problem stringlengths 11 4.31k | solution stringlengths 146 10.2k | messages listlengths 2 2 |
|---|---|---|
Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC.$ if $OQ$ is 3 inches, then $OP$, in inches, is:
$\text{(A) } 3\quad \text{(B) } \frac{9}{2}\quad \text{(C) } 6\quad \text{(D) } 9\quad \text{(E) } \text{undetermined}$ | To solve this problem, we need to recall some properties of medians and the centroid in a triangle. The centroid \(O\) of a triangle is the point where its three medians intersect. The centroid divides each median into a ratio of \(2:1\), with the longer segment being closer to the vertex.
Given that \(OQ = 3\) inches... | [
{
"content": "Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC.$ if $OQ$ is 3 inches, then $OP$, in inches, is:\n$\\text{(A) } 3\\quad \\text{(B) } \\frac{9}{2}\\quad \\text{(C) } 6\\quad \\text{(D) } 9\\quad \\text{(E) } \\text{undetermined}$",
"role": "user"
},
{
"conten... |
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\ln (\tan t) \\
y=\frac{1}{\sin ^{2} t}
\end{array}\right.
$$ | To find the derivative \( \frac{dy}{dx} \) given the parametric equations \( x = \ln (\tan t) \) and \( y = \frac{1}{\sin^2 t} \), we can use the chain rule for differentiation. Specifically, we need to compute \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \) and then use the relationship:
\[ \frac{dy}{dx} = \frac{\frac{dy... | [
{
"content": "Find the derivative $y_{x}^{\\prime}$.\n\n$$\n\\left\\{\\begin{array}{l}\nx=\\ln (\\tan t) \\\\\ny=\\frac{1}{\\sin ^{2} t}\n\\end{array}\\right.\n$$",
"role": "user"
},
{
"content": "<think>To find the derivative \\( \\frac{dy}{dx} \\) given the parametric equations \\( x = \\ln (\\tan... |
There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there were 2013 non-empty boxes. How many boxes wer... | To solve the problem, let's break it down into more manageable steps:
1. Initially, there were 2013 empty boxes.
2. In each operation, you place 13 new boxes into a non-nested box. This reduces the number of empty boxes by 1 and increases the total number of boxes by 13.
3. We need to count how many operations are req... | [
{
"content": "There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there were 2013 non-empty box... |
In the Cartesian coordinate system, O is the origin. It is known that $\overrightarrow {p}=(-1, 2)$, A(8, 0), B(n, t), and C(ksinθ, t), where $0 \leq \theta \leq \frac{\pi}{2}$
(1) If $\overrightarrow {AB} \perp \overrightarrow {p}$ and $|\overrightarrow {AB}|= \sqrt {5}|\overrightarrow {OA}|$, find the vector $\over... | Let's solve the problem step-by-step using Python and Sympy.
### Part 1: Finding the vector \(\overrightarrow{OB}\)
Given:
- \(\overrightarrow{p} = (-1, 2)\)
- \(A = (8, 0)\)
- \(B = (n, t)\)
Conditions:
1. \(\overrightarrow{AB} \perp \overrightarrow{p}\)
2. \(|\overrightarrow{AB}| = \sqrt{5}|\overrightarrow{OA}|\)
... | [
{
"content": "In the Cartesian coordinate system, O is the origin. It is known that $\\overrightarrow {p}=(-1, 2)$, A(8, 0), B(n, t), and C(ksinθ, t), where $0 \\leq \\theta \\leq \\frac{\\pi}{2}$ \n(1) If $\\overrightarrow {AB} \\perp \\overrightarrow {p}$ and $|\\overrightarrow {AB}|= \\sqrt {5}|\\overrighta... |
There are 211 students and four different types of chocolates, with more than 633 pieces of each type of chocolate. Each student can take up to three pieces of chocolate, or none at all. Grouping students based on both the type and number of chocolates they take, the maximum number of students in any group is at least ... | To solve this problem, we must determine the maximum number of students that can be in any single group given the constraints:
- There are 211 students.
- There are four different types of chocolates.
- Each student can take up to three pieces of chocolate, or none at all.
Firstly, we need to consider all possible ... | [
{
"content": "There are 211 students and four different types of chocolates, with more than 633 pieces of each type of chocolate. Each student can take up to three pieces of chocolate, or none at all. Grouping students based on both the type and number of chocolates they take, the maximum number of students in ... |
Given an geometric sequence $\{a_{n}\}$ that satisfies $a_{1}+a_{3}=5$, $a_{2}+a_{4}=\frac{5}{2}$, find the maximum value of the product $a_{1}a_{2}...a_{n}$. | To solve the given problem, we first need to understand the conditions of the geometric sequence \( \{a_n\} \).
A geometric sequence can be expressed as:
\[ a_n = a_1 \cdot r^{n-1} \]
where \( a_1 \) is the first term and \( r \) is the common ratio.
Given the conditions:
1. \( a_1 + a_3 = 5 \)
2. \( a_2 + a_4 = \fr... | [
{
"content": "Given an geometric sequence $\\{a_{n}\\}$ that satisfies $a_{1}+a_{3}=5$, $a_{2}+a_{4}=\\frac{5}{2}$, find the maximum value of the product $a_{1}a_{2}...a_{n}$.",
"role": "user"
},
{
"content": "<think>To solve the given problem, we first need to understand the conditions of the geome... |
There are two taps, $A$ and $B$, in a pool. If both taps $A$ and $B$ are opened simultaneously, it takes 30 minutes to fill the pool. Now, if both taps $A$ and $B$ are opened for 10 minutes, and then tap $A$ is closed with tap $B$ continuing to fill for another 40 minutes, the pool will also be filled. How many minutes... | To solve this problem, we need to determine the rates at which taps \(A\) and \(B\) fill the pool. Let's denote:
- \( R_A \) as the rate at which tap \(A\) fills the pool (fraction of the pool filled per minute).
- \( R_B \) as the rate at which tap \(B\) fills the pool (fraction of the pool filled per minute).
From t... | [
{
"content": "There are two taps, $A$ and $B$, in a pool. If both taps $A$ and $B$ are opened simultaneously, it takes 30 minutes to fill the pool. Now, if both taps $A$ and $B$ are opened for 10 minutes, and then tap $A$ is closed with tap $B$ continuing to fill for another 40 minutes, the pool will also be fi... |
A carpenter took a wooden square and cut 4 smaller equal squares from it, each with an area of $9\%$ of the area of the large square. The remaining area of the original square was $256 \mathrm{~cm}^{2}$. Find the side length of the original square in centimeters. | Let's start by denoting the side length of the original square as \( s \) cm.
The area of the original square is:
\[ A = s^2 \]
Each of the 4 smaller squares has an area of \( 9\% \) of the area of the original square:
\[ \text{Area of each small square} = 0.09 \times A = 0.09 \times s^2 \]
Since there are 4 such sm... | [
{
"content": "A carpenter took a wooden square and cut 4 smaller equal squares from it, each with an area of $9\\%$ of the area of the large square. The remaining area of the original square was $256 \\mathrm{~cm}^{2}$. Find the side length of the original square in centimeters.",
"role": "user"
},
{
... |
A random variable \( X \) is defined by the distribution function
$$
F(x)=\left\{\begin{array}{lc}
0 & \text{if } x \leq 0 \\
x / 2 & \text{if } 0 < x \leq 2 \\
1 & \text{if } x > 2
\end{array}\right.
$$
Find the probability that, as a result of the trial, the random variable \( X \) takes a value in the interval \(... | To find the probability that the random variable \( X \) takes a value in the interval \( (1, 2) \), we need to evaluate the cumulative distribution function (CDF) \( F(x) \) at the boundaries of the interval and then find the difference between them.
The cumulative distribution function \( F(x) \) is given by:
\[
F(x... | [
{
"content": "\nA random variable \\( X \\) is defined by the distribution function\n\n$$\nF(x)=\\left\\{\\begin{array}{lc}\n0 & \\text{if } x \\leq 0 \\\\\nx / 2 & \\text{if } 0 < x \\leq 2 \\\\\n1 & \\text{if } x > 2\n\\end{array}\\right.\n$$\n\nFind the probability that, as a result of the trial, the random ... |
Find the volume of the solid bounded by the surfaces
$$
x=17 \sqrt{2 y}, \quad x=2 \sqrt{2 y}, \quad z=1 / 2-y, \quad z=0
$$ | To find the volume of the solid bounded by the given surfaces, we break the problem into manageable parts:
1. **Understand the Boundaries:**
- \(x = 17 \sqrt{2y}\)
- \(x = 2 \sqrt{2y}\)
- \(z = \frac{1}{2} - y\)
- \(z = 0\)
2. **Set Up the Triple Integral:**
We'll use a triple integral to find the volu... | [
{
"content": "Find the volume of the solid bounded by the surfaces\n$$\nx=17 \\sqrt{2 y}, \\quad x=2 \\sqrt{2 y}, \\quad z=1 / 2-y, \\quad z=0\n$$",
"role": "user"
},
{
"content": "<think>To find the volume of the solid bounded by the given surfaces, we break the problem into manageable parts:\n\n1.... |
Given the binomial distribution $\xi\sim B(4, \frac{1}{2})$, the value of the variance $D\xi$ is $\_\_\_\_\_\_$. | To solve for the variance of a binomial distribution \(\xi \sim B(n, p)\), we can use the well-known formula for the variance of a binomial distribution:
\[ D\xi = n \cdot p \cdot (1 - p) \]
In this problem, we are given:
- \( n = 4 \)
- \( p = \frac{1}{2} \)
Let's calculate the variance using Python and SymPy.
```... | [
{
"content": "Given the binomial distribution $\\xi\\sim B(4, \\frac{1}{2})$, the value of the variance $D\\xi$ is $\\_\\_\\_\\_\\_\\_$.",
"role": "user"
},
{
"content": "<think>To solve for the variance of a binomial distribution \\(\\xi \\sim B(n, p)\\), we can use the well-known formula for the v... |
If sets \( M \) and \( N \) contain \( m \) and \( n \) elements respectively, how many possible mappings from \( M \) to \( N \) can be established?
A. \( m+n \)
B. \( m \cdot n \)
C. \( m^{n} \)
D. \( n^{m} \) | To solve this problem, we need to understand that a mapping from set \( M \) to set \( N \) is a function that assigns each element in \( M \) to exactly one element in \( N \).
Given:
- Set \( M \) has \( m \) elements.
- Set \( N \) has \( n \) elements.
For each element in \( M \), there are \( n \) possible choic... | [
{
"content": "If sets \\( M \\) and \\( N \\) contain \\( m \\) and \\( n \\) elements respectively, how many possible mappings from \\( M \\) to \\( N \\) can be established?\nA. \\( m+n \\)\nB. \\( m \\cdot n \\)\nC. \\( m^{n} \\)\nD. \\( n^{m} \\)",
"role": "user"
},
{
"content": "<think>To solve... |
Given that $a$, $b$, and $c$ are positive real numbers satisfying $a+b+c=1$, find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac {1}{c}$$. | To solve this problem, we'll use the method of Lagrange multipliers to find the minimum value of the function \( f(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \) subject to the constraint \( g(a, b, c) = a + b + c - 1 = 0 \).
Here's how we can do it step-by-step:
1. **Define the function and constraint:**
-... | [
{
"content": "Given that $a$, $b$, and $c$ are positive real numbers satisfying $a+b+c=1$, find the minimum value of $$\\frac {1}{a}+ \\frac {1}{b}+ \\frac {1}{c}$$.",
"role": "user"
},
{
"content": "<think>To solve this problem, we'll use the method of Lagrange multipliers to find the minimum value... |
Evaluate the integral $$\int_{0}^{2}(2-|1-x|)dx=\_\_\_\_\_\_.$$ | To evaluate the integral \(\int_{0}^{2}(2-|1-x|) \, dx\), we need to break it into manageable parts. The absolute value function \(|1 - x|\) will need special handling because it changes its expression depending on whether \(x\) is less than or greater than 1.
Let's break it down step-by-step:
1. **Analyze the functi... | [
{
"content": "Evaluate the integral $$\\int_{0}^{2}(2-|1-x|)dx=\\_\\_\\_\\_\\_\\_.$$",
"role": "user"
},
{
"content": "<think>To evaluate the integral \\(\\int_{0}^{2}(2-|1-x|) \\, dx\\), we need to break it into manageable parts. The absolute value function \\(|1 - x|\\) will need special handling ... |
If \(3x - 9 = 12\), then the value of \(6x\) is
(A) 42
(B) 24
(C) 6
(D) 32
(E) 52 | Let's start by solving the equation \(3x - 9 = 12\) step by step, and then we can determine the value of \(6x\).
1. Add 9 to both sides of the equation to isolate the term with \(x\):
\[ 3x - 9 + 9 = 12 + 9 \]
\[ 3x = 21 \]
2. Divide both sides by 3 to solve for \(x\):
\[ x = \frac{21}{3} \]
\[ x = 7 \]
... | [
{
"content": "If \\(3x - 9 = 12\\), then the value of \\(6x\\) is\n(A) 42\n(B) 24\n(C) 6\n(D) 32\n(E) 52",
"role": "user"
},
{
"content": "<think>Let's start by solving the equation \\(3x - 9 = 12\\) step by step, and then we can determine the value of \\(6x\\).\n\n1. Add 9 to both sides of the equa... |
In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimming pool, including a trip to the sauna, for 250 rubles to warm up. However, now that summer has arrived, there is no longer a need to visi... | Let's break down the problem:
1. Katya used to buy a pass to the swimming pool and the sauna for 250 rubles.
2. She needs to save up 275 rubles (the cost of the headphones) by reducing her expenses on swimming pool visits.
3. Visiting only the swimming pool costs 200 rubles more than visiting the sauna.
Let's denote ... | [
{
"content": "In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimming pool, including a trip to the sauna, for 250 rubles to warm up. However, now that summer has arrived, there is no... |
Given $f(x)= \begin{cases} \cos \pi x, & (x < 1) \\ f(x-1)-1, & (x > 1) \end{cases}$, find $f(\frac{1}{3})+f(\frac{4}{3})$. | To solve the problem, we need to evaluate the function \( f(x) \) at two points: \( x = \frac{1}{3} \) and \( x = \frac{4}{3} \). We will then sum these two values.
1. **Evaluate \( f(\frac{1}{3}) \)**:
Since \( \frac{1}{3} < 1 \), we use the first piece of the piecewise function:
\[
f\left(\frac{1}{3}\right)... | [
{
"content": "Given $f(x)= \\begin{cases} \\cos \\pi x, & (x < 1) \\\\ f(x-1)-1, & (x > 1) \\end{cases}$, find $f(\\frac{1}{3})+f(\\frac{4}{3})$.",
"role": "user"
},
{
"content": "<think>To solve the problem, we need to evaluate the function \\( f(x) \\) at two points: \\( x = \\frac{1}{3} \\) and \... |
There is a five-digit positive odd number \( x \). In \( x \), all 2s are replaced with 5s, and all 5s are replaced with 2s, while other digits remain unchanged. This creates a new five-digit number \( y \). If \( x \) and \( y \) satisfy the equation \( y = 2(x + 1) \), what is the value of \( x \)?
(Chinese Junior Hi... | Let's denote the given five-digit positive odd number by \( x \), and the new number formed by the described replacement of digits by \( y \). According to the problem, \( y \) is related to \( x \) by the equation:
\[ y = 2(x + 1) \]
To solve this, we need to consider the effect of the digit transformation on each pl... | [
{
"content": "There is a five-digit positive odd number \\( x \\). In \\( x \\), all 2s are replaced with 5s, and all 5s are replaced with 2s, while other digits remain unchanged. This creates a new five-digit number \\( y \\). If \\( x \\) and \\( y \\) satisfy the equation \\( y = 2(x + 1) \\), what is the va... |
(5 points) A line passing through the focus of the parabola $y^2=4x$ intersects the parabola at points A$(x_1, y_1)$ and B$(x_2, y_2)$. If $x_1+x_2=6$, then $|AB|=$ . | To solve the problem, we need to use the fact that the focus of the parabola \(y^2 = 4x\) is at the point \((1, 0)\). A line passing through this point intersects the parabola at points \(A(x_1, y_1)\) and \(B(x_2, y_2)\). We are given that \(x_1 + x_2 = 6\) and we need to find the distance \(|AB|\).
Let's break down ... | [
{
"content": "(5 points) A line passing through the focus of the parabola $y^2=4x$ intersects the parabola at points A$(x_1, y_1)$ and B$(x_2, y_2)$. If $x_1+x_2=6$, then $|AB|=$ .",
"role": "user"
},
{
"content": "<think>To solve the problem, we need to use the fact that the focus of the parabola \... |
Given $a \gt 1$, $b \gt 1$, when $b$ changes, the minimum value of $log_{a}b+log_{b}({a}^{2}+12)$ is $4$, then $a=\_\_\_\_\_\_$. | To solve the given problem, we need to find the value of \(a\) such that the minimum value of \( \log_a b + \log_b (a^2 + 12) \) is 4.
Steps to solve the problem:
1. **Rewrite the expression using the change of base formula:**
\[
\log_a b + \log_b (a^2 + 12) = \frac{\log b}{\log a} + \frac{\log (a^2 + 12)}{\lo... | [
{
"content": "Given $a \\gt 1$, $b \\gt 1$, when $b$ changes, the minimum value of $log_{a}b+log_{b}({a}^{2}+12)$ is $4$, then $a=\\_\\_\\_\\_\\_\\_$.",
"role": "user"
},
{
"content": "<think>To solve the given problem, we need to find the value of \\(a\\) such that the minimum value of \\( \\log_a ... |
On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\textsc{am}$, and the sunset as $8:15\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
$\textbf{(A)}\hspace{.05in}5:10\textsc{pm}\quad... | To determine the correct sunset time, we'll start with the provided information:
- Length of daylight: 10 hours and 24 minutes
- Correct sunrise time: 6:57 AM
We'll convert the length of daylight into minutes and add it to the sunrise time to find the accurate sunset time.
First, let's convert 10 hours and 24 minutes... | [
{
"content": "On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\\textsc{am}$, and the sunset as $8:15\\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?\n$\\textbf{(A)}\\hspace... |
If \( a = \frac{1}{2} \) and \( (a + 1)(b + 1) = 2 \), then \( \operatorname{arctg} a + \operatorname{arctg} b \) in radians is equal to:
(A) \(\frac{\pi}{2}\).
(B) \(\frac{\pi}{3}\).
(C) \(\frac{\pi}{4}\).
(D) \(\frac{\pi}{5}\).
(E) \(\frac{\pi}{6}\). | To solve the problem, we need to find \( \operatorname{arctg} a + \operatorname{arctg} b \) in radians given that \( a = \frac{1}{2} \) and \( (a + 1)(b + 1) = 2 \). The range of values for \( \arctan \) indicates that we are looking for a specific angle.
Let's break this down into manageable steps:
1. **Substitute \... | [
{
"content": "If \\( a = \\frac{1}{2} \\) and \\( (a + 1)(b + 1) = 2 \\), then \\( \\operatorname{arctg} a + \\operatorname{arctg} b \\) in radians is equal to:\n(A) \\(\\frac{\\pi}{2}\\).\n(B) \\(\\frac{\\pi}{3}\\).\n(C) \\(\\frac{\\pi}{4}\\).\n(D) \\(\\frac{\\pi}{5}\\).\n(E) \\(\\frac{\\pi}{6}\\).",
"role... |
In an arithmetic sequence $\{a_{n}\}$, given $a_{3}+a_{13}=12$, then the sum of the first $15$ terms of this sequence $S_{15}=$____. | To find the sum of the first 15 terms of the arithmetic sequence given that \(a_3 + a_{13} = 12\), let’s break down the problem into smaller steps and solve it step-by-step using Python and sympy.
1. **Arithmetic Sequence Definition**:
- The \(n\)-th term of an arithmetic sequence can be expressed as:
\[
... | [
{
"content": "In an arithmetic sequence $\\{a_{n}\\}$, given $a_{3}+a_{13}=12$, then the sum of the first $15$ terms of this sequence $S_{15}=$____.",
"role": "user"
},
{
"content": "<think>To find the sum of the first 15 terms of the arithmetic sequence given that \\(a_3 + a_{13} = 12\\), let’s bre... |
Given a sector with a radius of 2 and an area of 4, find the length of the arc corresponding to the central angle of the sector. | To find the length of the arc of a sector given the radius and the area of the sector, we can follow these steps:
1. **Understand the Relationship:**
The area \(A\) of a sector with a central angle \(\theta\) (in radians) and radius \(r\) is given by:
\[
A = \frac{1}{2} r^2 \theta
\]
We are given \(A = ... | [
{
"content": "Given a sector with a radius of 2 and an area of 4, find the length of the arc corresponding to the central angle of the sector.",
"role": "user"
},
{
"content": "<think>To find the length of the arc of a sector given the radius and the area of the sector, we can follow these steps:\n\... |
Let the solution set of the inequality $|x-a| < 1$ with respect to $x$ be $A$, and $2 \in A$. Find the value of the positive integer $a$. | To solve this problem, we need to follow a structured approach:
1. **Understand the inequality**: Given the inequality \( |x - a| < 1 \), we can derive the range of \(x\) that satisfies this inequality.
2. **Translate the condition**: The set \(A\) is defined as the set of all \(x\) satisfying \( |x - a| < 1 \). We ar... | [
{
"content": "Let the solution set of the inequality $|x-a| < 1$ with respect to $x$ be $A$, and $2 \\in A$. Find the value of the positive integer $a$.",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to follow a structured approach:\n\n1. **Understand the inequality**: Given... |
Triangle $ABC$ has vertices $A(0,8)$, $B(2,0)$, $C(8,0)$. A vertical line intersects $AC$ at $R$ and $\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\triangle RSC$ is 12.5, determine the positive difference of the $x$ and $y$ coordinates of point $R$. | To solve the problem, we need to find points \( R \) and \( S \) on lines \( AC \) and \( BC \), respectively, such that the area of triangle \( RSC \) is 12.5. Let's proceed by first determining the equations of lines \( AC \) and \( BC \).
1. **Find the equations of lines \( AC \) and \( BC \)**:
- Line \( AC \):... | [
{
"content": "Triangle $ABC$ has vertices $A(0,8)$, $B(2,0)$, $C(8,0)$. A vertical line intersects $AC$ at $R$ and $\\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\\triangle RSC$ is 12.5, determine the positive difference of the $x$ and $y$ coordinates of point $R$.",
"role": "user"
},
... |
For a natural number $A$, define the product of its digits in base 10 as $p(A)$. Find all values of $A$ that satisfy the equation $A = 1.5 \cdot p(A)$. | To find all natural numbers \( A \) that satisfy the equation \( A = 1.5 \cdot p(A) \), where \( p(A) \) is the product of the digits of \( A \), we can take the following approach:
1. Iterate through possible values of \( A \).
2. For each value of \( A \), compute the product of its digits \( p(A) \).
3. Check if \(... | [
{
"content": "For a natural number $A$, define the product of its digits in base 10 as $p(A)$. Find all values of $A$ that satisfy the equation $A = 1.5 \\cdot p(A)$.",
"role": "user"
},
{
"content": "<think>To find all natural numbers \\( A \\) that satisfy the equation \\( A = 1.5 \\cdot p(A) \\),... |
If \( A + B = 5 \), then the value of \( B - 3 + A \) is:
(A) 2
(B) 8
(C) 7
(D) 15
(E) 13 | To solve this problem, we can use basic algebraic manipulation. Given that \( A + B = 5 \), we need to find the value of \( B - 3 + A \).
Let's break this down step-by-step:
1. Start with the given equation:
\[ A + B = 5 \]
2. Substitute \( A + B \) in the expression \( B - 3 + A \):
\[ B - 3 + A = A + B - 3 \... | [
{
"content": "If \\( A + B = 5 \\), then the value of \\( B - 3 + A \\) is:\n(A) 2\n(B) 8\n(C) 7\n(D) 15\n(E) 13",
"role": "user"
},
{
"content": "<think>To solve this problem, we can use basic algebraic manipulation. Given that \\( A + B = 5 \\), we need to find the value of \\( B - 3 + A \\).\n\nL... |
For any natural number, let $S(n)$ denote sum of digits of $n$ . Find the number of $3$ digit numbers for which $S(S(n)) = 2$ . | To solve the problem of determining the number of three-digit numbers \( n \) for which \( S(S(n)) = 2 \), we will follow these steps:
1. Generate all three-digit numbers (i.e., numbers from 100 to 999).
2. Calculate the sum of the digits of each number, \( S(n) \).
3. Calculate the sum of the digits of \( S(n) \), de... | [
{
"content": "For any natural number, let $S(n)$ denote sum of digits of $n$ . Find the number of $3$ digit numbers for which $S(S(n)) = 2$ .",
"role": "user"
},
{
"content": "<think>To solve the problem of determining the number of three-digit numbers \\( n \\) for which \\( S(S(n)) = 2 \\), ... |
Place 10 identical balls into three boxes numbered 1, 2, and 3 (all 10 balls must be placed), with the requirement that the number of balls in each box is not less than the number on the box. How many ways are there to do this? (Please answer with a number.) | To solve this problem, we need to find the number of ways to distribute 10 identical balls into three boxes numbered 1, 2, and 3 such that the number of balls in each box is not less than the number on the box.
Let \( x_1 \), \( x_2 \), and \( x_3 \) be the number of balls in boxes 1, 2, and 3 respectively. The constr... | [
{
"content": "Place 10 identical balls into three boxes numbered 1, 2, and 3 (all 10 balls must be placed), with the requirement that the number of balls in each box is not less than the number on the box. How many ways are there to do this? (Please answer with a number.)",
"role": "user"
},
{
"cont... |
In a math class, the teacher gave the students 2 questions. Scoring is as follows: getting one question correct earns 10 points, partially correct earns 5 points, and completely wrong or unanswered earns 0 points. After grading, the teacher found that there were students with each possible score, and for each score, th... | The possible scores for each student are based on the combination of getting the two questions either correct, partially correct, or wrong. To understand the scores, we need to examine all possible combinations:
1. Both correct: \(10 + 10 = 20\) points
2. One correct and one partially correct: \(10 + 5 = 15\) points
3... | [
{
"content": "In a math class, the teacher gave the students 2 questions. Scoring is as follows: getting one question correct earns 10 points, partially correct earns 5 points, and completely wrong or unanswered earns 0 points. After grading, the teacher found that there were students with each possible score, ... |
Simplify first, then evaluate: $(2m^{2}-3mn+8)-(5mn-4m^{2}+8)$, where $m=2$, $n=1$. | Sure, let's break down the problem into manageable steps:
1. **Simplify the expression:**
\[
(2m^2 - 3mn + 8) - (5mn - 4m^2 + 8)
\]
Combine like terms.
2. **Substitute the values \(m = 2\) and \(n = 1\) into the simplified expression and evaluate the result.
Let's implement this step-by-step using Python... | [
{
"content": "Simplify first, then evaluate: $(2m^{2}-3mn+8)-(5mn-4m^{2}+8)$, where $m=2$, $n=1$.",
"role": "user"
},
{
"content": "<think>Sure, let's break down the problem into manageable steps:\n\n1. **Simplify the expression:**\n \\[\n (2m^2 - 3mn + 8) - (5mn - 4m^2 + 8)\n \\]\n Combine ... |
Given the function $f(x) = \begin{cases} \log_{2}(x+3), & x \geqslant 0 \\ x^{2}, & x < 0 \end{cases}$, find $f(f(-1))=$ \_\_\_\_\_\_. | To solve this problem, we need to evaluate the composite function \( f(f(-1)) \) given the piecewise function \( f(x) \). Here's a step-by-step plan:
1. **Evaluate \( f(-1) \)**:
Since \(-1 < 0\), we use the second part of the piecewise function \( f(x) = x^2 \). Thus, \( f(-1) = (-1)^2 = 1 \).
2. **Evaluate \( f(... | [
{
"content": "Given the function $f(x) = \\begin{cases} \\log_{2}(x+3), & x \\geqslant 0 \\\\ x^{2}, & x < 0 \\end{cases}$, find $f(f(-1))=$ \\_\\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to evaluate the composite function \\( f(f(-1)) \\) given the piece... |
The number $\text{N}$ is between $9$ and $17$. The [average](https://artofproblemsolving.com/wiki/index.php/Average) of $6$, $10$, and $\text{N}$ could be
$\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$ | To solve the problem, we'll use Python and SymPy to check which of the given average values (8, 10, 12, 14, 16) can be achieved by choosing \( \text{N} \) between 9 and 17.
Here's the step-by-step process:
1. **Define the Average Equation:**
The average of 6, 10, and \( \text{N} \) is given by:
\[
\text{Aver... | [
{
"content": "The number $\\text{N}$ is between $9$ and $17$. The [average](https://artofproblemsolving.com/wiki/index.php/Average) of $6$, $10$, and $\\text{N}$ could be\n$\\text{(A)}\\ 8 \\qquad \\text{(B)}\\ 10 \\qquad \\text{(C)}\\ 12 \\qquad \\text{(D)}\\ 14 \\qquad \\text{(E)}\\ 16$",
"role": "user"
... |
Let $D$ be a point in the plane of triangle $ABC$. If $\overrightarrow{AD}=3\overrightarrow{BD}$ and $\overrightarrow{CD}=λ\overrightarrow{CA}+μ\overrightarrow{CB}$, then $\lambda -\mu =\_\_\_\_\_\_$. | To solve the given problem, we can use vector algebra to find the values of \(\lambda\) and \(\mu\). Let's break this down into manageable steps and solve it using Python and SymPy.
Given:
1. \(\overrightarrow{AD} = 3\overrightarrow{BD}\)
2. \(\overrightarrow{CD} = \lambda \overrightarrow{CA} + \mu \overrightarrow{CB}... | [
{
"content": "Let $D$ be a point in the plane of triangle $ABC$. If $\\overrightarrow{AD}=3\\overrightarrow{BD}$ and $\\overrightarrow{CD}=λ\\overrightarrow{CA}+μ\\overrightarrow{CB}$, then $\\lambda -\\mu =\\_\\_\\_\\_\\_\\_$.",
"role": "user"
},
{
"content": "<think>To solve the given problem, we ... |
The number of solutions in positive integers of $2x+3y=763$ is:
$\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 254\qquad \textbf{(C)}\ 128 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 0$ | To solve the problem of finding the number of solutions in positive integers \((x, y)\) for the equation \(2x + 3y = 763\), we need to follow these steps:
1. **Rewrite the equation**: Express \(y\) in terms of \(x\):
\[ 3y = 763 - 2x \]
\[ y = \frac{763 - 2x}{3} \]
2. **Conditions for \(y\) to be an integer**: For \(... | [
{
"content": "The number of solutions in positive integers of $2x+3y=763$ is:\n$\\textbf{(A)}\\ 255 \\qquad \\textbf{(B)}\\ 254\\qquad \\textbf{(C)}\\ 128 \\qquad \\textbf{(D)}\\ 127 \\qquad \\textbf{(E)}\\ 0$",
"role": "user"
},
{
"content": "<think>To solve the problem of finding the number of sol... |
Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 3AD = 6BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $C$ and $D$? Express your answer as a common fraction.
[asy]
draw((0,0)--(12,.0000000001));
dot((0,0)); dot((4,0)); dot((10,0)); dot(... | To solve the problem, we will begin by setting up an equation using the given relationships between \(AB\), \(AD\), and \(BC\). We need to determine the lengths of all segments involved, understand their positions on the line segment, and then calculate the probability that a randomly selected point lies between \(C\) ... | [
{
"content": "Points $A$, $B$, $C$, and $D$ are located on $\\overline{AB}$ such that $AB = 3AD = 6BC$. If a point is selected at random on $\\overline{AB}$, what is the probability that it is between $C$ and $D$? Express your answer as a common fraction.\n\n[asy]\n\ndraw((0,0)--(12,.0000000001));\ndot((0,0)); ... |
Given that $F$ is the focus of the parabola $C$: $y^{2}=4x$, and a line passing through $F$ with a slope of $\sqrt{3}$ intersects $C$ at points $A$ and $B$. If $|FA| > |FB|$, then the value of $\dfrac{|FA|}{|FB|}$ is \_\_\_\_\_\_. | To solve the problem, we need to determine the ratio \(\dfrac{|FA|}{|FB|}\), where \(F\) is the focus of the parabola \(C\): \(y^2 = 4x\), and \(A\) and \(B\) are points of intersection of the parabola with a line passing through \(F\) with a slope of \(\sqrt{3}\).
The steps are as follows:
1. **Identify the focus**:... | [
{
"content": "Given that $F$ is the focus of the parabola $C$: $y^{2}=4x$, and a line passing through $F$ with a slope of $\\sqrt{3}$ intersects $C$ at points $A$ and $B$. If $|FA| > |FB|$, then the value of $\\dfrac{|FA|}{|FB|}$ is \\_\\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "<think>To sol... |
Petya tells Kolya: "If you think of a quadratic trinomial that has roots and tell me only the leading coefficient and the distance between the roots, then I will guess the ordinate of the vertex in its graph." Kolya thinks that Petya is wrong: after all, to define a quadratic trinomial, you need to know three numbers. ... | To determine who is right, we need to analyze what information is needed to fully define a quadratic trinomial and whether the provided information (the leading coefficient and the distance between the roots) is sufficient to find the ordinate of the vertex.
A quadratic trinomial can be expressed in the standard form:... | [
{
"content": "Petya tells Kolya: \"If you think of a quadratic trinomial that has roots and tell me only the leading coefficient and the distance between the roots, then I will guess the ordinate of the vertex in its graph.\" Kolya thinks that Petya is wrong: after all, to define a quadratic trinomial, you need... |
The sum of the digits of the integer equal to \( 777777777777777^2 - 222222222222223^2 \) is
(A) 148
(B) 84
(C) 74
(D) 69
(E) 79 | To solve the problem, we can use the algebraic identity for the difference of squares, which states:
\[
a^2 - b^2 = (a - b)(a + b)
\]
Given the integers \( a = 777777777777777 \) and \( b = 222222222222223 \):
\[
777777777777777^2 - 222222222222223^2 = (777777777777777 - 222222222222223)(777777777777777 + 2222222222... | [
{
"content": "The sum of the digits of the integer equal to \\( 777777777777777^2 - 222222222222223^2 \\) is\n(A) 148\n(B) 84\n(C) 74\n(D) 69\n(E) 79",
"role": "user"
},
{
"content": "<think>To solve the problem, we can use the algebraic identity for the difference of squares, which states:\n\n\\[\n... |
Given that $f(x)$ is an odd function and an increasing function defined on $\mathbb{R}$. The sequence ${x_n}$ is an arithmetic sequence with a common difference of $2$, and it satisfies $f(x_7) + f(x_8) = 0$. Find the value of $x_{2017}$. | To solve this problem, we need to carefully understand the given conditions and use properties of odd functions and arithmetic sequences.
1. **Properties of Odd Functions:**
Since \(f(x)\) is an odd function, it satisfies the condition \(f(-x) = -f(x)\).
2. **Arithmetic Sequence:**
The sequence \(\{x_n\}\) is a... | [
{
"content": "Given that $f(x)$ is an odd function and an increasing function defined on $\\mathbb{R}$. The sequence ${x_n}$ is an arithmetic sequence with a common difference of $2$, and it satisfies $f(x_7) + f(x_8) = 0$. Find the value of $x_{2017}$.",
"role": "user"
},
{
"content": "<think>To so... |
If $5^x=100$, what is the value of $5^{x+2}$? | We start by solving for \( x \) given the equation \( 5^x = 100 \).
To do this, we take the logarithm of both sides (using natural logarithm, \(\ln\)):
\[ \ln(5^x) = \ln(100) \]
Using the logarithm power rule \(\ln(a^b) = b \ln(a)\), we get:
\[ x \ln(5) = \ln(100) \]
\[ x = \frac{\ln(100)}{\ln(5)} \]
Now we can ca... | [
{
"content": "If $5^x=100$, what is the value of $5^{x+2}$?",
"role": "user"
},
{
"content": "<think>We start by solving for \\( x \\) given the equation \\( 5^x = 100 \\).\n\nTo do this, we take the logarithm of both sides (using natural logarithm, \\(\\ln\\)):\n\n\\[ \\ln(5^x) = \\ln(100) \\]\n\nU... |
Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$ . For example, if $r=32.86$ , then $\{r\}=0.86$ and $[r]=32$ . What is the sum of all positive numbers $r$ satisfying $25\{r\}+[r]=125$ ? | To solve the problem, we need to find all positive numbers \( r \) that satisfy the equation:
\[ 25\{r\} + [r] = 125 \]
where \( [r] \) is the integer part and \( \{r\} \) is the fractional part of \( r \).
Let:
- \( [r] = n \) (an integer)
- \( \{r\} = f \) (a decimal such that \( 0 \leq f < 1 \)).
Substituting t... | [
{
"content": "Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\\{r\\}$ the fractional part of $r$ . For example, if $r=32.86$ , then $\\{r\\}=0.86$ and $[r]=32$ . What is the sum of all positive numbers $r$ satisfying $25\\{r\\}+[r]=125$ ?",
"role": "user"
... |
The function $f(x) = (x+a)(x-2)$ is an even function. Find the real number $a$. | To determine the value of \(a\) for which the function \(f(x)=(x+a)(x-2)\) is an even function, we need to use the property of even functions. A function \(f(x)\) is even if \(f(x) = f(-x)\) for all \(x\).
Let's proceed step-by-step:
1. **Define the function \(f(x)\):**
\[
f(x) = (x+a)(x-2)
\]
2. **Express ... | [
{
"content": "The function $f(x) = (x+a)(x-2)$ is an even function. Find the real number $a$.",
"role": "user"
},
{
"content": "<think>To determine the value of \\(a\\) for which the function \\(f(x)=(x+a)(x-2)\\) is an even function, we need to use the property of even functions. A function \\(f(x)... |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, there is a line $l: \begin{cases}x=t \\ y= \sqrt{5}+2t\end{cases} (t$ is the parameter$)$, with the origin $O$ as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. The polar ... | Let's tackle the problem step by step using Python and SymPy.
### 1. Cartesian Equation of Curve \( C \)
The polar equation given is \(\rho^2 \cos 2\theta + 4 = 0\).
1.1. Convert the polar equation to Cartesian coordinates:
- \(\rho = \sqrt{x^2 + y^2}\)
- \(\theta = \tan^{-1} \left( \frac{y}{x} \right)\)
- \(\cos 2\t... | [
{
"content": "Elective 4-4: Coordinate System and Parametric Equations\n\nIn the Cartesian coordinate system $xOy$, there is a line $l: \\begin{cases}x=t \\\\ y= \\sqrt{5}+2t\\end{cases} (t$ is the parameter$)$, with the origin $O$ as the pole and the positive $x$-axis as the polar axis, a polar coordinate syst... |
What is the least three-digit positive integer that has 2, 5 and 7 as factors? | To solve this problem, we need to find the least three-digit positive integer that has 2, 5, and 7 as factors.
A number that has 2, 5, and 7 as factors must be a multiple of their product.
First, let's calculate the product of 2, 5, and 7:
\[ 2 \times 5 \times 7 = 70 \]
Next, we need to find the smallest three-dig... | [
{
"content": "What is the least three-digit positive integer that has 2, 5 and 7 as factors?",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to find the least three-digit positive integer that has 2, 5, and 7 as factors. \n\nA number that has 2, 5, and 7 as factors must be a ... |
By the first of September, Vlad bought several ballpoint and gel pens. He noticed that if all the purchased pens were gel pens, he would have paid 4 times more than he actually did. Conversely, if all the pens were ballpoint pens, the purchase would have cost him half of the actual amount. How many times more expensive... | Let's denote:
- \( x \) as the number of ballpoint pens
- \( y \) as the number of gel pens
- \( b \) as the price of one ballpoint pen
- \( g \) as the price of one gel pen
- \( A \) as the actual total amount Vlad paid for both types of pens
According to the problem, we have the following equations:
1. If all the pe... | [
{
"content": "By the first of September, Vlad bought several ballpoint and gel pens. He noticed that if all the purchased pens were gel pens, he would have paid 4 times more than he actually did. Conversely, if all the pens were ballpoint pens, the purchase would have cost him half of the actual amount. How man... |
A bag contains 60 colored balls, with 6 balls of each color. Blindly drawing balls from the bag, in order to ensure that there are three balls of the same color among the drawn balls, at least 21 balls need to be drawn. | To solve this problem, we can simulate the process of drawing balls from the bag until we have three balls of the same color. We will do this iteratively using Python to ensure accuracy.
Let’s perform the steps necessary to solve the problem:
1. Simulate drawing balls from the bag.
2. Keep track of the count of each... | [
{
"content": "A bag contains 60 colored balls, with 6 balls of each color. Blindly drawing balls from the bag, in order to ensure that there are three balls of the same color among the drawn balls, at least 21 balls need to be drawn.",
"role": "user"
},
{
"content": "<think>To solve this problem, we... |
Let $O$ be the center of an equilateral triangle $ABC$ with a side length of 10. Point $K$ divides the median $BM$ of triangle $BOC$ in the ratio 3:1, starting from point $B$. Which is larger: $BO$ or $BK$? | To solve this problem, we need to determine the lengths of \(BO\) and \(BK\) and compare them. Let's break it down step by step.
1. **Determine the Position of Points:**
- \(O\) is the centroid of the equilateral triangle \(ABC\).
- \(M\) is the midpoint of side \(AC\).
- Point \(K\) divides the median \(BM\)... | [
{
"content": "Let $O$ be the center of an equilateral triangle $ABC$ with a side length of 10. Point $K$ divides the median $BM$ of triangle $BOC$ in the ratio 3:1, starting from point $B$. Which is larger: $BO$ or $BK$?",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to dete... |
Find the dot product of $\begin{pmatrix} 3 \\ -4 \\ -3 \end{pmatrix}$ and $\begin{pmatrix} -5 \\ 2 \\ 1 \end{pmatrix}.$ | ```python
import numpy as np
# Define the vectors
vector1 = np.array([3, -4, -3])
vector2 = np.array([-5, 2, 1])
# Calculate the dot product
dot_product = np.dot(vector1, vector2)
print(dot_product)
```
```output
-26
```
The dot product of \(\begin{pmatrix} 3 \\ -4 \\ -3 \end{pmatrix}\) and \(\begin{pmatrix} -5 \\ 2 ... | [
{
"content": "Find the dot product of $\\begin{pmatrix} 3 \\\\ -4 \\\\ -3 \\end{pmatrix}$ and $\\begin{pmatrix} -5 \\\\ 2 \\\\ 1 \\end{pmatrix}.$",
"role": "user"
},
{
"content": "<think>```python\nimport numpy as np\n\n# Define the vectors\nvector1 = np.array([3, -4, -3])\nvector2 = np.array([-5, 2... |
Find the focal length of the hyperbola $\frac{x^{2}}{m^{2}+12} - \frac{y^{2}}{4-m^{2}} = 1$. | To find the focal length of the hyperbola \(\frac{x^{2}}{m^{2}+12} - \frac{y^{2}}{4-m^{2}} = 1\), we need to determine the values of \(a\) and \(b\) such that the equation is in the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). The focal length \(2c\) is found using the relationship \(c^2 = a^2 + b^2\).
Her... | [
{
"content": "Find the focal length of the hyperbola $\\frac{x^{2}}{m^{2}+12} - \\frac{y^{2}}{4-m^{2}} = 1$.",
"role": "user"
},
{
"content": "<think>To find the focal length of the hyperbola \\(\\frac{x^{2}}{m^{2}+12} - \\frac{y^{2}}{4-m^{2}} = 1\\), we need to determine the values of \\(a\\) and \... |
Consider the set \( S = \{1, 2, \cdots, 15\} \). Let \( A = \{a_1, a_2, a_3\} \) be a subset of \( S \), with \( (a_1, a_2, a_3) \) satisfying:
\[
1 \leq a_1 < a_2 < a_3 \leq 15, \quad a_3 - a_2 \leq 6.
\]
Determine the number of subsets \( A \) that satisfy these conditions. | To determine the number of subsets \(A = \{a_1, a_2, a_3\}\) from \(S = \{1, 2, \cdots, 15\}\) that satisfy the conditions:
- \(1 \leq a_1 < a_2 < a_3 \leq 15\)
- \(a_3 - a_2 \leq 6\)
We can use a computational approach to iterate through all possible combinations of \(a_1, a_2,\) and \(a_3\) and count the valid subs... | [
{
"content": "Consider the set \\( S = \\{1, 2, \\cdots, 15\\} \\). Let \\( A = \\{a_1, a_2, a_3\\} \\) be a subset of \\( S \\), with \\( (a_1, a_2, a_3) \\) satisfying:\n\\[\n1 \\leq a_1 < a_2 < a_3 \\leq 15, \\quad a_3 - a_2 \\leq 6.\n\\]\nDetermine the number of subsets \\( A \\) that satisfy these conditio... |
In order to understand the academic level of students in a certain high school, the education department conducted stratified sampling by grade, selecting 100 students from a total of 2400 students in the school. If there are 840 students in the first grade of the school, then the number of first-grade students that sh... | To solve this problem, we need to determine how many first-grade students should be selected in the stratified sampling to maintain the proportional representation of first graders in the sample.
Here's the step-by-step breakdown:
1. **Determine the proportion of first-grade students in the entire school:**
The to... | [
{
"content": "In order to understand the academic level of students in a certain high school, the education department conducted stratified sampling by grade, selecting 100 students from a total of 2400 students in the school. If there are 840 students in the first grade of the school, then the number of first-... |
The diagram shows an arc \( PQ \) of a circle with center \( O \) and radius 8. Angle \( QOP \) is a right angle, the point \( M \) is the midpoint of \( OP \), and \( N \) lies on the arc \( PQ \) so that \( MN \) is perpendicular to \( OP \). Which of the following is closest to the length of the perimeter of triangl... | To solve this problem, we can calculate the perimeter of triangle \( PNM \) step-by-step. Here are the steps:
1. **Identify important points and coordinates:**
- Since \( \angle QOP \) is a right angle and \( O \) is the center of the circle with radius 8, points \( P \) and \( Q \) can be placed on the circle suc... | [
{
"content": "The diagram shows an arc \\( PQ \\) of a circle with center \\( O \\) and radius 8. Angle \\( QOP \\) is a right angle, the point \\( M \\) is the midpoint of \\( OP \\), and \\( N \\) lies on the arc \\( PQ \\) so that \\( MN \\) is perpendicular to \\( OP \\). Which of the following is closest t... |
Find a four-digit number that is a perfect square, knowing that the first two digits, as well as the last two digits, are each equal to each other. | To solve this problem, we need to find a four-digit number that is a perfect square and whose format is ABBA, where A and B are digits.
Let's break down the steps:
1. A four-digit number ranges from 1000 to 9999.
2. We need to check if the number is a perfect square.
3. The condition that the format is ABBA will also ... | [
{
"content": "Find a four-digit number that is a perfect square, knowing that the first two digits, as well as the last two digits, are each equal to each other.",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to find a four-digit number that is a perfect square and whose for... |
A coat is priced at 800 yuan, and a pair of pants is 40% cheaper than a coat. The selling price of a pair of pants is yuan. | To solve this problem, we'll follow these steps:
1. **Understand the Problem Statement:**
- The price of a coat is 800 yuan.
- A pair of pants is 40% cheaper than a coat.
2. **Calculate the Selling Price of Pants:**
- The selling price of a pair of pants is 60% of the price of a coat (since it is 40% cheaper... | [
{
"content": "A coat is priced at 800 yuan, and a pair of pants is 40% cheaper than a coat. The selling price of a pair of pants is yuan.",
"role": "user"
},
{
"content": "<think>To solve this problem, we'll follow these steps:\n\n1. **Understand the Problem Statement:**\n - The price of a coa... |
$C_{99}^{98} + C_{99}^{97} =$ （Answer in digits） | To solve the problem \( C_{99}^{98} + C_{99}^{97} \), we need to understand that these are binomial coefficients. Binomial coefficients are given by:
\[
C_n^k = \frac{n!}{k!(n-k)!}
\]
We can use the properties of binomial coefficients to simplify the expression:
\[
C_{99}^{98} + C_{99}^{97}
\]
Using the Pascal's ru... | [
{
"content": "$C_{99}^{98} + C_{99}^{97} =$ （Answer in digits）",
"role": "user"
},
{
"content": "<think>To solve the problem \\( C_{99}^{98} + C_{99}^{97} \\), we need to understand that these are binomial coefficients. Binomial coefficients are given by:\n\n\\[\nC_n^k = \\frac{n!}{k!(n-k)!}\n\\]\n\... |
Given \( S_{n}=1+2+3+\cdots+n \) for \( n \in \mathbf{N} \), find the maximum value of the function \( f(n)=\frac{S_{n}}{(n+32) S_{n+1}} \). | To solve this problem, we need to find the maximum value of the function \( f(n)=\frac{S_{n}}{(n+32) S_{n+1}} \), where \( S_{n} = 1 + 2 + 3 + \cdots + n \).
Let's proceed with the following steps:
1. Calculate the sum \( S_n \) and \( S_{n+1} \).
2. Substitute \( S_n \) and \( S_{n+1} \) into the function \( f(n) \)... | [
{
"content": "Given \\( S_{n}=1+2+3+\\cdots+n \\) for \\( n \\in \\mathbf{N} \\), find the maximum value of the function \\( f(n)=\\frac{S_{n}}{(n+32) S_{n+1}} \\).",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to find the maximum value of the function \\( f(n)=\\frac{S_{n}... |
Calculate:<br/>$(1)-{3}^{3}×(-\frac{1}{3})+|-2|÷{(-\frac{1}{2})}^{2}$;<br/>$(2)7-12×(\frac{2}{3}-\frac{3}{4}+\frac{5}{6})$. | Let's break down the problem into two parts and solve each one step-by-step using Python and SymPy.
### Part 1:
\[
-(3^3) \times \left(-\frac{1}{3}\right) + |-2| \div \left(-\frac{1}{2}\right)^2
\]
### Part 2:
\[
7 - 12 \times \left(\frac{2}{3} - \frac{3}{4} + \frac{5}{6}\right)
\]
We'll use SymPy to compute these e... | [
{
"content": "Calculate:<br/>$(1)-{3}^{3}×(-\\frac{1}{3})+|-2|÷{(-\\frac{1}{2})}^{2}$;<br/>$(2)7-12×(\\frac{2}{3}-\\frac{3}{4}+\\frac{5}{6})$.",
"role": "user"
},
{
"content": "<think>Let's break down the problem into two parts and solve each one step-by-step using Python and SymPy.\n\n### Part 1:\n... |
Fill a $3 \times 3$ grid with three 1s, three 2s, and three 3s, ensuring that each row and column contains no repeated numbers. How many different ways are there to fill the grid? | Sure, let's break down this problem into smaller, more manageable steps and implement them in Python using the sympy library where applicable.
1. **Understand the Problem:**
We need to fill a \(3 \times 3\) grid with three 1s, three 2s, and three 3s, ensuring that each row and column contains no repeated numbers. ... | [
{
"content": "Fill a $3 \\times 3$ grid with three 1s, three 2s, and three 3s, ensuring that each row and column contains no repeated numbers. How many different ways are there to fill the grid?",
"role": "user"
},
{
"content": "<think>Sure, let's break down this problem into smaller, more manageabl... |
If $a$, $b$, $c$, $d$ are integers, $b$ is a positive integer, and satisfy $a+b=c$, $b+c=d$, $c+d=a$, then the maximum value of $a-2b+3c-4d$ is ______. | To solve the given problem, we need to find the maximum value of the expression \(a - 2b + 3c - 4d\) under the conditions that \( a, b, c, d \) are integers, \( b \) is a positive integer, and they satisfy the equations:
1. \(a + b = c\)
2. \(b + c = d\)
3. \(c + d = a\)
Let's break it down step-by-step:
1. **Expres... | [
{
"content": "If $a$, $b$, $c$, $d$ are integers, $b$ is a positive integer, and satisfy $a+b=c$, $b+c=d$, $c+d=a$, then the maximum value of $a-2b+3c-4d$ is ______.",
"role": "user"
},
{
"content": "<think>To solve the given problem, we need to find the maximum value of the expression \\(a - 2b + 3... |
A man travels 10 km at a speed of 4 km/h and another 10 km at a speed of 6 km/h. If the average speed of the whole journey is \( x \) km/h, find \( x \). | To find the average speed for the whole journey, we need to use the formula for the average speed, which is the total distance traveled divided by the total time taken.
Here are the steps:
1. Calculate the total distance traveled.
2. Calculate the time taken for each part of the journey.
3. Sum the times to get the t... | [
{
"content": "A man travels 10 km at a speed of 4 km/h and another 10 km at a speed of 6 km/h. If the average speed of the whole journey is \\( x \\) km/h, find \\( x \\).",
"role": "user"
},
{
"content": "<think>To find the average speed for the whole journey, we need to use the formula for the ave... |
Let $\{a_i\}_{i=0}^\infty$ be a sequence of real numbers such that \[\sum_{n=1}^\infty\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\cdots\] for all $|x|<1$ . Find $a_{1000}$ .
*Proposed by David Altizio* | To solve this problem, we start by analyzing the given series and re-expressing it in a form that will help us identify the coefficients \(a_n\) of the power series.
Given:
\[
\sum_{n=1}^\infty \frac{x^n}{1-x^n} = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots
\]
We will first expand \(\frac{x^n}{1-x^n}\) as a geometric se... | [
{
"content": "Let $\\{a_i\\}_{i=0}^\\infty$ be a sequence of real numbers such that \\[\\sum_{n=1}^\\infty\\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\\cdots\\] for all $|x|<1$ . Find $a_{1000}$ .\n\n*Proposed by David Altizio*",
"role": "user"
},
{
"content": "<think>To solve this problem, we ... |
Given two skew lines a and b forming an angle of 50°, and P is a fixed point in space, then the number of lines passing through point P and forming an angle of 30° with both lines a and b is exactly $\boxed{2}$. | To solve this problem using reasoning and Python code, let's go through the following steps:
1. **Understand the Problem:**
We need to determine how many lines pass through a fixed point \(P\) and form an angle of 30° with both of the given skew lines \(a\) and \(b\).
2. **Geometric Interpretation:**
When a lin... | [
{
"content": "Given two skew lines a and b forming an angle of 50°, and P is a fixed point in space, then the number of lines passing through point P and forming an angle of 30° with both lines a and b is exactly $\\boxed{2}$.",
"role": "user"
},
{
"content": "<think>To solve this problem using reas... |
Answer the following questions as required.
1. Given $10^{m}=12$ and $10^{n}=3$, find the value of $10^{m-n}$.
2. If $a+3b=3$, find the value of $3^{a}\times 27^{b}$.
3. Given $8\times 2^{m}\div 16^{m}=2^{6}$, find the value of $m$. | Sure, let's solve each part step-by-step using Python and SymPy.
### Part 1: Given \(10^{m} = 12\) and \(10^{n} = 3\), find the value of \(10^{m-n}\).
To find \(10^{m-n}\):
\[ 10^{m-n} = \frac{10^m}{10^n} = \frac{12}{3} = 4 \]
### Part 2: If \(a + 3b = 3\), find the value of \(3^{a} \times 27^{b}\).
Note that \(27\... | [
{
"content": "Answer the following questions as required.\n1. Given $10^{m}=12$ and $10^{n}=3$, find the value of $10^{m-n}$.\n2. If $a+3b=3$, find the value of $3^{a}\\times 27^{b}$.\n3. Given $8\\times 2^{m}\\div 16^{m}=2^{6}$, find the value of $m$.",
"role": "user"
},
{
"content": "<think>Sure, ... |
Let \( a_{1}, a_{2}, \cdots, a_{2006} \) be 2006 positive integers (they can be the same) such that \( \frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}} \) are all different from each other. What is the minimum number of distinct numbers in \( a_{1}, a_{2}, \cdots, a_{2006} \)? | To solve the problem, we need to find the minimum number of distinct values in a sequence of positive integers \( a_1, a_2, \ldots, a_{2006} \) such that the ratios \( \frac{a_1}{a_2}, \frac{a_2}{a_3}, \cdots, \frac{a_{2005}}{a_{2006}} \) are all different.
Let's denote the sequence by \( a_i \). Since each ratio \( \... | [
{
"content": "Let \\( a_{1}, a_{2}, \\cdots, a_{2006} \\) be 2006 positive integers (they can be the same) such that \\( \\frac{a_{1}}{a_{2}}, \\frac{a_{2}}{a_{3}}, \\cdots, \\frac{a_{2005}}{a_{2006}} \\) are all different from each other. What is the minimum number of distinct numbers in \\( a_{1}, a_{2}, \\cd... |
If the equation $x^{m+1}-2=1$ with respect to $x$ is a linear equation, then the value of $m$ is ______. | To determine the value of \( m \) such that the equation \( x^{m+1} - 2 = 1 \) is a linear equation with respect to \( x \), we need to analyze what makes the equation linear.
A linear equation in \( x \) is of the form \( ax + b = 0 \), where \( a \) and \( b \) are constants. For an equation to be linear in \( x \)... | [
{
"content": "If the equation $x^{m+1}-2=1$ with respect to $x$ is a linear equation, then the value of $m$ is ______.",
"role": "user"
},
{
"content": "<think>To determine the value of \\( m \\) such that the equation \\( x^{m+1} - 2 = 1 \\) is a linear equation with respect to \\( x \\), we need t... |
For what smallest natural number \( a \) are there exactly 50 perfect squares in the numerical interval \( (a, 3a) \)? | To find the smallest natural number \( a \) for which there are exactly 50 perfect squares in the interval \( (a, 3a) \), we need to determine how many perfect squares lie between \( a \) and \( 3a \).
A perfect square is of the form \( n^2 \), where \( n \) is an integer. To find the range of \( n \), we focus on the... | [
{
"content": "For what smallest natural number \\( a \\) are there exactly 50 perfect squares in the numerical interval \\( (a, 3a) \\)?",
"role": "user"
},
{
"content": "<think>To find the smallest natural number \\( a \\) for which there are exactly 50 perfect squares in the interval \\( (a, 3a) \... |
The motto "Green waters and green mountains are golden mountains and silver mountains" encourages Hope High School to organize tree planting activities for students every year. This year, the school bought a batch of tree seedlings and formed tree planting teams. If each team plants 5 trees, there will be 6 extra seedl... | To solve the problem, we need to set up a system of equations based on the information given:
1. If each team plants 5 trees, there will be 6 extra seedlings left.
\[
N = 5T + 6
\]
where \(N\) is the total number of seedlings and \(T\) is the number of teams.
2. If each team plants 6 trees, there will be ... | [
{
"content": "The motto \"Green waters and green mountains are golden mountains and silver mountains\" encourages Hope High School to organize tree planting activities for students every year. This year, the school bought a batch of tree seedlings and formed tree planting teams. If each team plants 5 trees, the... |
How many 4-digit positive multiples of 4 can be formed from the digits 0, 1, 2, 3, 4, 5, 6 such that each digit appears without repetition? | To solve the problem, let's break it down into smaller steps and then use Python code to verify the solution.
1. **Understanding the Conditions**:
- The number must be a 4-digit number.
- It must be a multiple of 4.
- Each digit must be from the set {0, 1, 2, 3, 4, 5, 6} without repetition.
2. **Rule for Mul... | [
{
"content": "How many 4-digit positive multiples of 4 can be formed from the digits 0, 1, 2, 3, 4, 5, 6 such that each digit appears without repetition?",
"role": "user"
},
{
"content": "<think>To solve the problem, let's break it down into smaller steps and then use Python code to verify the solut... |
Simplify first, then evaluate: $-7x+2(x^{2}-1)-(2x^{2}-x+3)$, where $x=1$. | Certainly! Let's break down the problem step-by-step and utilize Python with SymPy to simplify and evaluate the expression.
1. **Expression Simplification:**
We start with the expression \(-7x + 2(x^2 - 1) - (2x^2 - x + 3)\).
2. **Substitute \(x=1\) into the simplified expression:**
Once simplified, we will sub... | [
{
"content": "Simplify first, then evaluate: $-7x+2(x^{2}-1)-(2x^{2}-x+3)$, where $x=1$.",
"role": "user"
},
{
"content": "<think>Certainly! Let's break down the problem step-by-step and utilize Python with SymPy to simplify and evaluate the expression.\n\n1. **Expression Simplification:**\n We st... |
How many three-digit numbers are there in which the digits 1, 2, and 3 each appear exactly once? | To solve the problem of finding how many three-digit numbers can be formed where the digits 1, 2, and 3 each appear exactly once, we can use the concept of permutations. Essentially, we want to count all the unique arrangements of the digits 1, 2, and 3.
We will do this by performing the following steps:
1. Generate ... | [
{
"content": "How many three-digit numbers are there in which the digits 1, 2, and 3 each appear exactly once?",
"role": "user"
},
{
"content": "<think>To solve the problem of finding how many three-digit numbers can be formed where the digits 1, 2, and 3 each appear exactly once, we can use the con... |
The least common multiple of two numbers that do not divide each other is 90, and their greatest common divisor is 6. Find these numbers. | To solve the problem, we will utilize the relationship between the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) of two numbers \(a\) and \(b\). This relationship is given by the formula:
\[
\text{LCM}(a, b) \times \text{GCD}(a, b) = a \times b
\]
We know that:
\[
\text{LCM}(a, b) = 90 \quad \tex... | [
{
"content": "The least common multiple of two numbers that do not divide each other is 90, and their greatest common divisor is 6. Find these numbers.",
"role": "user"
},
{
"content": "<think>To solve the problem, we will utilize the relationship between the Least Common Multiple (LCM) and the Grea... |
Let $f(x)$ be a function defined on the set of real numbers $R$ and satisfying the following conditions:
$$
f(10 + x) = f(10 - x), \quad f(20 - x) = -f(20 + x) .
$$
Then $f(x)$ is:
(A) An even function and a periodic function.
(B) An even function but not a periodic function.
(C) An odd function and a periodic functio... | To solve this problem, we need to determine the properties of the function \( f(x) \) based on the given conditions:
1. \( f(10 + x) = f(10 - x) \)
2. \( f(20 - x) = -f(20 + x) \)
We will analyze these conditions step-by-step to determine whether \( f(x) \) is an even function, an odd function, and/or a periodic funct... | [
{
"content": "Let $f(x)$ be a function defined on the set of real numbers $R$ and satisfying the following conditions:\n$$\nf(10 + x) = f(10 - x), \\quad f(20 - x) = -f(20 + x) .\n$$\n\nThen $f(x)$ is:\n(A) An even function and a periodic function.\n(B) An even function but not a periodic function.\n(C) An odd ... |
Calculate: $-{3}^{2}+|-5|-18\times{(-\frac{1}{3})}^{2}$. | To solve the problem \(-3^2 + |-5| - 18 \times \left(-\frac{1}{3}\right)^2\), we can follow these steps:
1. **Simplify the individual components:**
- \( -3^2 \) : Calculate the square of 3 and then apply the negative sign.
- \( |-5| \) : Calculate the absolute value of -5.
- \( \left(-\frac{1}{3}\right)^2 \) ... | [
{
"content": "Calculate: $-{3}^{2}+|-5|-18\\times{(-\\frac{1}{3})}^{2}$.",
"role": "user"
},
{
"content": "<think>To solve the problem \\(-3^2 + |-5| - 18 \\times \\left(-\\frac{1}{3}\\right)^2\\), we can follow these steps:\n\n1. **Simplify the individual components:**\n - \\( -3^2 \\) : Calculat... |
Eva the magician has a deck of 100 cards, numbered from 1 to 100, with each card displaying the same number on both of its sides. Initially, she has shuffled her cards arbitrarily and stacked them. Then she performs the following operations: if the card with number $k$ is on the top of the stack, she takes the first $k... | To determine if any sequence of such operations is necessarily finite, we need to analyze the operation's effect on the deck and identify if and how it leads to a termination condition.
Here's a step-by-step process:
1. **Define the Operation:**
- If the card with number \( k \) is on the top of the stack, take th... | [
{
"content": "Eva the magician has a deck of 100 cards, numbered from 1 to 100, with each card displaying the same number on both of its sides. Initially, she has shuffled her cards arbitrarily and stacked them. Then she performs the following operations: if the card with number $k$ is on the top of the stack, ... |
If the sum of the first three terms of an arithmetic sequence $\{a_n\}$ is $S_3 = 9$ and $a_1 = 1$, then $a_2$ equals ____. | To solve this problem, we need to find the second term \(a_2\) of an arithmetic sequence given that the sum of the first three terms is 9 and the first term \(a_1\) is 1.
Steps to solve the problem:
1. Write down the general formula for the \(n\)-th term of an arithmetic sequence.
2. Use the given information to set u... | [
{
"content": "If the sum of the first three terms of an arithmetic sequence $\\{a_n\\}$ is $S_3 = 9$ and $a_1 = 1$, then $a_2$ equals ____.",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to find the second term \\(a_2\\) of an arithmetic sequence given that the sum of the fi... |
From the numbers 1, 2, 3, and 4, form a four-digit number where the first digit is 1 and only two digits are the same. There are $\boxed{\_\_\_\_}$ such four-digit numbers. | To solve this problem, we need to count how many four-digit numbers can be formed using the digits 1, 2, 3, and 4, where the first digit is 1 and exactly two of the digits are the same. Let's break this problem down into smaller steps:
1. **Identify the format of the number:**
Since the first digit is fixed at 1, t... | [
{
"content": "From the numbers 1, 2, 3, and 4, form a four-digit number where the first digit is 1 and only two digits are the same. There are $\\boxed{\\_\\_\\_\\_}$ such four-digit numbers.",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to count how many four-digit numbers... |
Three cans of juice fill $\frac{2}{3}$ of a one-litre jug. How many cans of juice are needed to completely fill 8 one-litre jugs? | First, we need to determine the volume of juice in one can by using the information that three cans of juice fill \(\frac{2}{3}\) of a one-litre jug.
Let's denote \(V_{\text{one can}}\) as the volume of juice in one can.
Given:
\[ 3 \times V_{\text{one can}} = \frac{2}{3} \text{ litres} \]
From this equation, we can... | [
{
"content": "Three cans of juice fill $\\frac{2}{3}$ of a one-litre jug. How many cans of juice are needed to completely fill 8 one-litre jugs?",
"role": "user"
},
{
"content": "<think>First, we need to determine the volume of juice in one can by using the information that three cans of juice fill ... |
The number $101$ is the smallest three-digit palindromic prime. What is the second-smallest one? | To find the second-smallest three-digit palindromic prime number, let's first understand the definition:
1. A three-digit palindromic number reads the same forward and backward. Thus, it has the form \( \overline{aba} \), where \( a \) and \( b \) are digits and \( a \neq 0 \).
2. A prime number is a natural number gr... | [
{
"content": "The number $101$ is the smallest three-digit palindromic prime. What is the second-smallest one?",
"role": "user"
},
{
"content": "<think>To find the second-smallest three-digit palindromic prime number, let's first understand the definition:\n\n1. A three-digit palindromic number read... |
Jeff bought 6 pairs of shoes and 4 jerseys for $560. Jerseys cost 1/4 price of one pair of shoes. Find the shoe's price total price. | Let's break down and solve the problem step by step and then use a Python code to verify our calculations.
1. Let's denote:
- The price of one pair of shoes as \( S \).
- The price of one jersey as \( J \).
2. Given:
- Jeff bought 6 pairs of shoes and 4 jerseys for $560.
- Jerseys cost \( \frac{1}{4} \) o... | [
{
"content": "Jeff bought 6 pairs of shoes and 4 jerseys for $560. Jerseys cost 1/4 price of one pair of shoes. Find the shoe's price total price.",
"role": "user"
},
{
"content": "<think>Let's break down and solve the problem step by step and then use a Python code to verify our calculations.\n\n1.... |
Coordinate System and Parametric Equations.
Given the curve $C$:
$$
\begin{cases}
x=\cos \theta \\
y=\sin \theta
\end{cases}
(\theta \text{ is the parameter}).
$$
$(1)$ Convert the parametric equations of $C$ into a standard equation;
$(2)$ If the coordinates of each point on $C$ undergo a scaling transformation... | To solve the given problem, we'll follow these steps:
1. **Convert the parametric equations of the curve \(C\) into a standard equation.**
2. **Apply the scaling transformation to obtain the curve \(C'\).**
3. **Find the maximum value of the product of the distances from any point on \(C'\) to the two coordinate axes.... | [
{
"content": "Coordinate System and Parametric Equations. \nGiven the curve $C$: \n$$\n\\begin{cases}\nx=\\cos \\theta \\\\\ny=\\sin \\theta\n\\end{cases}\n(\\theta \\text{ is the parameter}).\n$$\n\n$(1)$ Convert the parametric equations of $C$ into a standard equation; \n$(2)$ If the coordinates of each po... |
Given $f(x)=|x-1|-|x|$, find the value of $f[f(\frac{1}{2})]$. | To solve the problem, we need to find the value of \( f(f(\frac{1}{2})) \) where \( f(x) = |x-1| - |x| \).
Let's break down the steps:
1. **Define the function \( f(x) \):**
\[
f(x) = |x - 1| - |x|
\]
2. **Compute \( f(\frac{1}{2}) \):**
Substitute \( x = \frac{1}{2} \) into the function.
3. **Compute \... | [
{
"content": "Given $f(x)=|x-1|-|x|$, find the value of $f[f(\\frac{1}{2})]$.",
"role": "user"
},
{
"content": "<think>To solve the problem, we need to find the value of \\( f(f(\\frac{1}{2})) \\) where \\( f(x) = |x-1| - |x| \\).\n\nLet's break down the steps:\n\n1. **Define the function \\( f(x) \... |
In the binomial expansion of $$(2x^{2}- \frac {1}{x})^{6}$$, the constant term is \_\_\_\_\_. | To find the constant term in the binomial expansion of \((2x^2 - \frac {1}{x})^6\), we need to identify the term in the expansion where the power of \(x\) is zero.
The binomial expansion of \((a + b)^n\) is given by:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
For the given expression \((2x^2 - \frac {... | [
{
"content": "In the binomial expansion of $$(2x^{2}- \\frac {1}{x})^{6}$$, the constant term is \\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "<think>To find the constant term in the binomial expansion of \\((2x^2 - \\frac {1}{x})^6\\), we need to identify the term in the expansion where the po... |
Oleg has 550 rubles and wants to give his mother tulips for March 8th. The number of tulips must be odd, and each tulip must be of a different shade. In the store, each tulip costs 49 rubles, and there are eleven different shades available. How many ways are there for Oleg to give the flowers to his mother? Provide the... | To solve this problem, we need to identify how many ways Oleg can buy an odd number of tulips such that the total cost does not exceed 550 rubles. Each tulip costs 49 rubles, and there are 11 different shades available. This limits Oleg to buying tulips where each has a different shade.
First, let's determine the maxi... | [
{
"content": "Oleg has 550 rubles and wants to give his mother tulips for March 8th. The number of tulips must be odd, and each tulip must be of a different shade. In the store, each tulip costs 49 rubles, and there are eleven different shades available. How many ways are there for Oleg to give the flowers to h... |
Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.
<details><summary>Note</summary>Note. Here (and generally in MathLinks) natural numbers supposed to be positive.</details> | To determine the natural number \( n \) such that \( n! \) (n factorial) ends in exactly 1987 zeros, we need to understand how trailing zeros in a factorial are produced.
A trailing zero is produced when 10 is a factor of \( n! \). Since \( 10 = 2 \times 5 \), we need to count the number of pairs of factors of 2 and ... | [
{
"content": "Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.\n\n<details><summary>Note</summary>Note. Here (and generally in MathLinks) natural numbers supposed to be positive.</details>",
"role": "user"
},
{
"content": "<think>To dete... |
Given an arithmetic sequence $\{a_n\}$, where $a_3+a_7=8$, find the sum of the first $9$ terms, $S_9=$ ______. | To solve this problem, we need to use the properties of an arithmetic sequence.
1. **Understand the Arithmetic Sequence:**
The general form of an arithmetic sequence is given by:
\[
a_n = a + (n-1)d
\]
where \(a\) is the first term and \(d\) is the common difference.
2. **Use the Given Information:**
... | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$, where $a_3+a_7=8$, find the sum of the first $9$ terms, $S_9=$ ______.",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to use the properties of an arithmetic sequence.\n\n1. **Understand the Arithmetic Sequence:**\n Th... |
The number $16^4+16^2+1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.
*2018 CCA Math Bonanza Lightning Round #3.1* | To determine the sum of the four distinct prime numbers that divide \(16^4 + 16^2 + 1\), we need to simplify the expression and factorize it.
Let's start simplifying:
\[
16^4 + 16^2 + 1
\]
We will substitute \( x = 16^2 \):
\[
x^2 + x + 1 \Rightarrow 16^4 + 16^2 + 1
\]
Now we need to factorize \( x^2 + x + 1 \).... | [
{
"content": "The number $16^4+16^2+1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.\n\n*2018 CCA Math Bonanza Lightning Round #3.1*",
"role": "user"
},
{
"content": "<think>To determine the sum of the four distinct prime numbers that divide \\(16^4 + 16^2 + 1\... |
Let $x$ be a value such that $8x^2 + 7x - 1 = 0$ and $24x^2+53x-7 = 0.$ What is the value of $x$? Express your answer as a simplified common fraction. | To solve the problem, we need to find the value of \( x \) that satisfies both quadratic equations:
1. \( 8x^2 + 7x - 1 = 0 \)
2. \( 24x^2 + 53x - 7 = 0 \)
Finding \( x \) involves solving these simultaneous quadratic equations.
First, let’s write a Python script to solve these equations.
We will use the `sympy` l... | [
{
"content": "Let $x$ be a value such that $8x^2 + 7x - 1 = 0$ and $24x^2+53x-7 = 0.$ What is the value of $x$? Express your answer as a simplified common fraction.",
"role": "user"
},
{
"content": "<think>To solve the problem, we need to find the value of \\( x \\) that satisfies both quadratic equ... |
Find the expected value of a random variable \( X \) that is uniformly distributed in the interval \((a, b)\). | To find the expected value of a random variable \(X\) that is uniformly distributed in the interval \((a, b)\), we can use the following steps:
1. **Understand the Uniform Distribution:**
A random variable \(X\) uniformly distributed in the interval \((a, b)\) has the probability density function (pdf):
\[
f(... | [
{
"content": "Find the expected value of a random variable \\( X \\) that is uniformly distributed in the interval \\((a, b)\\).",
"role": "user"
},
{
"content": "<think>To find the expected value of a random variable \\(X\\) that is uniformly distributed in the interval \\((a, b)\\), we can use the... |
Let the sequence \\(\{a_n\}\) be defined by \\(a_n = 2n - 7 (n \in \mathbb{N}^*)\), then the value of \\(|a_1| + |a_2| + \ldots + |a_{15}| = \) ______. | To solve the problem, we need to calculate the sum of the absolute values of the first 15 terms of the sequence defined by \(a_n = 2n - 7\).
Let's break this down into manageable steps:
1. **Generate the sequence \(a_n = 2n - 7\) for \(n = 1\) to \(n = 15\).**
2. **Calculate the absolute value of each term in the seq... | [
{
"content": "Let the sequence \\\\(\\{a_n\\}\\) be defined by \\\\(a_n = 2n - 7 (n \\in \\mathbb{N}^*)\\), then the value of \\\\(|a_1| + |a_2| + \\ldots + |a_{15}| = \\) ______.",
"role": "user"
},
{
"content": "<think>To solve the problem, we need to calculate the sum of the absolute values of th... |
If $a=\sqrt{5}+1$ and $b=\sqrt{5}-1$, find the value of the following algebraic expressions:
1. $a^{2}b+ab^{2}$
2. $a^{2}-ab+b^{2}$. | Let's solve each algebraic expression step-by-step using Python and SymPy.
1. **Expression \(a^{2}b + ab^{2}\)**:
Given \(a = \sqrt{5} + 1\) and \(b = \sqrt{5} - 1\), we need to find the value of \(a^{2}b + ab^{2}\).
2. **Expression \(a^{2} - ab + b^{2}\)**:
We will use the same values for \(a\) and \(b\) to fi... | [
{
"content": "If $a=\\sqrt{5}+1$ and $b=\\sqrt{5}-1$, find the value of the following algebraic expressions:\n1. $a^{2}b+ab^{2}$\n2. $a^{2}-ab+b^{2}$.",
"role": "user"
},
{
"content": "<think>Let's solve each algebraic expression step-by-step using Python and SymPy.\n\n1. **Expression \\(a^{2}b + ab... |
What is the smallest positive integer $n$ such that
$$
\sqrt{n}-\sqrt{n-1}<0.01 ?
$$ | To solve the problem of finding the smallest positive integer \( n \) such that:
$$
\sqrt{n} - \sqrt{n-1} < 0.01
$$
we can follow these steps:
1. Re-arrange and simplify the inequality to find an approximate range for \( n \).
2. Create a Python function that will iterate over positive integers until the condition i... | [
{
"content": "What is the smallest positive integer $n$ such that\n\n$$\n\\sqrt{n}-\\sqrt{n-1}<0.01 ?\n$$",
"role": "user"
},
{
"content": "<think>To solve the problem of finding the smallest positive integer \\( n \\) such that:\n\n$$\n\\sqrt{n} - \\sqrt{n-1} < 0.01\n$$\n\nwe can follow these steps... |
Convert the binary number $10011_{(2)}$ to a decimal number equals \_\_\_\_\_. | To convert the binary number \(10011_{(2)}\) to a decimal number, you can follow these steps:
1. Understand that each digit in a binary number represents a power of 2, starting from the rightmost digit which represents \(2^0\).
2. Sum the products of each binary digit with its corresponding power of 2.
The binary num... | [
{
"content": "Convert the binary number $10011_{(2)}$ to a decimal number equals \\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "<think>To convert the binary number \\(10011_{(2)}\\) to a decimal number, you can follow these steps:\n\n1. Understand that each digit in a binary number represents a ... |
Let's assume that Olga Pavlovna has \( x \) liters, and Maria Petrovna has \( y \) liters of jam left. The numbers \( x \) and \( y \) are randomly and independently chosen from the interval from 0 to 1. Consider a random point with coordinates \((x, y)\) from the unit square \( F \). The event \( A \) "Olga Pavlovna a... | To solve the problem, we need to compute the area of the region defined by the inequality \(1 \leq x + y < 1.5\) within the unit square \(F\). The probability of the event \(A\) is the ratio of the area of this region \(S_A\) to the area of the unit square \(S_F\).
Let's break down the problem into manageable parts an... | [
{
"content": "Let's assume that Olga Pavlovna has \\( x \\) liters, and Maria Petrovna has \\( y \\) liters of jam left. The numbers \\( x \\) and \\( y \\) are randomly and independently chosen from the interval from 0 to 1. Consider a random point with coordinates \\((x, y)\\) from the unit square \\( F \\). ... |
Vasya and Tolya exchanged badges. Before the exchange, Vasya had 5 more badges than Tolya. After Vasya exchanged 24% of his badges for 20% of Tolya's badges, Vasya had one badge fewer than Tolya. How many badges did each boy have before the exchange? | Let's denote the number of badges Vasya had before the exchange as \(v\) and the number of badges Tolya had before the exchange as \(t\).
We can form the following equations based on the problem statement:
1. Vasya had 5 more badges than Tolya:
\[ v = t + 5 \]
2. After the exchange, Vasya gave 24% of his badges to T... | [
{
"content": "Vasya and Tolya exchanged badges. Before the exchange, Vasya had 5 more badges than Tolya. After Vasya exchanged 24% of his badges for 20% of Tolya's badges, Vasya had one badge fewer than Tolya. How many badges did each boy have before the exchange?",
"role": "user"
},
{
"content": "<... |
Mrs. Oaklyn buys handmade rugs at $40 each and sells them at $60 each. If she bought 20 rugs, calculate the profit she will make from selling the rugs. | To solve this problem, we need to follow these steps:
1. Calculate the total cost of buying 20 rugs.
2. Calculate the total revenue from selling 20 rugs.
3. Determine the profit by subtracting the total cost from the total revenue.
Let's use Python to perform the calculations and ensure the accuracy of the results.
`... | [
{
"content": "Mrs. Oaklyn buys handmade rugs at $40 each and sells them at $60 each. If she bought 20 rugs, calculate the profit she will make from selling the rugs.",
"role": "user"
},
{
"content": "<think>To solve this problem, we need to follow these steps:\n\n1. Calculate the total cost of buyin... |
On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process conti... | The problem involves finding the smallest number of students that can be arranged in rows of different sizes for 12 consecutive days. Each day's row count must be distinct, and on June 13, it should be impossible to find a new way to arrange the students.
To find the smallest possible number of students, consider that... | [
{
"content": "On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each r... |
A book that is to be recorded onto compact discs takes $412$ minutes to read aloud. Each disc can hold up to $56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?
$\mathrm{(A)}\ 50.2 ... | To determine the number of minutes of reading each disc will contain, we first need to identify the smallest number of discs required to store 412 minutes of audio, given that each disc can hold up to 56 minutes.
To find the minimum number of discs needed, we will use the ceiling function:
\[
\text{number of discs} =... | [
{
"content": "A book that is to be recorded onto compact discs takes $412$ minutes to read aloud. Each disc can hold up to $56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain... |
Yura left home for school 5 minutes later than Lena, but he walked twice as fast as her. How long after Yura started walking will he catch up to Lena? | To solve this problem, we can set up the following equations and solve them step by step.
1. **Define Variables:**
- Let \( t \) be the time (in minutes) after Yura starts walking that he catches up to Lena.
- Let \( v \) be Lena's walking speed (in units of distance per minute).
- Since Yura walks twice as f... | [
{
"content": "Yura left home for school 5 minutes later than Lena, but he walked twice as fast as her. How long after Yura started walking will he catch up to Lena?",
"role": "user"
},
{
"content": "<think>To solve this problem, we can set up the following equations and solve them step by step.\n\n1... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.