetrahedron, expressed in the form \(n = a\sqrt{b} + c\... | 157 | Geometry | other | Yes | Yes | aops_forum | false |
Triangle $ABC$ satisfies $\tan A \cdot \tan B = 3$ and $AB = 5$. Let $G$ and $O$ be the centroid and circumcenter of $ABC$ respectively. The maximum possible area of triangle $CGO$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $a$ and $c$ relatively prime and $b$ not divisible by ... | 1. **Define the problem and given conditions:**
- We are given a triangle \(ABC\) with \(\tan A \cdot \tan B = 3\) and \(AB = 5\).
- We need to find the maximum possible area of triangle \(CGO\), where \(G\) is the centroid and \(O\) is the circumcenter of \(ABC\).
2. **Establish the relationship between angles ... | 100 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Adam and Bettie are playing a game. They take turns generating a random number between $0$ and $127$ inclusive. The numbers they generate are scored as follows:
$\bullet$ If the number is zero, it receives no points.
$\bullet$ If the number is odd, it receives one more point than the number one less than it.
$\bullet$ ... | 1. **Understanding the scoring system**:
- If the number is zero, it receives no points.
- If the number is odd, it receives one more point than the number one less than it.
- If the number is even, it receives the same score as the number with half its value.
2. **Binary representation and scoring**:
- Ea... | 429 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The function y$ = x^2$ is graphed in the $xy$-plane. A line from every point on the parabola is drawn to the point $(0,-10, a)$ in three-dimensional space. The locus of points where the lines intersect the $xz$-plane forms a closed path with area $\pi$. Given that $a = \frac{p\sqrt{q}}{r}$ for positive integers $p$, $q... | 1. **Identify the given points and the equation of the parabola:**
- The point \( A \) is given as \( (0, -10, a) \).
- The parabola is given by \( y = x^2 \).
2. **Parameterize a point \( P \) on the parabola:**
- Let \( P \) be a point on the parabola, parameterized as \( (\lambda, \lambda^2, 0) \).
3. **F... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The value of $x$ which satisfies
$$1 +\log_x \left( \lfloor x \rfloor \right) = 2 \log_x \left(\sqrt3\{x\}\right)$$
can be written in the form $\frac{a+\sqrt{b}}{c}$ , where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a + b + c$.
Note: $\lfloor x \rfloor$ denotes... | 1. Given the equation:
\[
1 + \log_x \left( \lfloor x \rfloor \right) = 2 \log_x \left(\sqrt{3}\{x\}\right)
\]
we need to find the value of \( x \) that satisfies this equation.
2. For the logarithms to be defined, we need \( x > 1 \). Let's take \( x \) to the power of both sides to simplify the equation:... | 26 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For a sequence $s = (s_1, s_2, . . . , s_n)$, define
$$F(s) =\sum^{n-1}_{i=1} (-1)^{i+1}(s_i - s_{i+1})^2.$$
Consider the sequence $S =\left(2^1, 2^2, . . . , 2^{1000}\right)$. Let $R$ be the sum of all $F(m)$ for all non-empty subsequences $m$ of $S$. Find the remainder when $R$ is divided by $1000$.
Note: A subseque... | 1. **Rewrite the total sum over all subsequences of \( S \):**
\[
R = \sum_{m \subseteq S} F(m) = \sum_{m \subseteq S} \sum_{i=1}^{|m|-1} (-1)^{i+1} (2^i - 2^{i+1})^2
\]
This can be further simplified to:
\[
R = \sum_{1 \leq i < j \leq 1000} (p_{i,j} - n_{i,j})(2^i - 2^j)^2
\]
where \( p_{i,j} \... | 500 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $b > 1$ is a real number where $\log_5 (\log_5 b + \log_b 125) = 2$. Find $log_5 \left(b^{\log_5 b}\right) + log_b \left(125^{\log_b 125}\right).$ | 1. Given the equation:
\[
\log_5 (\log_5 b + \log_b 125) = 2
\]
We can rewrite this as:
\[
\log_5 (\log_5 b + \log_b 125) = 2 \implies \log_5 b + \log_b 125 = 5^2 = 25
\]
2. We need to find:
\[
\log_5 \left(b^{\log_5 b}\right) + \log_b \left(125^{\log_b 125}\right)
\]
3. Notice that:
... | 619 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Call a positive integer $x$ with non-zero digits [i]fruity [/i] if it satisfies $E(x) = 24$ where $E(x)$ is the number of trailing zeros in the product of the digits of $x$ defined over the positive integers. Determine the remainder when the $30$th smallest fruity number is divided by $1000$. (Trailing zeros are consec... | 1. **Understanding the Problem**:
- We need to find the 30th smallest positive integer \( x \) such that \( E(x) = 24 \), where \( E(x) \) is the number of trailing zeros in the product of the digits of \( x \).
- Trailing zeros in a number are produced by factors of 10, which are the product of 2 and 5. Therefor... | 885 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\omega_1$ be a circle of radius $1$ that is internally tangent to a circle $\omega_2$ of radius $2$ at point $A$. Suppose $\overline{AB}$ is a chord of $\omega_2$ with length $2\sqrt3$ that intersects $\omega_1$ at point $C\ne A$. If the tangent line of $\omega_1$ at $C$ intersects $\omega_2$ at points $D$ and $E$... | 1. **Define the centers and radii:**
Let \( O_1 \) be the center of the circle \( \omega_1 \) with radius 1, and \( O_2 \) be the center of the circle \( \omega_2 \) with radius 2. The circles are internally tangent at point \( A \).
2. **Use homothety:**
Since \( \omega_1 \) and \( \omega_2 \) are internally ta... | 63 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an isosceles triangle with $AB = AC = 4$ and $BC = 5$. Two circles centered at $B$ and $C$ each have radius $2$, and the line through the midpoint of $\overline{BC}$ perpendicular to $\overline{AC}$ intersects the two circles in four different points. If the greatest possible distance between any two of th... | 1. **Identify the coordinates of points \( B \) and \( C \):**
- Since \( \triangle ABC \) is isosceles with \( AB = AC = 4 \) and \( BC = 5 \), we can place \( B \) and \( C \) symmetrically about the y-axis.
- Let \( B = (-\frac{5}{2}, 0) \) and \( C = (\frac{5}{2}, 0) \).
2. **Find the coordinates of point \(... | 451 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $f(n)$ denotes the number of divisors of $2024^{2024}$ that are either less than $n$ or share at least one prime factor with $n$, find the remainder when $$\sum^{2024^{2024}}_{n=1} f(n)$$ is divided by $1000$. | 1. **Define the functions and the problem**:
Let \( f(n) \) denote the number of divisors of \( 2024^{2024} \) that are either less than \( n \) or share at least one prime factor with \( n \). We need to find the remainder when
\[
\sum_{n=1}^{2024^{2024}} f(n)
\]
is divided by 1000.
2. **Understanding... | 224 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Stars can fall in one of seven stellar classifications. The constellation Leo contains $9$ stars and $10$ line segments, as shown in the diagram, with stars connected by line segments having distinct stellar classifications. Let $n$ be the number of valid stellar classifications of the $9$ stars. Compute the number of ... | 1. **Understanding the problem**: We need to find the number of valid stellar classifications for the 9 stars in the constellation Leo, where each star has a distinct classification from its connected stars. Then, we need to compute the number of positive integer divisors of this number.
2. **Analyzing the pentagon su... | 160 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $S_n$ is the set of positive divisors of $n$, and denote $|X|$ as the number of elements in a set $X$. Let $\xi$ be the set of positive integers $n$ where $|S_n| = 2m$ is even, and $S_n$ can be partitioned evenly into pairs $\{a_i, b_i\}$ for integers $1 \le i \le m$ such that the following conditions hold:
$\b... | 1. **Understanding the Problem:**
- We need to find the number of positive divisors \( d \) of \( 24! \) such that \( d \in \xi \).
- The set \( \xi \) contains positive integers \( n \) where:
- \( |S_n| = 2m \) is even.
- \( S_n \) can be partitioned into pairs \(\{a_i, b_i\}\) such that \( a_i \) and... | 64 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Here I shall collect for the sake of collecting in separate threads the geometry problems those posting in multi-problems threads inside aops. I shall create post collections from these threads also. Geometry from USA contests are collected [url=https://artofproblemsolving.com/community/c2746635_geometry_from_usa_conte... | **
1. **Understanding the Geometry of the Hexagon:**
A regular hexagon can be divided into 6 equilateral triangles. Each side of the hexagon is of length 1.
2. **Identifying the Points:**
The largest distance between any two points in a regular hexagon is the distance between two opposite vertices. This is beca... | 2 | Geometry | other | Yes | Yes | aops_forum | false |
You are playing a game called "Hovse."
Initially you have the number $0$ on a blackboard.
If at any moment the number $x$ is written on the board, you can either:
$\bullet$ replace $x$ with $3x + 1$
$\bullet$ replace $x$ with $9x + 1$
$\bullet$ replace $x$ with $27x + 3$
$\bullet$ or replace $x$ with $\left \lfloor \f... | 1. **Understanding the Problem:**
- We start with the number \(0\) on the blackboard.
- We can perform one of the following operations on a number \(x\):
- Replace \(x\) with \(3x + 1\)
- Replace \(x\) with \(9x + 1\)
- Replace \(x\) with \(27x + 3\)
- Replace \(x\) with \(\left\lfloor \frac{x... | 127 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Calculate the sum of matrix commutators $[A, [B, C]] + [B, [C, A]] + [C, [A, B]]$, where $[A, B] = AB-BA$ | 1. **Express the commutators in terms of matrix products:**
\[
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = [A, (BC - CB)] + [B, (CA - AC)] + [C, (AB - BA)]
\]
2. **Expand each commutator:**
\[
[A, (BC - CB)] = A(BC - CB) - (BC - CB)A = ABC - ACB - BCA + CBA
\]
\[
[B, (CA - AC)] = B(CA - AC) - (CA... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given two vectors $v = (v_1,\dots,v_n)$ and $w = (w_1\dots,w_n)$ in $\mathbb{R}^n$, lets define $v*w$ as the matrix in which the element of row $i$ and column $j$ is $v_iw_j$. Supose that $v$ and $w$ are linearly independent. Find the rank of the matrix $v*w - w*v.$
| 1. **Define the matrix \( A = v * w - w * v \):**
Given two vectors \( v = (v_1, v_2, \ldots, v_n) \) and \( w = (w_1, w_2, \ldots, w_n) \) in \( \mathbb{R}^n \), the matrix \( v * w \) is defined such that the element in the \( i \)-th row and \( j \)-th column is \( v_i w_j \). Similarly, the matrix \( w * v \) is... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For $n \geq 3$, let $(b_0, b_1,..., b_{n-1}) = (1, 1, 1, 0, ..., 0).$ Let $C_n =
(c_{i, j})$ the $n \times n$ matrix defined by $c_{i, j} = b _{(j -i) \mod n}$. Show
that $\det (C_n) = 3$ if $n$ is not a multiple of 3 and $\det (C_n) = 0$ if $n$
is a multiple of 3. | To show that $\det(C_n) = 3$ if $n$ is not a multiple of 3 and $\det(C_n) = 0$ if $n$ is a multiple of 3, we will analyze the structure of the matrix $C_n$ and use properties of determinants.
1. **Matrix Definition and Initial Setup:**
The matrix $C_n = (c_{i,j})$ is defined by $c_{i,j} = b_{(j-i) \mod n}$, where $... | 0 | Algebra | proof | Yes | Yes | aops_forum | false |
For each positive integer $n$ let $A_n$ be the $n \times n$ matrix such that its $a_{ij}$
entry is equal to ${i+j-2 \choose j-1}$ for all $1\leq i,j \leq n.$ Find the determinant of $A_n$.
| 1. We start by examining the given matrix \( A_n \) for small values of \( n \) to identify any patterns. For \( n = 2 \), \( n = 3 \), and \( n = 4 \), the matrices are:
\[
A_2 = \begin{bmatrix}
1 & 1 \\
1 & 2
\end{bmatrix}
\]
\[
A_3 = \begin{bmatrix}
1 & 1 & 1 \\
1 & 2 & 3 \\
1 & 3 & ... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given a $3 \times 3$ symmetric real matrix $A$, we define $f(A)$ as a $3 \times 3$ matrix with the same eigenvectors of $A$ such that if $A$ has eigenvalues $a$, $b$, $c$, then $f(A)$ has eigenvalues $b+c$, $c+a$, $a+b$ (in that order). We define a sequence of symmetric real $3\times3$ matrices $A_0, A_1, A_2, \ldots$ ... | 1. **Spectral Decomposition of \( A \)**:
Given a symmetric real matrix \( A \), we can use spectral decomposition to write:
\[
A = Q \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} Q^T = Q D Q^T
\]
where \( Q \) is an orthogonal matrix and \( D \) is a diagonal matrix with eigenvalues ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, $\sigma(n)$ denotes the sum of the positive divisors of $n$. Determine $$\limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}}$$
[b]Note:[/b] Given a sequence ($a_n$) of real numbers, we say that $\limsup\limits_{n\rightarrow \infty} a_n = +\infty$ if ($a_n$) is n... | To determine the value of
$$ \limsup_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}}, $$
we start by using the formula for the sum of the divisors function $\sigma(n)$ when $n$ is expressed as a product of prime powers. Let $n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}$, then
$$ \sigma(n) ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The positive divisors of a positive integer $n$ are written in increasing order starting with 1.
\[1=d_1<d_2<d_3<\cdots<n\]
Find $n$ if it is known that:
[b]i[/b]. $\, n=d_{13}+d_{14}+d_{15}$
[b]ii[/b]. $\,(d_5+1)^3=d_{15}+1$ | 1. **Given Conditions:**
- The positive divisors of \( n \) are written in increasing order: \( 1 = d_1 < d_2 < d_3 < \cdots < n \).
- \( n = d_{13} + d_{14} + d_{15} \).
- \( (d_5 + 1)^3 = d_{15} + 1 \).
2. **Initial Assumptions and Inequalities:**
- Assume \( d_{15} \leq \frac{n}{3} \), \( d_{14} \leq \f... | 1998 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says $0$. In each turn except the first the possible moves are determined from the previous number $n$ in the following way: write
\[n =\sum_{m\in O_n}2^m;\]
the valid numbers are the elements $m$ of $O_n$. That ... | 1. **Step 1:** We prove that all odd natural numbers are in \( B \).
Note that \( n \) is odd if and only if \( 0 \in O_n \). Hence if Arnaldo says \( n \) in his first turn, Bernaldo can immediately win by saying \( 0 \). Therefore, all odd numbers are in \( B \).
2. **Step 2:** We prove the inequality \( f(n) > ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$.
Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$. | 1. **Understanding the Group \( A \)**:
The group \( A \) consists of all roots of unity of degree \( 2006^k \) for all positive integers \( k \). This means \( A \) is the set of all complex numbers \( z \) such that \( z^{2006^k} = 1 \) for some positive integer \( k \).
2. **Idempotent Homomorphisms**:
We nee... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let n be a non-negative integer. Define the [i]decimal digit product[/i] \(D(n)\) inductively as follows:
- If \(n\) has a single decimal digit, then let \(D(n) = n\).
- Otherwise let \(D(n) = D(m)\), where \(m\) is the product of the decimal digits of \(n\).
Let \(P_k(1)\) be the probability that \(D(i) = 1\) where... | 1. **Understanding the Definition of \(D(n)\)**:
- If \(n\) has a single decimal digit, then \(D(n) = n\).
- Otherwise, \(D(n)\) is defined as \(D(m)\), where \(m\) is the product of the decimal digits of \(n\).
2. **Characterizing \(D(n) = 1\)**:
- We claim that \(D(n) = 1\) if and only if \(n\) is a repunit... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of $2$.
[i]Proposed by Harun Khan[/i] | To find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of 2, we will proceed with the following steps:
1. **Claim 1: \(m\) is odd.**
*Proof:* Suppose \(m\) is even. Then \(m\) can be written as \(2^k \cdot n\) where \(n\... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let the series $$s(n,x)=\sum \limits_{k= 0}^n \frac{(1-x)(1-2x)(1-3x)\cdots(1-nx)}{n!}$$ Find a real set on which this series is convergent, and then compute its sum. Find also $$\lim \limits_{(n,x)\to (\infty ,0)} s(n,x)$$ | 1. **Rewrite the series**: The given series is
\[
s(n,x) = \sum_{k=0}^n \frac{(1-x)(1-2x)(1-3x)\cdots(1-kx)}{k!}
\]
We need to find the set of \( x \) for which this series is convergent and then compute its sum.
2. **Analyze the general term**: The general term of the series is
\[
a_k = \frac{(1-x... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_n=\left(1+\frac{1}{n}\right)^n\left((2n-1)!F_n\right)^{\frac{1}{n}}$. Find $\lim \limits_{n \to \infty}(f_{n+1} - f_n)$ where $F_n$ denotes the $n$th Fibonacci number (given by $F_0 = 0$, $F_1 = 1$, and by $F_{n+1} = F_n + F_{n-1}$ for all $n \geq 1$ | To find the limit \(\lim_{n \to \infty}(f_{n+1} - f_n)\), we start by analyzing the given expression for \(f_n\):
\[ f_n = \left(1 + \frac{1}{n}\right)^n \left((2n-1)! F_n\right)^{\frac{1}{n}} \]
1. **Examine the first part of \(f_n\):**
\[ \left(1 + \frac{1}{n}\right)^n \]
As \(n \to \infty\), we know from the... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
[list=1]
[*] Prove that $$\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0$$
[*] Calculate $$\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)$$
[/list] | 1. **Prove that**
\[
\lim_{n \to \infty} \left(n + \frac{1}{4} - \zeta(3) - \zeta(5) - \cdots - \zeta(2n+1)\right) = 0
\]
To prove this, we need to analyze the behavior of the series involving the Riemann zeta function. We start by considering the series:
\[
\sum_{k=1}^{\infty} (\zeta(2k+1) - 1)
... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
[list=1]
[*] If $a$, $b$, $c$, $d > 0$, show inequality:$$\left(\tan^{-1}\left(\frac{ad-bc}{ac+bd}\right)\right)^2\geq 2\left(1-\frac{ac+bd}{\sqrt{\left(a^2+b^2\right)\left(c^2+d^2\right)}}\right)$$
[*] Calculate $$\lim \limits_{n \to \infty}n^{\alpha}\left(n- \sum \limits_{k=1}^n\frac{n^+k^2-k}{\sqrt{\left(n^2+k^2\rig... | ### Problem 1:
Given \( a, b, c, d > 0 \), show the inequality:
\[ \left(\tan^{-1}\left(\frac{ad - bc}{ac + bd}\right)\right)^2 \geq 2\left(1 - \frac{ac + bd}{\sqrt{(a^2 + b^2)(c^2 + d^2)}}\right) \]
1. **Express the given fraction in terms of trigonometric identities:**
\[
\left| \frac{ad - bc}{ac + bd} \right|... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A\ne0$. | 1. **Understanding the matrix $A$:**
The matrix $A = (a_{ij})$ is defined such that each element $a_{ij}$ is the remainder when $i^j + j^i$ is divided by $3$. This means $a_{ij} = (i^j + j^i) \mod 3$ for $i, j = 1, 2, \ldots, n$.
2. **Analyzing the pattern for $i = 7$:**
For $i = 7$, we need to check the values ... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $z=z(x,y)$ be implicit function with two variables from $2sin(x+2y-3z)=x+2y-3z$. Find $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$. | 1. Consider the mapping \( F(x,y,z(x,y)) := 2\sin(x+2y-3z) - (x+2y-3z) \) and suppose that \( F(x,y,z(x,y)) = 0 \) implicitly defines \( z \) as a continuously differentiable function of \( x \) and \( y \). This means \( F \in \mathcal{C}^{1} \) over its domain.
2. By the Implicit Function Theorem, if \( F_z \neq 0 \... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{x_n\}_{n=1}^\infty$ be a sequence such that $x_1=25$, $x_n=\operatorname{arctan}(x_{n-1})$. Prove that this sequence has a limit and find it. | 1. **Initial Sequence Definition and Properties**:
- Given the sequence $\{x_n\}_{n=1}^\infty$ with $x_1 = 25$ and $x_n = \arctan(x_{n-1})$ for $n \geq 2$.
- We need to show that this sequence has a limit and find it.
2. **Monotonicity and Boundedness**:
- For $x > 0$, we have $0 < \arctan(x) < x$. This is be... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $\operatorname{cif}(x)$ denote the sum of the digits of the number $x$ in the decimal system. Put $a_1=1997^{1996^{1997}}$, and $a_{n+1}=\operatorname{cif}(a_n)$ for every $n>0$. Find $\lim_{n\to\infty}a_n$. | 1. **Understanding the problem**: We need to find the limit of the sequence \(a_n\) defined by \(a_1 = 1997^{1996^{1997}}\) and \(a_{n+1} = \operatorname{cif}(a_n)\), where \(\operatorname{cif}(x)\) denotes the sum of the digits of \(x\).
2. **Modulo 9 property**: The sum of the digits of a number \(x\) in the decimal... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be a set of positive integers such that for any $x,y\in A$,
$$x>y\implies x-y\ge\frac{xy}{25}.$$Find the maximal possible number of elements of the set $A$. | 1. Given the inequality for any \( x, y \in A \) where \( x > y \):
\[
x - y \ge \frac{xy}{25}
\]
We start by manipulating this inequality to make it more manageable. Multiply both sides by 25:
\[
25(x - y) \ge xy
\]
Rearrange the terms to isolate \( xy \):
\[
25x - 25y \ge xy
\]
Rew... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $(a_n)$ is a sequence of real numbers such that the series
$$\sum_{n=1}^\infty\frac{a_n}n$$is convergent. Show that the sequence
$$b_n=\frac1n\sum^n_{j=1}a_j$$is convergent and find its limit. | 1. Given that the series $\sum_{n=1}^\infty \frac{a_n}{n}$ is convergent, we need to show that the sequence $b_n = \frac{1}{n} \sum_{j=1}^n a_j$ is convergent and find its limit.
2. Define $s_n := \sum_{i=1}^n \frac{a_i}{i}$ for $n \geq 0$ and let $s := \lim_{n \to \infty} s_n$. Since $\sum_{n=1}^\infty \frac{a_n}{n}$... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $0<a<b$ and let $f:[a,b]\to\mathbb R$ be a continuous function with $\int^b_af(t)dt=0$. Show that
$$\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.$$ | 1. **Claim:** $\int_0^\infty \frac{e^{-t} - e^{-\lambda t}}{t} dt = \ln \lambda$ for $\lambda > 0$.
**Proof:** Let $F(\lambda,t) = \frac{e^{-t} - e^{-\lambda t}}{t}$ for $\lambda, t \in \mathbb{R}_+$. By the Mean Value Theorem (MVT), we have:
\[
|F(\lambda, t)| = \left| \frac{e^{-t} - e^{-\lambda t}}{t} \righ... | 0 | Calculus | proof | Yes | Yes | aops_forum | false |
Let $$a_n = \frac{1\cdot3\cdot5\cdot\cdots\cdot(2n-1)}{2\cdot4\cdot6\cdot\cdots\cdot2n}.$$
(a) Prove that $\lim_{n\to \infty}a_n$ exists.
(b) Show that $$a_n = \frac{\left(1-\frac1{2^2}\right)\left(1-\frac1{4^2}\right)\left(1-\frac1{6^2}\right)\cdots\left(1-\frac{1}{(2n)^2}\right)}{(2n+1)a_n}.$$
(c) Find $\lim_{n\to\in... | ### Part (a): Prove that $\lim_{n\to \infty}a_n$ exists.
1. **Expression for \(a_n\):**
\[
a_n = \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots 2n}
\]
2. **Simplify the product:**
\[
a_n = \prod_{k=1}^n \frac{2k-1}{2k}
\]
3. **Rewrite each term:**
\[
\frac{2k-1}{2k} = 1 - \fr... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Define $f(x,y)=\frac{xy}{x^2+y^2\ln(x^2)^2}$ if $x\ne0$, and $f(0,y)=0$ if $y\ne0$. Determine whether $\lim_{(x,y)\to(0,0)}f(x,y)$ exists, and find its value is if the limit does exist. | To determine whether \(\lim_{(x,y)\to(0,0)} f(x,y)\) exists, we need to analyze the behavior of the function \(f(x,y)\) as \((x,y)\) approaches \((0,0)\).
1. **Define the function and consider the limit:**
\[
f(x,y) = \frac{xy}{x^2 + y^2 \ln(x^2)^2} \quad \text{if} \quad x \ne 0
\]
\[
f(0,y) = 0 \quad \... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Hey,
This problem is from the VTRMC 2006.
3. Recall that the Fibonacci numbers $ F(n)$ are defined by $ F(0) \equal{} 0$, $ F(1) \equal{} 1$ and $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$ for $ n \geq 2$. Determine the last digit of $ F(2006)$ (e.g. the last digit of 2006 is 6).
As, I and a fri... | To determine the last digit of \( F(2006) \), we need to find \( F(2006) \mod 10 \). We will use the periodicity of the Fibonacci sequence modulo 10.
1. **Establish the periodicity of Fibonacci numbers modulo 10:**
- The Fibonacci sequence modulo 10 is periodic. This means that after a certain number of terms, the ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f ( x , y ) = ( x + y ) / 2 , g ( x , y ) = \sqrt { x y } , h ( x , y ) = 2 x y / ( x + y ) $, and let $$ S = \{ ( a , b ) \in \mathrm { N } \times \mathrm { N } | a \neq b \text { and } f( a , b ) , g ( a , b ) , h ( a , b ) \in \mathrm { N } \} $$ where $\mathbb{N}$ denotes the positive integers. Find the minim... | 1. **Define the functions and set \( S \):**
\[
f(x, y) = \frac{x + y}{2}, \quad g(x, y) = \sqrt{xy}, \quad h(x, y) = \frac{2xy}{x + y}
\]
\[
S = \{ (a, b) \in \mathbb{N} \times \mathbb{N} \mid a \neq b \text{ and } f(a, b), g(a, b), h(a, b) \in \mathbb{N} \}
\]
2. **Express \( g(a, b) \) in terms of... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Example 1. Calculate $8 \frac{1}{7} + 43.8 - 13.947 + 0.00375$ ( $\frac{1}{7}$ is an exact number, all others are approximate numbers) | The number with the least decimal places is 43.8, so the other numbers only need to be truncated to the hundredth place, and the result is accurate to the tenth place.
$$
\begin{array}{l}
\frac{1}{7}+43.8-13.947+0.00375 \\
\approx 0.14+43.8-13.95 \approx 30.0
\end{array}
$$ | 30.0 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
Example 1. Using the ten digits from $0-9$,
(1) How many 3-digit numbers without repeated digits can be formed?
(2) How many of these numbers are between $300-700$ and do not end in zero? | (1) Since different arrangements of three digits form different numbers (ordered), this is a permutation problem. We can arrange them in the order of hundreds, tens, and units. The first digit cannot be 0, so there are $\mathrm{P}_{\mathrm{P}}^{1}$ ways to arrange the first digit. The next two digits can be arranged fr... | 648 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false |
Example 1. Given $f(n)=n^{4}+n^{3}+n^{2}+n+1$, find the remainder of $f\left(2^{5}\right)$ divided by $f(2)$. | $$
\begin{aligned}
f(2) & =2^{4}+2^{3}+2^{2}+2+1 \\
& =(11111)_{2}
\end{aligned}
$$
(The subscript 2 outside the parentheses indicates a binary number, the same applies below).
$$
\begin{array}{l}
f\left(2^{5}\right)=2^{20}+2^{15}+2^{10}+2^{5}+1 \\
=(100001000010000100001)_{2} \\
=(1111100 \cdots 0)_{15 \text { ones }}... | 5 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
Example 2. Find the remainder when $2^{22}-2^{20}+2^{18}-\cdots+2^{2}-1$ is divided by
81. | Solve $2^{22}-2^{20}+2^{18}-\cdots+2^{2}-1$
$$
\begin{array}{l}
=\left(2^{22}-2^{20}+2^{18}-2^{16}+2^{14}-\right. \\
\left.2^{12}\right)+\left(2^{10}-2^{8}+2^{6}-2^{4}+2^{2}-1\right) \\
=\left(2^{10}-2^{8}+2^{6}-2^{4}+2^{2}-1\right) \\
\left(2^{12}+1\right)
\end{array}
$$
Since $9=(1001)_{2}, 81=9 \times 9$,
and $2^{1... | 2 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false |
Example 2. Study the maximum and minimum values of the function $\mathrm{y}=\frac{1}{\mathrm{x}(1-\mathrm{x})}$ on $(0,1)$.
| Given $x \in (0,1)$, therefore $x>0, 1-x>0$.
When $x \rightarrow +0$, $1-x \rightarrow 1, x(1-x) \rightarrow +0$,
When $x \rightarrow 1-0$, $1-x \rightarrow +0, x(1-x) \rightarrow +0$,
Therefore, $y(+0)=y(1-0)=+\infty$. From (v),
it is known that $y$ has only a minimum value on $(0,1)$. Also, $y' = \frac{2x-1}{x^2(1-x... | 4 | Calculus | math-word-problem | Yes | Yes | cn_contest | false |
Example 4. Study the maximum and minimum values of the function $y=x^{2}-6 x+4 \ln x$ on $(0,2.5)$. | Since $y(+0)=-\infty, \therefore y$ has no minimum value on $(0,2.5)$. Also, $y^{\prime}=2 x-6+\frac{4}{x}=\frac{2\left(x^{2}-3 x+2\right)}{x}$
$$
=\frac{2(x-1)(x-2)}{x},
$$
the critical points are $x_{1}=1, x_{2}=2$. Comparing $y(1)=-5, y(2)=4 \ln 2 - 2 \doteq -5.2, y(2.5-0)$
$=y(2.5)=4 \ln \frac{5}{2}-\frac{35}{4}=$... | -5 | Calculus | math-word-problem | Yes | Yes | cn_contest | false |
Four. The distance between location A and location B is 120 kilometers. A car travels from location A to location B at a speed of 30 kilometers per hour. 1. Write the function relationship between the distance \( s \) (kilometers) the car is from location B and the time \( t \) (hours) since it left location A, and det... | 4. Solution:
1) The function relationship between $\mathrm{s}$ and $\mathrm{t}$ is $s=120-30 t$.
$$
(0 \leqslant \mathrm{t} \leqslant 4)
$$
2) Draw point A $(0, 120)$ and point B $(4,0)$,
then line segment $\mathrm{AB}$ is
the required graph (Figure 5).
From the graph, we can see: When $t=2.5$, $s=45$, that is, 2.5 ho... | 45 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
Eight. For what value of $\mathrm{k}$, the roots of the equation $3\left(\mathrm{x}^{2}+3 \mathrm{x}+4\right)$ $=x(2-x)(2+k)$ are the sides of a right triangle
保持了源文本的换行和格式。 | Eight, Solution: The original equation is rearranged as
$$
(\mathrm{k}+5) \mathrm{x}^{2}-(2 \mathrm{k}-5) \mathrm{x}+12=0 \text {. }
$$
Let the two roots of the equation be $x_{1}$ and $x_{2}$, then
$$
\begin{array}{l}
x_{1}+x_{2}=\frac{2 k-5}{k+5}, \\
x_{1} \cdot x_{2}=\frac{12}{k+5} .
\end{array}
$$
Since $x_{1}$ a... | 20 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
$2 . \operatorname{tg} \frac{\pi}{8}-\operatorname{ctg} \frac{\pi}{8}$ 的值是 $\qquad$ -
The value of $2 . \operatorname{tg} \frac{\pi}{8}-\operatorname{ctg} \frac{\pi}{8}$ is $\qquad$ - | 2. $\operatorname{tg} \frac{\pi}{8}-\operatorname{ctg} \frac{\pi}{8}=-2$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
2. $\operatorname{tan} \frac{\pi}{8}-\operatorname{cot} \frac{\pi}{8}=-2$. | -2 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
Example. Solve the following problems:
1. Without using tables, find the value of $\lg ^{3} 2+\lg ^{3} 5+3 \lg 2 \cdot \lg 5$;
2. Simplify: $\frac{1-\operatorname{tg} \theta}{1+\operatorname{tg} \theta}$;
3. Solve the system of equations $\left\{\begin{array}{l}\frac{x}{y}+\frac{y}{x}=\frac{25}{12}, \\ x^{2}+y^{2}=7\en... | 1. Original expression $=\mathbf{t g}^{3} 2+\lg ^{3} 5+3 \lg 2 \cdot 1 \mathrm{~g} 5$
$$
\begin{aligned}
& (\lg 2+\operatorname{Ig} 5) \\
= & (\lg 2+\lg 5)^{2}=1,
\end{aligned}
$$
2. Original expression $=\frac{\operatorname{tg} 45^{\circ}-\operatorname{tg} \theta}{1+\operatorname{tg} 45^{\circ} \operatorname{tg} \thet... | 1 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
2. Please fill in the four boxes below with the numbers $1$, $9$, $8$, and $3$. How should you fill them to get the maximum product? How should you fill them to get the minimum product?
$\square \square \times \square \square=$ ? | \begin{aligned} \text { 2. } & \text { Maximum: } 91 \times 83=7553 \text {, } \\ \text { Minimum: } & 18 \times 39=702 .\end{aligned} | 7553 | Logic and Puzzles | math-word-problem | Yes | Yes | cn_contest | false |
2. Given, $\odot \mathrm{O}_{1}$ and $\odot \mathrm{O}_{2}$ intersect at $\mathrm{A}$ and $\mathrm{B}$, a tangent line $\mathrm{AC}$ is drawn from point $\mathrm{A}$ to $\odot \mathrm{O}_{2}$, $\angle \mathrm{CAB}=45^{\circ}$, the radius of $\odot \mathrm{O}_{2}$ is $5 \sqrt{2} \mathrm{~cm}$, find the length of $\mathr... | 2. Solution: As shown in the figure
Draw diameter $\mathrm{AD}$, and connect $\mathrm{BD}$.
$$
\begin{array}{c}
\because \mathrm{AC} \text { is tangent to } \odot \text { at } \mathrm{A}, \text { and } \because \angle \mathrm{CAB}=45^{\circ}, \\
\therefore \angle \mathrm{D}=\angle \mathrm{CAB}=45^{\circ} . \\
\mathrm{... | 10 | Geometry | math-word-problem | Yes | Yes | cn_contest | false |
Five, a factory has 350 tons of coal in stock, planning to burn it all within a certain number of days. Due to improvements in the boiler, 2 tons are saved each day, allowing the stock of coal to last 15 days longer than originally planned, with 25 tons remaining. How many tons of coal were originally planned to be bur... | Five, Solution: Let the original plan be to burn $x$ tons of coal per day, then the actual daily consumption is $(x-2)$ tons.
$$
\frac{350-25}{x-2}-\frac{350}{x}=15 \text {. }
$$
Simplifying, we get $3 x^{2}-x-140=0$.
Solving, we get $\mathrm{x}_{1}=7, \mathrm{x}_{2}=-\frac{20}{3}$.
Upon verification, $\mathrm{x}_{1},... | 7 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
Seven, insert numbers between 55 and 555 so that they form an arithmetic sequence. The last number inserted equals the coefficient of the $x^{3}$ term in the expansion of $\left(\sqrt{\bar{x}}+\frac{1}{x}\right)^{1}$.
Keep the original text's line breaks and format, and output the translation result directly. | Seven, Solution: The general term formula of the expansion of $\left(\sqrt{\mathbf{x}}+\frac{1}{\mathbf{x}}\right)^{15}$ is
$$
\begin{array}{l}
T_{r+1}=C_{15}^{x}(\sqrt{\mathbf{x}})^{15-x}\left(\frac{1}{x}\right)^{\prime} \\
=C_{15}^{2} x^{\frac{15-3 r}{2}}, \\
\text { Let } \frac{15-3 r}{2}=3, \quad \text { then } r=3... | 4 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
Example 2. For what value of the real number $\mathrm{x}$, does $\sqrt{4 \mathrm{x}^{2}-4 \mathrm{x}+1}$ $-\sqrt{x^{2}-2 x+1}(0 \leqslant x \leqslant 2)$ attain its maximum and minimum values? And find the maximum and minimum values. | Let $y=\sqrt{4 x^{2}-4 x+1}-\sqrt{x^{2}-2 x+1}$
$$
\begin{aligned}
x & \sqrt{(2 x-1)^{2}}=\sqrt{(x-1)^{2}} \\
\text { Then } y & =|2 x-1|-|x-1|, \quad(0 \leqslant x \leqslant 2)
\end{aligned}
$$
$$
\therefore y=\left\{\begin{array}{ll}
-x, & \text { (when } 0 \leqslant x \leqslant \frac{1}{2} \text { ) } \\
3 x-2, & \t... | 2 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
Example 21. How many three-digit numbers can be formed using $0,1,2,3,4,5$ without repeating any digit? | Solution (Method 1 - "Take all and subtract the unwanted"): $\mathrm{P}_{\mathrm{G}}^{\mathrm{s}}-\mathrm{P}_{5}^{2}=100$ (items).
(Solution 2 - "Take only the wanted"): $5 \mathrm{P}_{5}^{2}=100$ (items). | 100 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false |
Example 24. Take 3 numbers from $1,3,5,7,9$, and 2 numbers from $2,4,6,8$, to form a five-digit even number without repeated digits. How many such numbers can be formed? | There are $\mathrm{C}_{4}^{1} \cdot \mathrm{C}_{3}^{1} \cdot \mathrm{C}_{5}^{3} \cdot \mathrm{P}_{4}^{4}=2880($ ways). | 2880 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false |
Example 2. Calculate $\left|\begin{array}{lll}1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos (\alpha+\beta) \\ \cos \beta & \cos (\alpha+\beta) & 1\end{array}\right|$. | $$
\begin{aligned}
\text { Original expression }= & 1+2 \cos \alpha \cos \beta \cos (\alpha+\beta)-\cos ^{2} \beta \\
& -\cos 2 \alpha-\cos ^{2}(\alpha+\beta) \\
= & 1+\cos (\alpha+\beta)[2 \cos \alpha \cos \beta \\
& -\cos (\alpha+\beta)]-\cos ^{2} \alpha-\cos ^{2} \beta \\
= & 1+\cos (\alpha+\beta) \cos (\alpha-\beta... | 0 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
1. A gathering has 1982 people attending, and among any 4 people, at least one person knows the other three. How many people at this gathering know all the attendees? | We prove that there are at least 1979 people who know all the attendees. In other words, at most three people do not know all the attendees.
Using proof by contradiction. Assume that at least four people do not know all the attendees. Let $A$ be one of them, and $A$ does not know $B$. At this point, apart from $A$ and... | 1979 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false |
Four, the height of a certain U is $\mathrm{CD}$, at point $\mathrm{A}$ due east of the mountain, the angle of elevation to the mountain peak is $60^{\circ}$. From point A, moving 300 meters in a direction 28 degrees west of south, reaching point $\mathrm{B}$, which is exactly southeast of the mountain. Find the height... | Four, Solution: As shown in Figure 6, $\because \angle \mathrm{BAE}=28^{\circ}$,
$$
\therefore \angle \mathrm{CAB}=90^{\circ}-28^{\circ}=
$$
$62^{\circ}$.
$$
\begin{array}{l}
\therefore \angle \mathrm{BCA}=45^{\circ}. \\
\therefore \angle \mathrm{ABC}=180^{\circ}-45^{\circ}- \\
62^{\circ}=73^{\circ}.
\end{array}
$$
$$
... | 703 | Geometry | math-word-problem | Yes | Yes | cn_contest | false |
Example 1. 6 students line up, where one of them does not stand at the head or the tail of the line, how many ways are there to arrange them? (Question 4, page 157) | Solution 1: The total number of possible arrangements for 6 people is $6!$; the number of possible arrangements with the restricted person at the head of the line is $5!$, and the number of possible arrangements with the restricted person at the tail of the line is also $5!$, which are not allowed; therefore, the numbe... | 480 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false |
Example 2. 8 different elements are arranged in two rows, with 4 elements in each row, where 2 specific elements must be in the front row, and 1 specific element must be in the back row. How many arrangements are possible? (Question 12 (2), page 158) | Example 2 solution. For 2 elements to be arranged in the front row, first arrange them, there are $\mathrm{P}_{4}^{2}$ arrangements, for a certain element to be arranged in the back row, also arrange it first, there are 4 arrangements, the remaining 5 elements can be arranged in the remaining 5 positions in any order, ... | 5760 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false |
Example 4. In a militia squad marching, the main and deputy machine gunners must be adjacent, and they must not stand at the head or the tail of the line. How many ways can the squad be arranged? | Solution: Still using the insertion method: The other 8 people can be arranged arbitrarily, with $8!$ arrangements. The main and deputy machine gunners can only be inserted into the 7 intervals formed by these 8 people, and the 2 people have a sequence, so there are $7 \times 2!$ ways to insert. Therefore, the total nu... | 564480 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false |
$$
\begin{array}{l}
\text { 2. Let } f(x)=x-p^{!}+\left|x-15^{\prime}+x-p-15\right|, \text { where } \\
\text { } 0<p<15 \text {. }
\end{array}
$$
Find: For $x$ in the interval $p \leqslant x \leqslant 15$, what is the minimum value of $f(x)$? | \begin{array}{l}\text { Solve } \because p \leqslant x \leqslant 15 \text{ and } p>0, \therefore p+15 \geqslant x . \\ \therefore f(x)=x-p+15-x+p+15-x=30-x, \\ \text { Also } \because x \leqslant 15, \\ \therefore f(x) \geqslant 30-15=15 . \\ \therefore \text { the minimum value of } f(x) \text { is } 15 .\end{array} | 15 | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
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