instruction_seed stringlengths 15 8.79k | response_seed stringlengths 15 8.79k | response stringlengths 96 53.2k | __original_row_idx int64 0 50k | conversations listlengths 2 2 |
|---|---|---|---|---|
Ten families have an average of 2 children per family. If exactly two of these families are childless, what is the average number of children in the families with children? Express your answer as a decimal to the nearest tenth. | Ten families have an average of 2 children per family. If exactly two of these families are childless, what is the average number of children in the families with children? Express your answer as a decimal to the nearest tenth. | To find the average number of children in the families that have children, we start with the given information:
1. There are 10 families total.
2. The average number of children per family is 2.
3. Out of these 10 families, 2 are childless.
First, we calculate the total number of children across all the families. The... | 10,200 | [
{
"from": "human",
"value": "Ten families have an average of 2 children per family. If exactly two of these families are childless, what is the average number of children in the families with children? Express your answer as a decimal to the nearest tenth."
},
{
"from": "gpt",
"value": "To find ... |
Find the maximum value of the expression $\frac{x}{(x-2)^2}$ for $x>2$. | Find the maximum value of the expression $\frac{x}{(x-2)^2}$ for $x>2$. | To find the maximum value of the expression
\[
y = \frac{x}{(x-2)^2}
\]
for \( x > 2 \), we start by finding the derivative of \( y \) with respect to \( x \) and setting it to zero to find critical points.
First, we rewrite the expression:
\[
y = x \cdot (x-2)^{-2}.
\]
Next, we will use the product and chain rul... | 10,201 | [
{
"from": "human",
"value": "Find the maximum value of the expression $\\frac{x}{(x-2)^2}$ for $x>2$."
},
{
"from": "gpt",
"value": "To find the maximum value of the expression \n\n\\[\ny = \\frac{x}{(x-2)^2}\n\\]\n\nfor \\( x > 2 \\), we start by finding the derivative of \\( y \\) with respect... |
If $\cos \theta + \sin \theta = \frac{5}{4},$ then find $\sin 2 \theta.$ | If $\cos \theta + \sin \theta = \frac{5}{4},$ then find $\sin 2 \theta.$ | We start with the equation given in the problem:
\[
\cos \theta + \sin \theta = \frac{5}{4}.
\]
Let's denote \( x = \sin \theta \) and \( y = \cos \theta \). This gives us:
\[
y + x = \frac{5}{4}.
\]
We know from the Pythagorean identity that:
\[
x^2 + y^2 = 1.
\]
From the first equation, we can express \( y \) i... | 10,202 | [
{
"from": "human",
"value": "If $\\cos \\theta + \\sin \\theta = \\frac{5}{4},$ then find $\\sin 2 \\theta.$"
},
{
"from": "gpt",
"value": "We start with the equation given in the problem:\n\n\\[\n\\cos \\theta + \\sin \\theta = \\frac{5}{4}.\n\\]\n\nLet's denote \\( x = \\sin \\theta \\) and \\... |
For how many $n=2,3,4,\ldots,99,100$ is the base-$n$ number $214234_n$ a multiple of $11$? | For how many $n=2,3,4,\ldots,99,100$ is the base-$n$ number $214234_n$ a multiple of $11$? | We start with the base-\( n \) number \( 214234_n \). Converting this number from base \( n \) to decimal, we have:
\[
214234_n = 2n^5 + 1n^4 + 4n^3 + 2n^2 + 3n^1 + 4n^0
\]
This expands to:
\[
2n^5 + n^4 + 4n^3 + 2n^2 + 3n + 4
\]
Next, we need to determine for which values of \( n \) in the range \( n = 2, 3, 4, \l... | 10,203 | [
{
"from": "human",
"value": "For how many $n=2,3,4,\\ldots,99,100$ is the base-$n$ number $214234_n$ a multiple of $11$?"
},
{
"from": "gpt",
"value": "We start with the base-\\( n \\) number \\( 214234_n \\). Converting this number from base \\( n \\) to decimal, we have:\n\n\\[\n214234_n = 2n^... |
A list of six positive integers has a median of 12 and a standard deviation of 15. What is the minimum possible value of the list's smallest element? | A list of six positive integers has a median of 12 and a standard deviation of 15. What is the minimum possible value of the list's smallest element? | Let the six integers be represented as \( a_1, a_2, a_3, a_4, a_5, a_6 \), ordered such that \( a_1 \leq a_2 \leq a_3 \leq a_4 \leq a_5 \leq a_6 \).
Given that the median of the list is 12, we can determine the values of \( a_3 \) and \( a_4 \):
\[
\frac{a_3 + a_4}{2} = 12 \implies a_3 + a_4 = 24
\]
Since \( a_3 \) an... | 10,204 | [
{
"from": "human",
"value": "A list of six positive integers has a median of 12 and a standard deviation of 15. What is the minimum possible value of the list's smallest element?"
},
{
"from": "gpt",
"value": "Let the six integers be represented as \\( a_1, a_2, a_3, a_4, a_5, a_6 \\), ordered s... |
The graph of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ has a tangent line at the point $(3,0)$. Find the sum of the $x$-coordinates of the points where the tangent line intersects the ellipse again. | The graph of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ has a tangent line at the point $(3,0)$. Find the sum of the $x$-coordinates of the points where the tangent line intersects the ellipse again. | To find the tangent line of the ellipse defined by the equation \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) at the point \( (3, 0) \), we first need to confirm that the point \( (3, 0) \) lies on the ellipse.
Substituting \( x = 3 \) and \( y = 0 \):
\[
\frac{3^2}{9} + \frac{0^2}{4} = \frac{9}{9} + 0 = 1
\]
Since this i... | 10,205 | [
{
"from": "human",
"value": "The graph of the ellipse $\\frac{x^2}{9} + \\frac{y^2}{4} = 1$ has a tangent line at the point $(3,0)$. Find the sum of the $x$-coordinates of the points where the tangent line intersects the ellipse again."
},
{
"from": "gpt",
"value": "To find the tangent line of t... |
Maria has a set of identical boxes, each of which can hold 2/5 of a liter of juice. If she fills a container with 3 liters of juice and pours it equally into the boxes, what fraction of a box will be left empty? | Maria has a set of identical boxes, each of which can hold 2/5 of a liter of juice. If she fills a container with 3 liters of juice and pours it equally into the boxes, what fraction of a box will be left empty? | First, we need to determine how many boxes Maria will use for the 3 liters of juice, given that each box can hold \( \frac{2}{5} \) of a liter.
To find out how many boxes are required for 3 liters, we can calculate the total number of boxes needed by dividing the total volume of juice by the capacity of one box:
\[
\... | 10,206 | [
{
"from": "human",
"value": "Maria has a set of identical boxes, each of which can hold 2/5 of a liter of juice. If she fills a container with 3 liters of juice and pours it equally into the boxes, what fraction of a box will be left empty?"
},
{
"from": "gpt",
"value": "First, we need to determ... |
Let $P$ be a point outside of an ellipse $E.$ A line is drawn from $P$ such that it is tangent to ellipse $E$ at point $T.$ Meanwhile, a secant from $P$ intersects $E$ at $A$ and $B,$ such that $PA < PB.$ If the area of the triangle $PAB$ is equal to the area of the triangle $PAT$, then what is the length of the major ... | Let $P$ be a point outside of an ellipse $E.$ A line is drawn from $P$ such that it is tangent to ellipse $E$ at point $T.$ Meanwhile, a secant from $P$ intersects $E$ at $A$ and $B,$ such that $PA < PB.$ If the area of the triangle $PAB$ is equal to the area of the triangle $PAT$, then what is the length of the major ... | Let the ellipse \( E \) be centered at the origin with semi-major axis \( a \) and semi-minor axis \( b \). The equation of the ellipse is given by
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.
\]
Let the coordinates of point \( P \) be \( P(x_0, y_0) \). The tangent line to the ellipse at point \( T(x_T, y_T) \) has th... | 10,207 | [
{
"from": "human",
"value": "Let $P$ be a point outside of an ellipse $E.$ A line is drawn from $P$ such that it is tangent to ellipse $E$ at point $T.$ Meanwhile, a secant from $P$ intersects $E$ at $A$ and $B,$ such that $PA < PB.$ If the area of the triangle $PAB$ is equal to the area of the triangle $PA... |
What is the absolute value of the difference between the cubes of 19 and 21? | What is the absolute value of the difference between the cubes of 19 and 21? | To find the absolute value of the difference between the cubes of 19 and 21, we start by calculating the cubes of each of those numbers.
First, we calculate the cube of 21:
\[
21^3 = 21 \times 21 \times 21
\]
Calculating \(21 \times 21\):
\[
21 \times 21 = 441
\]
Now we multiply 441 by 21 to find \(21^3\):
\[
21^3... | 10,208 | [
{
"from": "human",
"value": "What is the absolute value of the difference between the cubes of 19 and 21?"
},
{
"from": "gpt",
"value": "To find the absolute value of the difference between the cubes of 19 and 21, we start by calculating the cubes of each of those numbers.\n\nFirst, we calculate... |
Let $z$ be a complex number such that $\arg(z^3) = \arg(z^2) + \frac{\pi}{3}$ and $|z^2 + z + 1| = 2$. Find $|z|$. | Let $z$ be a complex number such that $\arg(z^3) = \arg(z^2) + \frac{\pi}{3}$ and $|z^2 + z + 1| = 2$. Find $|z|$. | Let \( z = re^{i\theta} \), where \( r = |z| \) and \( \theta = \arg(z) \).
First, we express \(\arg(z^3)\) and \(\arg(z^2)\):
\[
\arg(z^3) = \arg(z^2) + \frac{\pi}{3}
\]
\[
\arg(z^3) = 3\theta, \quad \arg(z^2) = 2\theta
\]
Therefore, we have:
\[
3\theta = 2\theta + \frac{\pi}{3}
\]
Solving for \(\theta\):
\[
3\theta... | 10,209 | [
{
"from": "human",
"value": "Let $z$ be a complex number such that $\\arg(z^3) = \\arg(z^2) + \\frac{\\pi}{3}$ and $|z^2 + z + 1| = 2$. Find $|z|$."
},
{
"from": "gpt",
"value": "Let \\( z = re^{i\\theta} \\), where \\( r = |z| \\) and \\( \\theta = \\arg(z) \\). \n\nFirst, we express \\(\\arg(z... |
A bookstore has 8 shelves of books. What is the greatest number of pairs of books that can be selected such that each pair consists of books from two different shelves and no two pairs are the same combination? | A bookstore has 8 shelves of books. What is the greatest number of pairs of books that can be selected such that each pair consists of books from two different shelves and no two pairs are the same combination? | To determine the greatest number of pairs of books that can be selected from a bookstore with 8 shelves, where each pair consists of one book from one shelf and another book from a different shelf, we can use the concept of combinations.
### Step-by-step analysis:
1. **Understanding the Problem**:
We need to selec... | 10,210 | [
{
"from": "human",
"value": "A bookstore has 8 shelves of books. What is the greatest number of pairs of books that can be selected such that each pair consists of books from two different shelves and no two pairs are the same combination?"
},
{
"from": "gpt",
"value": "To determine the greatest... |
A bookshelf has 50 books, each with a different number of pages. The books are arranged in such a way that the number of pages in each book is one more than the number of pages in the previous book. If the first book has 1 page and the last book has 50 pages, how many books are there on the bookshelf that have a number... | A bookshelf has 50 books, each with a different number of pages. The books are arranged in such a way that the number of pages in each book is one more than the number of pages in the previous book. If the first book has 1 page and the last book has 50 pages, how many books are there on the bookshelf that have a number... | To solve the problem, we start with the arrangement of the books. The first book has 1 page, and the subsequent books increase the number of pages by 1 for each book. The pages in each of the 50 books can be listed as follows:
- Pages in book 1: \(1\)
- Pages in book 2: \(2\)
- Pages in book 3: \(3\)
- ...
- Pages in ... | 10,211 | [
{
"from": "human",
"value": "A bookshelf has 50 books, each with a different number of pages. The books are arranged in such a way that the number of pages in each book is one more than the number of pages in the previous book. If the first book has 1 page and the last book has 50 pages, how many books are ... |
What is the smallest base-10 integer that can be represented as $AA_3$ and $BB_9$, where $A$ and $B$ are valid digits in their respective bases? | What is the smallest base-10 integer that can be represented as $AA_3$ and $BB_9$, where $A$ and $B$ are valid digits in their respective bases? | To solve for the smallest base-10 integer \( n \) that can be represented as \( AA_3 \) and \( BB_9 \), we interpret the expressions in their respective bases.
**Step 1: Represent \( AA_3 \)**
In base 3, \( AA_3 \) means that both digits are \( A \):
\[
AA_3 = A \cdot 3^1 + A \cdot 3^0 = A \cdot 3 + A = A(3 + 1) = 4... | 10,212 | [
{
"from": "human",
"value": "What is the smallest base-10 integer that can be represented as $AA_3$ and $BB_9$, where $A$ and $B$ are valid digits in their respective bases?"
},
{
"from": "gpt",
"value": "To solve for the smallest base-10 integer \\( n \\) that can be represented as \\( AA_3 \\)... |
Two fair dice are rolled. What is the probability that the difference between the numbers on the two dice is 3? Express your answer as a common fraction. | Two fair dice are rolled. What is the probability that the difference between the numbers on the two dice is 3? Express your answer as a common fraction. | When rolling two fair dice, each die can show an integer from 1 to 6. The total possible outcomes when rolling two dice is given by:
\[
6 \times 6 = 36
\]
Next, we need to determine the number of outcomes where the difference between the numbers on the two dice is exactly 3. We can denote the numbers on the first and... | 10,213 | [
{
"from": "human",
"value": "Two fair dice are rolled. What is the probability that the difference between the numbers on the two dice is 3? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "When rolling two fair dice, each die can show an integer from 1 to 6. The total po... |
In a magical forest, there are four identical-looking doors, each leading to a different room. Three of the rooms contain a treasure chest, while the fourth room contains a dragon. A brave adventurer randomly opens two of the doors and finds two treasure chests. What is the probability that the remaining unopened doors... | In a magical forest, there are four identical-looking doors, each leading to a different room. Three of the rooms contain a treasure chest, while the fourth room contains a dragon. A brave adventurer randomly opens two of the doors and finds two treasure chests. What is the probability that the remaining unopened doors... | To solve the problem, we start by outlining the setup clearly. We have four doors: \(D_1\), \(D_2\), \(D_3\), and \(D_4\). Among these doors, three contain a treasure chest (let's label them \(T\)) and one contains a dragon (\(R\)).
Let's denote the doors and their contents as follows:
- \(D_1\) - contains \(T\) (tr... | 10,214 | [
{
"from": "human",
"value": "In a magical forest, there are four identical-looking doors, each leading to a different room. Three of the rooms contain a treasure chest, while the fourth room contains a dragon. A brave adventurer randomly opens two of the doors and finds two treasure chests. What is the prob... |
A recipe for 30 cookies requires two cups of flour among its ingredients. Eduardo wants to bake five dozen cookies. How many cups of flour will he need to use in his adjusted recipe? | A recipe for 30 cookies requires two cups of flour among its ingredients. Eduardo wants to bake five dozen cookies. How many cups of flour will he need to use in his adjusted recipe? | First, let's determine how many cookies Eduardo wants to bake. He wants to bake five dozen cookies:
\[
5 \text{ dozen} = 5 \times 12 = 60 \text{ cookies}
\]
Now, we know the original recipe makes 30 cookies with 2 cups of flour. We need to find out how many cups of flour are required for 60 cookies.
To do this, we ... | 10,215 | [
{
"from": "human",
"value": "A recipe for 30 cookies requires two cups of flour among its ingredients. Eduardo wants to bake five dozen cookies. How many cups of flour will he need to use in his adjusted recipe?"
},
{
"from": "gpt",
"value": "First, let's determine how many cookies Eduardo wants... |
In a certain kingdom, there are three types of coins: gold, silver, and copper. The king has decreed that no gold coin can be followed by a silver coin, no silver coin can be followed by a copper coin, and no copper coin can be followed by a gold coin in any sequence of coins. How many sequences of seven coins are ther... | In a certain kingdom, there are three types of coins: gold, silver, and copper. The king has decreed that no gold coin can be followed by a silver coin, no silver coin can be followed by a copper coin, and no copper coin can be followed by a gold coin in any sequence of coins. How many sequences of seven coins are ther... | Let \( G \) represent gold coins, \( S \) represent silver coins, and \( C \) represent copper coins. The conditions given for sequencing the coins can be summarized as follows:
1. \( G \) cannot be followed by \( S \).
2. \( S \) cannot be followed by \( C \).
3. \( C \) cannot be followed by \( G \).
In terms of va... | 10,216 | [
{
"from": "human",
"value": "In a certain kingdom, there are three types of coins: gold, silver, and copper. The king has decreed that no gold coin can be followed by a silver coin, no silver coin can be followed by a copper coin, and no copper coin can be followed by a gold coin in any sequence of coins. H... |
Let two non-collinear vectors $\mathbf{u}$ and $\mathbf{v}$ be given in a plane. Suppose that $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are three non-collinear points on this plane such that the projection of $\mathbf{u}$ onto $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are equal. What is the relationship between t... | Let two non-collinear vectors $\mathbf{u}$ and $\mathbf{v}$ be given in a plane. Suppose that $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are three non-collinear points on this plane such that the projection of $\mathbf{u}$ onto $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are equal. What is the relationship between t... | Given two non-collinear vectors \(\mathbf{u}\) and \(\mathbf{v}\) in a plane, and three non-collinear points \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\), we know that the projection of \(\mathbf{u}\) onto each of these points is equal.
The projection of vector \(\mathbf{u}\) onto a point \(\mathbf{p}\) can be ... | 10,217 | [
{
"from": "human",
"value": "Let two non-collinear vectors $\\mathbf{u}$ and $\\mathbf{v}$ be given in a plane. Suppose that $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ are three non-collinear points on this plane such that the projection of $\\mathbf{u}$ onto $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf... |
Find the number of ordered pairs of integers $(a,b)$ with $2\le a<b\le 200$ such that $(-1)^a + (-1)^b$ is a positive integer. | Find the number of ordered pairs of integers $(a,b)$ with $2\le a<b\le 200$ such that $(-1)^a + (-1)^b$ is a positive integer. | To find the number of ordered pairs of integers \( (a, b) \) with \( 2 \le a < b \le 200 \) such that \( (-1)^a + (-1)^b \) is a positive integer, we first analyze the expression \( (-1)^a + (-1)^b \).
The term \( (-1)^x \) will take the following values based on whether \( x \) is even or odd:
- If \( x \) is even, ... | 10,218 | [
{
"from": "human",
"value": "Find the number of ordered pairs of integers $(a,b)$ with $2\\le a<b\\le 200$ such that $(-1)^a + (-1)^b$ is a positive integer."
},
{
"from": "gpt",
"value": "To find the number of ordered pairs of integers \\( (a, b) \\) with \\( 2 \\le a < b \\le 200 \\) such that... |
Let $a,$ $b,$ and $c$ be nonnegative numbers such that $a+b+c = 10.$ Find the maximum value of
\[a^{1/3}b^{2/3} + b^{1/3}c^{2/3} + c^{1/3}a^{2/3}.\] | Let $a,$ $b,$ and $c$ be nonnegative numbers such that $a+b+c = 10.$ Find the maximum value of
\[a^{1/3}b^{2/3} + b^{1/3}c^{2/3} + c^{1/3}a^{2/3}.\] | Let \( S = a^{1/3}b^{2/3} + b^{1/3}c^{2/3} + c^{1/3}a^{2/3} \). We want to maximize \( S \) subject to the constraint \( a + b + c = 10 \) and \( a, b, c \geq 0 \).
Using the method of Lagrange multipliers, we define our function \( S \) and the constraint function \( g(a, b, c) = a + b + c - 10 \). We set up the equa... | 10,219 | [
{
"from": "human",
"value": "Let $a,$ $b,$ and $c$ be nonnegative numbers such that $a+b+c = 10.$ Find the maximum value of\n\\[a^{1/3}b^{2/3} + b^{1/3}c^{2/3} + c^{1/3}a^{2/3}.\\]"
},
{
"from": "gpt",
"value": "Let \\( S = a^{1/3}b^{2/3} + b^{1/3}c^{2/3} + c^{1/3}a^{2/3} \\). We want to maximi... |
Ashleigh traveled from City A to City B at an average speed of 45 miles per hour. On her return trip, she took the same route, but due to traffic, she traveled at an average speed of 30 miles per hour. If the total time spent traveling was 7 hours, what is the distance between City A and City B? | Ashleigh traveled from City A to City B at an average speed of 45 miles per hour. On her return trip, she took the same route, but due to traffic, she traveled at an average speed of 30 miles per hour. If the total time spent traveling was 7 hours, what is the distance between City A and City B? | Let the distance between City A and City B be \( d \) miles.
On the way to City B, Ashleigh traveled at an average speed of 45 miles per hour. The time taken for this part of the trip can be expressed as:
\[
t_1 = \frac{d}{45}
\]
On her return trip to City A, she traveled at an average speed of 30 miles per hour. T... | 10,220 | [
{
"from": "human",
"value": "Ashleigh traveled from City A to City B at an average speed of 45 miles per hour. On her return trip, she took the same route, but due to traffic, she traveled at an average speed of 30 miles per hour. If the total time spent traveling was 7 hours, what is the distance between C... |
A deck of 52 playing cards is shuffled, and two cards are drawn without replacement. What is the probability that the second card drawn is of a different suit than the first card drawn? Express your answer as a common fraction. | A deck of 52 playing cards is shuffled, and two cards are drawn without replacement. What is the probability that the second card drawn is of a different suit than the first card drawn? Express your answer as a common fraction. | To determine the probability that the second card drawn is of a different suit than the first card drawn, we can analyze the situation as follows:
1. **Identify total number of suits**: There are 4 suits in a standard deck of 52 playing cards (hearts, diamonds, clubs, spades).
2. **Draw the first card**: When the fir... | 10,221 | [
{
"from": "human",
"value": "A deck of 52 playing cards is shuffled, and two cards are drawn without replacement. What is the probability that the second card drawn is of a different suit than the first card drawn? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "To deter... |
To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 400 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? | To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 400 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? | To express 400 as a sum of distinct powers of 2, we start by determining the binary representation of 400.
To convert 400 to binary, we can start by finding the largest power of 2 less than or equal to 400:
\[
2^0 = 1, \quad 2^1 = 2, \quad 2^2 = 4, \quad 2^3 = 8, \quad 2^4 = 16, \quad 2^5 = 32, \quad 2^6 = 64, \quad ... | 10,222 | [
{
"from": "human",
"value": "To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 400 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?"... |
The sum of the squares of three consecutive integers is 7805. What is the product of the first and last integers? | The sum of the squares of three consecutive integers is 7805. What is the product of the first and last integers? | Let the three consecutive integers be represented as \( x - 1 \), \( x \), and \( x + 1 \).
The sum of the squares of these integers can be expressed as:
\[
(x - 1)^2 + x^2 + (x + 1)^2
\]
Expanding each square, we have:
\[
(x - 1)^2 = x^2 - 2x + 1,
\]
\[
x^2 = x^2,
\]
\[
(x + 1)^2 = x^2 + 2x + 1.
\]
Now, if we add... | 10,223 | [
{
"from": "human",
"value": "The sum of the squares of three consecutive integers is 7805. What is the product of the first and last integers?"
},
{
"from": "gpt",
"value": "Let the three consecutive integers be represented as \\( x - 1 \\), \\( x \\), and \\( x + 1 \\).\n\nThe sum of the square... |
Evaluate $\lfloor (15.6)^2 \rfloor - \lfloor 15.6 \rfloor \cdot \lfloor 15.6 \rfloor.$ | Evaluate $\lfloor (15.6)^2 \rfloor - \lfloor 15.6 \rfloor \cdot \lfloor 15.6 \rfloor.$ | To evaluate \(\lfloor (15.6)^2 \rfloor - \lfloor 15.6 \rfloor \cdot \lfloor 15.6 \rfloor\), we first need to compute \(\lfloor 15.6 \rfloor\).
Calculating \(\lfloor 15.6 \rfloor\):
\[
\lfloor 15.6 \rfloor = 15
\]
Next, we compute \(\lfloor 15.6 \rfloor \cdot \lfloor 15.6 \rfloor\):
\[
\lfloor 15.6 \rfloor \cdot \lflo... | 10,224 | [
{
"from": "human",
"value": "Evaluate $\\lfloor (15.6)^2 \\rfloor - \\lfloor 15.6 \\rfloor \\cdot \\lfloor 15.6 \\rfloor.$"
},
{
"from": "gpt",
"value": "To evaluate \\(\\lfloor (15.6)^2 \\rfloor - \\lfloor 15.6 \\rfloor \\cdot \\lfloor 15.6 \\rfloor\\), we first need to compute \\(\\lfloor 15.6... |
Find the remainder when $2^{2001}$ is divided by $11$. | Find the remainder when $2^{2001}$ is divided by $11$. | To find the remainder of \( 2^{2001} \) when divided by \( 11 \), we can use Fermat's Little Theorem which states that if \( p \) is a prime and \( a \) is an integer not divisible by \( p \), then:
\[
a^{p-1} \equiv 1 \mod p.
\]
In this case, let \( a = 2 \) and \( p = 11 \). According to Fermat's Little Theorem, si... | 10,225 | [
{
"from": "human",
"value": "Find the remainder when $2^{2001}$ is divided by $11$."
},
{
"from": "gpt",
"value": "To find the remainder of \\( 2^{2001} \\) when divided by \\( 11 \\), we can use Fermat's Little Theorem which states that if \\( p \\) is a prime and \\( a \\) is an integer not di... |
A rectangular pool with a length of 10 meters and a width of 5 meters is filled with water to a depth of 2 meters. The water is then pumped into a large cylindrical tank with a height of 3 meters. What is the radius of the cylindrical tank, in meters? | A rectangular pool with a length of 10 meters and a width of 5 meters is filled with water to a depth of 2 meters. The water is then pumped into a large cylindrical tank with a height of 3 meters. What is the radius of the cylindrical tank, in meters? | To find the radius of the cylindrical tank, we first need to determine the volume of water in the rectangular pool and then relate that volume to the volume of the cylindrical tank.
1. **Calculate the volume of water in the rectangular pool**:
The formula for the volume \( V \) of a rectangular prism (pool) is give... | 10,226 | [
{
"from": "human",
"value": "A rectangular pool with a length of 10 meters and a width of 5 meters is filled with water to a depth of 2 meters. The water is then pumped into a large cylindrical tank with a height of 3 meters. What is the radius of the cylindrical tank, in meters?"
},
{
"from": "gpt"... |
The parabolas $y = x^2$ and $x = y^2$ intersect at two points. All four points obtained by reflecting these two points about the x-axis and y-axis lie on a circle of radius $r$. Find $r^2$. | The parabolas $y = x^2$ and $x = y^2$ intersect at two points. All four points obtained by reflecting these two points about the x-axis and y-axis lie on a circle of radius $r$. Find $r^2$. | To find the points of intersection of the parabolas \( y = x^2 \) and \( x = y^2 \), we set \( y = x^2 \) into \( x = y^2 \).
Substituting \( y \):
\[
x = (x^2)^2 = x^4.
\]
This leads to the equation:
\[
x^4 - x = 0.
\]
Factoring out \( x \), we have:
\[
x(x^3 - 1) = 0.
\]
This gives us the solutions:
\[
x = 0 \quad \... | 10,227 | [
{
"from": "human",
"value": "The parabolas $y = x^2$ and $x = y^2$ intersect at two points. All four points obtained by reflecting these two points about the x-axis and y-axis lie on a circle of radius $r$. Find $r^2$."
},
{
"from": "gpt",
"value": "To find the points of intersection of the pa... |
Let $g(x) = |x-1|-|x+1|$. Find the value of $g(2)+g(-3)$. | Let $g(x) = |x-1|-|x+1|$. Find the value of $g(2)+g(-3)$. | To evaluate \( g(2) + g(-3) \), we first need to compute \( g(2) \) and \( g(-3) \) using the function defined as \( g(x) = |x-1| - |x+1| \).
### Step 1: Calculate \( g(2) \)
First, we find \( g(2) \):
\[
g(2) = |2-1| - |2+1|
\]
Calculating each absolute value separately:
\[
|2-1| = |1| = 1
\]
\[
|2+1| = |3| = 3
\... | 10,228 | [
{
"from": "human",
"value": "Let $g(x) = |x-1|-|x+1|$. Find the value of $g(2)+g(-3)$."
},
{
"from": "gpt",
"value": "To evaluate \\( g(2) + g(-3) \\), we first need to compute \\( g(2) \\) and \\( g(-3) \\) using the function defined as \\( g(x) = |x-1| - |x+1| \\).\n\n### Step 1: Calculate \\(... |
The points $(0,4)$ and $(1,3)$ lie on a circle whose center is on the $x$-axis. What is the radius of the circle?
The vertices of an equilateral triangle lie on a circle whose center is on the line $y=x$. What is the length of the side of the triangle? | The points $(0,4)$ and $(1,3)$ lie on a circle whose center is on the $x$-axis. What is the radius of the circle?
The vertices of an equilateral triangle lie on a circle whose center is on the line $y=x$. What is the length of the side of the triangle? | To find the radius of the circle whose center is on the \(x\)-axis and passes through the points \((0, 4)\) and \((1, 3)\), we denote the center of the circle as \((a, 0)\) where \(a\) is the \(x\)-coordinate.
The radius \(r\) of the circle can be defined as the distance from the center to any point on the circle. Usi... | 10,229 | [
{
"from": "human",
"value": "The points $(0,4)$ and $(1,3)$ lie on a circle whose center is on the $x$-axis. What is the radius of the circle?\n\nThe vertices of an equilateral triangle lie on a circle whose center is on the line $y=x$. What is the length of the side of the triangle?"
},
{
"from": "... |
Let $f(x)=Ax^2+Bx+C$ be a quadratic function. Suppose we want to find a linear function $g(x)$ such that the graph of $y=g(f(x))$ is tangent to the graph of $y=x^2$ exactly at two points. What is the value of $g(-1)$? | Let $f(x)=Ax^2+Bx+C$ be a quadratic function. Suppose we want to find a linear function $g(x)$ such that the graph of $y=g(f(x))$ is tangent to the graph of $y=x^2$ exactly at two points. What is the value of $g(-1)$? | To find a linear function \( g(x) = mx + b \) such that \( y = g(f(x)) \) is tangent to the graph \( y = x^2 \) at exactly two points, we start by defining the function \( h(x) = g(f(x)) - x^2 \). Our goal is to ensure that \( h(x) \) has exactly two roots, which indicates that the curves are tangent at those points.
... | 10,230 | [
{
"from": "human",
"value": "Let $f(x)=Ax^2+Bx+C$ be a quadratic function. Suppose we want to find a linear function $g(x)$ such that the graph of $y=g(f(x))$ is tangent to the graph of $y=x^2$ exactly at two points. What is the value of $g(-1)$?"
},
{
"from": "gpt",
"value": "To find a linear f... |
The sequence $(b_n)$ is defined recursively by $b_0=1$, $b_1=1+i$, and $b_n=b_{n-1}b_{n-2}^3$ for $n\geq 2$, where $i$ is the imaginary unit. What is the smallest positive integer $m$ such that the sum $b_1+b_2+\cdots+b_m$ is a real number? | The sequence $(b_n)$ is defined recursively by $b_0=1$, $b_1=1+i$, and $b_n=b_{n-1}b_{n-2}^3$ for $n\geq 2$, where $i$ is the imaginary unit. What is the smallest positive integer $m$ such that the sum $b_1+b_2+\cdots+b_m$ is a real number? | To find the smallest positive integer \( m \) such that the sum \( b_1 + b_2 + \cdots + b_m \) is a real number, we first need to compute the terms \( b_n \) of the sequence recursively defined as follows:
- \( b_0 = 1 \)
- \( b_1 = 1 + i \)
- \( b_n = b_{n-1} b_{n-2}^3 \) for \( n \geq 2 \)
We will compute the first... | 10,231 | [
{
"from": "human",
"value": "The sequence $(b_n)$ is defined recursively by $b_0=1$, $b_1=1+i$, and $b_n=b_{n-1}b_{n-2}^3$ for $n\\geq 2$, where $i$ is the imaginary unit. What is the smallest positive integer $m$ such that the sum $b_1+b_2+\\cdots+b_m$ is a real number?"
},
{
"from": "gpt",
"va... |
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 10.$ Find the maximum value of
\[\sqrt{3a + 2} + \sqrt{3b + 2} + \sqrt{3c + 2}.\] | Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 10.$ Find the maximum value of
\[\sqrt{3a + 2} + \sqrt{3b + 2} + \sqrt{3c + 2}.\] | We seek to maximize the expression
\[
S = \sqrt{3a + 2} + \sqrt{3b + 2} + \sqrt{3c + 2}
\]
under the constraint \(a + b + c = 10\) where \(a, b, c \geq 0\).
To facilitate our calculations, we can first rewrite the expression:
\[
S = \sqrt{3a + 2} + \sqrt{3b + 2} + \sqrt{3c + 2}
\]
Defining \(f(x) = \sqrt{3x + 2}\... | 10,232 | [
{
"from": "human",
"value": "Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 10.$ Find the maximum value of\n\\[\\sqrt{3a + 2} + \\sqrt{3b + 2} + \\sqrt{3c + 2}.\\]"
},
{
"from": "gpt",
"value": "We seek to maximize the expression \n\n\\[\nS = \\sqrt{3a + 2} + \\sqrt{3b... |
Given points $A(1,3)$ and $B(x,6)$ in a coordinate plane, for what value of $x$ is the slope of the line through $A$ and $B$ equal to the negative reciprocal of the slope of the line through $(0,0)$ and $(3,4)$? | Given points $A(1,3)$ and $B(x,6)$ in a coordinate plane, for what value of $x$ is the slope of the line through $A$ and $B$ equal to the negative reciprocal of the slope of the line through $(0,0)$ and $(3,4)$? | To solve the problem, we first find the slope of the line through points \( (0, 0) \) and \( (3, 4) \).
The slope \( m_1 \) is calculated using the formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Applying this to points \( (0, 0) \) and \( (3, 4) \)... | 10,233 | [
{
"from": "human",
"value": "Given points $A(1,3)$ and $B(x,6)$ in a coordinate plane, for what value of $x$ is the slope of the line through $A$ and $B$ equal to the negative reciprocal of the slope of the line through $(0,0)$ and $(3,4)$?"
},
{
"from": "gpt",
"value": "To solve the problem, we... |
A jar is filled with red, orange and yellow jelly beans. The probability of randomly selecting a red jelly bean from this jar is $0.2$, and the probability of randomly selecting an orange jelly bean from this jar is $0.5$. What is the probability of randomly selecting a yellow jelly bean from this jar? | A jar is filled with red, orange and yellow jelly beans. The probability of randomly selecting a red jelly bean from this jar is $0.2$, and the probability of randomly selecting an orange jelly bean from this jar is $0.5$. What is the probability of randomly selecting a yellow jelly bean from this jar? | To find the probability of randomly selecting a yellow jelly bean from the jar, we can use the fact that the total probability of selecting any jelly bean must equal 1.
We know the following probabilities:
- Probability of selecting a red jelly bean, \( P(\text{red}) = 0.2 \)
- Probability of selecting an orange jel... | 10,234 | [
{
"from": "human",
"value": "A jar is filled with red, orange and yellow jelly beans. The probability of randomly selecting a red jelly bean from this jar is $0.2$, and the probability of randomly selecting an orange jelly bean from this jar is $0.5$. What is the probability of randomly selecting a yellow... |
Given a point in spherical coordinates $(\rho, \theta, \phi) = (4, \frac{\pi}{3}, \frac{\pi}{4})$, convert it to rectangular coordinates $(x, y, z)$. Assume $\rho > 0$, $0 \leq \theta < 2\pi$, and $0 \leq \phi \leq \pi$. | Given a point in spherical coordinates $(\rho, \theta, \phi) = (4, \frac{\pi}{3}, \frac{\pi}{4})$, convert it to rectangular coordinates $(x, y, z)$. Assume $\rho > 0$, $0 \leq \theta < 2\pi$, and $0 \leq \phi \leq \pi$. | To convert the spherical coordinates \((\rho, \theta, \phi) = (4, \frac{\pi}{3}, \frac{\pi}{4})\) to rectangular coordinates \((x, y, z)\), we use the following conversion formulas:
\[
x = \rho \sin \phi \cos \theta
\]
\[
y = \rho \sin \phi \sin \theta
\]
\[
z = \rho \cos \phi
\]
Let's calculate each component step... | 10,235 | [
{
"from": "human",
"value": "Given a point in spherical coordinates $(\\rho, \\theta, \\phi) = (4, \\frac{\\pi}{3}, \\frac{\\pi}{4})$, convert it to rectangular coordinates $(x, y, z)$. Assume $\\rho > 0$, $0 \\leq \\theta < 2\\pi$, and $0 \\leq \\phi \\leq \\pi$."
},
{
"from": "gpt",
"value": "... |
At a local store, a special promotion is going on: if you buy three boxes of cereal, you get a fourth box of the same type for free. Each box of cereal costs $2.50. If Sara has $21, how many boxes of cereal can she buy in total, considering the promotion? | At a local store, a special promotion is going on: if you buy three boxes of cereal, you get a fourth box of the same type for free. Each box of cereal costs $2.50. If Sara has $21, how many boxes of cereal can she buy in total, considering the promotion? | To determine how many boxes of cereal Sara can buy with her $21 and the special promotion in mind, we start by calculating how many boxes she can buy without consideration of the promotion.
1. **Calculate the number of boxes she can buy outright:**
- Each box costs $2.50.
- Divide her total money by the price of... | 10,236 | [
{
"from": "human",
"value": "At a local store, a special promotion is going on: if you buy three boxes of cereal, you get a fourth box of the same type for free. Each box of cereal costs $2.50. If Sara has $21, how many boxes of cereal can she buy in total, considering the promotion?"
},
{
"from": "... |
In a certain town, there are three types of public transportation: buses, trams, and trains. The buses run every 15 minutes, the trams run every 20 minutes, and the trains run every 30 minutes. If you arrive at a random time, what is the probability that you will have to wait less than 5 minutes for each of the three t... | In a certain town, there are three types of public transportation: buses, trams, and trains. The buses run every 15 minutes, the trams run every 20 minutes, and the trains run every 30 minutes. If you arrive at a random time, what is the probability that you will have to wait less than 5 minutes for each of the three t... | To find the probability that you will have to wait less than 5 minutes for each type of transportation, we need to analyze the timing of the buses, trams, and trains based on their respective schedules.
1. **Buses** run every 15 minutes. This means:
- A bus arrives at \(0\), \(15\), \(30\), \(45\) minutes past the ... | 10,237 | [
{
"from": "human",
"value": "In a certain town, there are three types of public transportation: buses, trams, and trains. The buses run every 15 minutes, the trams run every 20 minutes, and the trains run every 30 minutes. If you arrive at a random time, what is the probability that you will have to wait le... |
Compute
\[\sum_{n = 2}^{10000} \frac{1}{n \sqrt{n - 1} + (n - 1) \sqrt{n}}.\]
Another problem similar to this one is:
Compute
\[\sum_{n = 3}^{10000} \frac{1}{n \sqrt{n - 2} + (n - 2) \sqrt{n}}.\] | Compute
\[\sum_{n = 2}^{10000} \frac{1}{n \sqrt{n - 1} + (n - 1) \sqrt{n}}.\]
Another problem similar to this one is:
Compute
\[\sum_{n = 3}^{10000} \frac{1}{n \sqrt{n - 2} + (n - 2) \sqrt{n}}.\] | To compute the sum
\[
\sum_{n = 2}^{10000} \frac{1}{n \sqrt{n - 1} + (n - 1) \sqrt{n}},
\]
we start by simplifying the term inside the summation. Consider the expression:
\[
n \sqrt{n - 1} + (n - 1) \sqrt{n}.
\]
We can factor this expression. Firstly, observe the term \( \sqrt{n} \cdot \sqrt{n - 1} \):
\[
\sqrt{n... | 10,238 | [
{
"from": "human",
"value": "Compute\n\\[\\sum_{n = 2}^{10000} \\frac{1}{n \\sqrt{n - 1} + (n - 1) \\sqrt{n}}.\\]\n\nAnother problem similar to this one is:\n\nCompute\n\\[\\sum_{n = 3}^{10000} \\frac{1}{n \\sqrt{n - 2} + (n - 2) \\sqrt{n}}.\\]"
},
{
"from": "gpt",
"value": "To compute the sum \... |
How many real numbers $b$ satisfy the condition that the quadratic equation $x^2 - 2bx + 3b = 0$ has only integer solutions for $x$? | How many real numbers $b$ satisfy the condition that the quadratic equation $x^2 - 2bx + 3b = 0$ has only integer solutions for $x$? | To determine how many real numbers \( b \) result in the quadratic equation
\[
x^2 - 2bx + 3b = 0
\]
having only integer solutions for \( x \), we will begin by applying Vieta's formulas to find the sum and product of the roots.
Let the two integer solutions be \( r_1 \) and \( r_2 \). According to Vieta's formula... | 10,239 | [
{
"from": "human",
"value": "How many real numbers $b$ satisfy the condition that the quadratic equation $x^2 - 2bx + 3b = 0$ has only integer solutions for $x$?"
},
{
"from": "gpt",
"value": "To determine how many real numbers \\( b \\) result in the quadratic equation \n\n\\[\nx^2 - 2bx + 3b =... |
Let $x$ and $y$ be positive real numbers. Find the maximum possible value of $$\frac{x^2y^2+1-\sqrt{x^4y^4+2x^2y^2+1}}{xy}.$$ | Let $x$ and $y$ be positive real numbers. Find the maximum possible value of $$\frac{x^2y^2+1-\sqrt{x^4y^4+2x^2y^2+1}}{xy}.$$ | Let \( z = xy \). We can rewrite the expression we want to maximize in terms of \( z \).
The expression is given by:
\[
E = \frac{x^2y^2 + 1 - \sqrt{x^4y^4 + 2x^2y^2 + 1}}{xy} = \frac{(xy)^2 + 1 - \sqrt{(xy)^4 + 2(xy)^2 + 1}}{xy} = \frac{z^2 + 1 - \sqrt{z^4 + 2z^2 + 1}}{z}.
\]
Next, we simplify \( \sqrt{z^4 + 2z^2 +... | 10,240 | [
{
"from": "human",
"value": "Let $x$ and $y$ be positive real numbers. Find the maximum possible value of $$\\frac{x^2y^2+1-\\sqrt{x^4y^4+2x^2y^2+1}}{xy}.$$"
},
{
"from": "gpt",
"value": "Let \\( z = xy \\). We can rewrite the expression we want to maximize in terms of \\( z \\).\n\nThe expressi... |
A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have? [asy]
/* AMC8 1998 #17 Problem */
pair A=(0,48), B=(0,0), C=(48,0), D=(48,48);
pair E=(24,72), F=(24,24)... | A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have? [asy]
/* AMC8 1998 #17 Problem */
pair A=(0,48), B=(0,0), C=(48,0), D=(48,48);
pair E=(24,72), F=(24,24)... | To determine the number of diagonals in a cube, we first analyze the structure and the vertices of the cube.
1. **Vertices of a Cube**: A cube has \(8\) vertices.
2. **Edges of a Cube**: A cube has \(12\) edges.
**Step 1: Total Line Segments Connecting Vertices**
We can connect any two vertices with a line segment. T... | 10,241 | [
{
"from": "human",
"value": "A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have? [asy]\n/* AMC8 1998 #17 Problem */\npair A=(0,48), B=(0,0), C=(... |
Find the value of $b$ for which the equation
$$2x^2+3x+b=0$$
has exactly one real root, and its multiplicity is 3. | Find the value of $b$ for which the equation
$$2x^2+3x+b=0$$
has exactly one real root, and its multiplicity is 3. | To have exactly one real root with multiplicity 3 for the quadratic equation \( 2x^2 + 3x + b = 0 \), the equation must be a perfect square. In fact, a quadratic can have a double root when the discriminant is zero.
The standard form of a quadratic equation \( ax^2 + bx + c = 0 \) yields a discriminant given by:
\[
... | 10,242 | [
{
"from": "human",
"value": "Find the value of $b$ for which the equation\n$$2x^2+3x+b=0$$\nhas exactly one real root, and its multiplicity is 3."
},
{
"from": "gpt",
"value": "To have exactly one real root with multiplicity 3 for the quadratic equation \\( 2x^2 + 3x + b = 0 \\), the equation mu... |
In a music competition, five judges will evaluate the performances of six contestants. Each judge will give a score to each contestant, and the contestant with the highest total score will win. If each judge can give only one of three possible scores (0, 1, or 2), how many different scoring combinations are possible fo... | In a music competition, five judges will evaluate the performances of six contestants. Each judge will give a score to each contestant, and the contestant with the highest total score will win. If each judge can give only one of three possible scores (0, 1, or 2), how many different scoring combinations are possible fo... | To determine how many different scoring combinations are possible for a single contestant evaluated by five judges where each judge can give a score of 0, 1, or 2, we will analyze the situation as follows.
Each judge has three distinct choices for scoring:
- 0 points
- 1 point
- 2 points
Since there are 5 judges and... | 10,243 | [
{
"from": "human",
"value": "In a music competition, five judges will evaluate the performances of six contestants. Each judge will give a score to each contestant, and the contestant with the highest total score will win. If each judge can give only one of three possible scores (0, 1, or 2), how many diffe... |
Let $S$ be the sum of all the real coefficients of the expansion of ${(1+ix)}^{2009}$. What is $\log_{2}(S)$? | Let $S$ be the sum of all the real coefficients of the expansion of ${(1+ix)}^{2009}$. What is $\log_{2}(S)$? | We start with the expression \((1 + ix)^{2009}\). To find the sum of all the real coefficients in the expansion, we can evaluate the polynomial at \(x = 1\) and \(x = -1\):
\[
(1 + ix)^{2009} = (1 + i)^{2009} \cdot (1 - i)^{2009}
\]
Next, we need to calculate \(1 + i\) and \(1 - i\):
\[
1 + i = \sqrt{2} \left( \cos\... | 10,244 | [
{
"from": "human",
"value": "Let $S$ be the sum of all the real coefficients of the expansion of ${(1+ix)}^{2009}$. What is $\\log_{2}(S)$?"
},
{
"from": "gpt",
"value": "We start with the expression \\((1 + ix)^{2009}\\). To find the sum of all the real coefficients in the expansion, we can ev... |
The lines $2x + 3y - 4 = 0$ and $bx - 2y + 1 = 0$ are perpendicular to each other. What is the value of $b$? | The lines $2x + 3y - 4 = 0$ and $bx - 2y + 1 = 0$ are perpendicular to each other. What is the value of $b$? | To determine the value of \( b \) such that the lines \( 2x + 3y - 4 = 0 \) and \( bx - 2y + 1 = 0 \) are perpendicular, we need to find the slopes of both lines and use the property that the product of their slopes must equal \(-1\).
First, we convert the first line to slope-intercept form (i.e., \( y = mx + c \)):
... | 10,245 | [
{
"from": "human",
"value": "The lines $2x + 3y - 4 = 0$ and $bx - 2y + 1 = 0$ are perpendicular to each other. What is the value of $b$?"
},
{
"from": "gpt",
"value": "To determine the value of \\( b \\) such that the lines \\( 2x + 3y - 4 = 0 \\) and \\( bx - 2y + 1 = 0 \\) are perpendicular, ... |
Find the largest integer value of $n$ such that $n^2-9n+18$ is negative. | Find the largest integer value of $n$ such that $n^2-9n+18$ is negative. | To determine the values of \( n \) for which the expression \( n^2 - 9n + 18 \) is negative, we start by finding the roots of the quadratic equation \( n^2 - 9n + 18 = 0 \).
We can use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \( a = 1 \), \( b = -9 \), and \( c = 18 \).
Calculating th... | 10,246 | [
{
"from": "human",
"value": "Find the largest integer value of $n$ such that $n^2-9n+18$ is negative."
},
{
"from": "gpt",
"value": "To determine the values of \\( n \\) for which the expression \\( n^2 - 9n + 18 \\) is negative, we start by finding the roots of the quadratic equation \\( n^2 - ... |
If $a \ge b > 1$, what is the largest possible value of $\log_a (a/b) + \log_b (b/a)$? | If $a \ge b > 1$, what is the largest possible value of $\log_a (a/b) + \log_b (b/a)$? | Let \( x = \log_a \left( \frac{a}{b} \right) + \log_b \left( \frac{b}{a} \right) \).
We start by simplifying the expressions using the properties of logarithms:
\[
\log_a\left( \frac{a}{b} \right) = \log_a a - \log_a b = 1 - \log_a b
\]
\[
\log_b\left( \frac{b}{a} \right) = \log_b b - \log_b a = 1 - \log_b a
\]
Sub... | 10,247 | [
{
"from": "human",
"value": "If $a \\ge b > 1$, what is the largest possible value of $\\log_a (a/b) + \\log_b (b/a)$?"
},
{
"from": "gpt",
"value": "Let \\( x = \\log_a \\left( \\frac{a}{b} \\right) + \\log_b \\left( \\frac{b}{a} \\right) \\).\n\nWe start by simplifying the expressions using ... |
Solve for $y$ in the equation $\frac{2y}{y+1} + \frac{3}{y-2} = 1.$ | Solve for $y$ in the equation $\frac{2y}{y+1} + \frac{3}{y-2} = 1.$ | To solve the equation
\[
\frac{2y}{y+1} + \frac{3}{y-2} = 1,
\]
we start by eliminating the fractions. We can accomplish this by multiplying both sides of the equation by the least common denominator, which is \((y+1)(y-2)\).
First, we rewrite the equation:
\[
\frac{2y (y-2)}{(y+1)(y-2)} + \frac{3(y+1)}{(y-2)(y+1... | 10,248 | [
{
"from": "human",
"value": "Solve for $y$ in the equation $\\frac{2y}{y+1} + \\frac{3}{y-2} = 1.$"
},
{
"from": "gpt",
"value": "To solve the equation \n\n\\[\n\\frac{2y}{y+1} + \\frac{3}{y-2} = 1,\n\\]\n\nwe start by eliminating the fractions. We can accomplish this by multiplying both sides o... |
The graphs of $2y + x + 3 = 0$ and $3y + ax + 2 = 0$ are perpendicular. Solve for $a.$ | The graphs of $2y + x + 3 = 0$ and $3y + ax + 2 = 0$ are perpendicular. Solve for $a.$ | To determine the value of \( a \) such that the lines given by the equations \( 2y + x + 3 = 0 \) and \( 3y + ax + 2 = 0 \) are perpendicular, we will first find the slopes of both lines and then use the condition for perpendicularity.
**Step 1: Find the slope of the first line.**
Starting with the equation of the fi... | 10,249 | [
{
"from": "human",
"value": "The graphs of $2y + x + 3 = 0$ and $3y + ax + 2 = 0$ are perpendicular. Solve for $a.$"
},
{
"from": "gpt",
"value": "To determine the value of \\( a \\) such that the lines given by the equations \\( 2y + x + 3 = 0 \\) and \\( 3y + ax + 2 = 0 \\) are perpendicular,... |
Let $g(x) = \cos(x) + \sin(x)$, where $x$ is in radians. What is the range of $g$? Express your answer in interval notation. | Let $g(x) = \cos(x) + \sin(x)$, where $x$ is in radians. What is the range of $g$? Express your answer in interval notation. | To find the range of the function \( g(x) = \cos x + \sin x \), we start by rewriting \( g(x) \).
We can use the sine of a sum formula:
\[
g(x) = \cos x + \sin x = \sqrt{2} \left( \frac{\cos x}{\sqrt{2}} + \frac{\sin x}{\sqrt{2}} \right)
\]
Noticing that \( \frac{1}{\sqrt{2}} \) gives us the cosine and sine of \( \... | 10,250 | [
{
"from": "human",
"value": "Let $g(x) = \\cos(x) + \\sin(x)$, where $x$ is in radians. What is the range of $g$? Express your answer in interval notation."
},
{
"from": "gpt",
"value": "To find the range of the function \\( g(x) = \\cos x + \\sin x \\), we start by rewriting \\( g(x) \\). \n\nW... |
A bag contains 10 marbles, 5 of which are blue and 5 of which are green. The marbles are drawn one by one, without replacement, and placed in a row. What is the probability that the first and last marbles drawn are of different colors? | A bag contains 10 marbles, 5 of which are blue and 5 of which are green. The marbles are drawn one by one, without replacement, and placed in a row. What is the probability that the first and last marbles drawn are of different colors? | To determine the probability that the first and last marbles drawn from the bag are of different colors, we start by observing the contents of the bag. The bag contains 5 blue marbles and 5 green marbles, making a total of 10 marbles.
First, we note the total number of ways to choose the first and last marbles, regard... | 10,251 | [
{
"from": "human",
"value": "A bag contains 10 marbles, 5 of which are blue and 5 of which are green. The marbles are drawn one by one, without replacement, and placed in a row. What is the probability that the first and last marbles drawn are of different colors?"
},
{
"from": "gpt",
"value": "... |
In triangle $PQR$, we have $\angle P = 90^\circ$, $QR = 20$, and $\tan R = 4\sin R$. What is $PR$? | In triangle $PQR$, we have $\angle P = 90^\circ$, $QR = 20$, and $\tan R = 4\sin R$. What is $PR$? | In triangle \( PQR \), where \( \angle P = 90^\circ \), we have \( QR = 20 \) and the relationship \( \tan R = 4 \sin R \). Let's denote \( PQ = a \), \( PR = b \), and \( QR = c = 20 \).
We start with the trigonometric identities:
\[
\tan R = \frac{PQ}{PR} = \frac{a}{b}, \quad \text{and} \quad \sin R = \frac{PQ}{QR}... | 10,252 | [
{
"from": "human",
"value": "In triangle $PQR$, we have $\\angle P = 90^\\circ$, $QR = 20$, and $\\tan R = 4\\sin R$. What is $PR$?"
},
{
"from": "gpt",
"value": "In triangle \\( PQR \\), where \\( \\angle P = 90^\\circ \\), we have \\( QR = 20 \\) and the relationship \\( \\tan R = 4 \\sin R \... |
Let $m$ be the number of positive integers less than $1000$ that are a multiple of $2$ but not $4$, and let $n$ be the number of positive integers less than $1000$ that are a multiple of $6$ but not $12$. Compute $(m-n)^2$. | Let $m$ be the number of positive integers less than $1000$ that are a multiple of $2$ but not $4$, and let $n$ be the number of positive integers less than $1000$ that are a multiple of $6$ but not $12$. Compute $(m-n)^2$. | To solve for \( m \) and \( n \), we start with \( m \), which counts the positive integers less than \( 1000 \) that are multiples of \( 2 \) but not \( 4 \).
1. **Counting multiples of \( 2 \)**:
The multiples of \( 2 \) less than \( 1000 \) can be calculated as follows:
\[
2k < 1000 \quad \Rightarrow \quad... | 10,253 | [
{
"from": "human",
"value": "Let $m$ be the number of positive integers less than $1000$ that are a multiple of $2$ but not $4$, and let $n$ be the number of positive integers less than $1000$ that are a multiple of $6$ but not $12$. Compute $(m-n)^2$."
},
{
"from": "gpt",
"value": "To solve for... |
Suppose $f(x)=\frac{1+x^2}{2x}$, and let $g(x)=f(f(x))$. Find $g(-2)$. | Suppose $f(x)=\frac{1+x^2}{2x}$, and let $g(x)=f(f(x))$. Find $g(-2)$. | To find \( g(-2) \), we start by evaluating \( f(-2) \) using the definition of the function \( f \).
Given \( f(x) = \frac{1+x^2}{2x} \), we compute:
\[
f(-2) = \frac{1 + (-2)^2}{2 \cdot (-2)} = \frac{1 + 4}{-4} = \frac{5}{-4} = -\frac{5}{4}
\]
Next, we need to evaluate \( g(x) = f(f(x)) \). Thus, we need to comput... | 10,254 | [
{
"from": "human",
"value": "Suppose $f(x)=\\frac{1+x^2}{2x}$, and let $g(x)=f(f(x))$. Find $g(-2)$."
},
{
"from": "gpt",
"value": "To find \\( g(-2) \\), we start by evaluating \\( f(-2) \\) using the definition of the function \\( f \\).\n\nGiven \\( f(x) = \\frac{1+x^2}{2x} \\), we compute:\n... |
Let a and b be the roots of the equation x^2+px+q=0. Given that the arithmetic mean of the squares of the roots is 6 and the geometric mean of the cubes of the roots is 5. Find p^2. | Let a and b be the roots of the equation x^2+px+q=0. Given that the arithmetic mean of the squares of the roots is 6 and the geometric mean of the cubes of the roots is 5. Find p^2. | Let \( a \) and \( b \) be the roots of the quadratic equation \( x^2 + px + q = 0 \). By Vieta's formulas, we know that:
\[
a + b = -p \quad \text{(sum of roots)}
\]
\[
ab = q \quad \text{(product of roots)}
\]
We are given that the arithmetic mean of the squares of the roots is 6. The arithmetic mean of \( a^2 \) a... | 10,255 | [
{
"from": "human",
"value": "Let a and b be the roots of the equation x^2+px+q=0. Given that the arithmetic mean of the squares of the roots is 6 and the geometric mean of the cubes of the roots is 5. Find p^2."
},
{
"from": "gpt",
"value": "Let \\( a \\) and \\( b \\) be the roots of the quadra... |
A company produces a special product code that consists of a three-character combination: a capital letter, followed by a symbol (!, @, #, $, or %), and then a lowercase letter. However, there is a restriction that certain combinations of letters and symbols are not allowed due to trademark issues. Specifically, the co... | A company produces a special product code that consists of a three-character combination: a capital letter, followed by a symbol (!, @, #, $, or %), and then a lowercase letter. However, there is a restriction that certain combinations of letters and symbols are not allowed due to trademark issues. Specifically, the co... | To determine the number of valid product codes that can be created under the given restrictions, we will follow these steps:
1. Calculate the total number of possible combinations without considering the restrictions.
2. Subtract the number of restricted combinations.
### Step 1: Calculate Total Combinations Without ... | 10,256 | [
{
"from": "human",
"value": "A company produces a special product code that consists of a three-character combination: a capital letter, followed by a symbol (!, @, #, $, or %), and then a lowercase letter. However, there is a restriction that certain combinations of letters and symbols are not allowed due ... |
In triangle $ABC$, $\angle B = 120^\circ$, $AB = 10$ cm, and the bisector of $\angle A$ intersects side $BC$ at point $D$. If $BD:DC = 2:5$, find the length of side $AC$. | In triangle $ABC$, $\angle B = 120^\circ$, $AB = 10$ cm, and the bisector of $\angle A$ intersects side $BC$ at point $D$. If $BD:DC = 2:5$, find the length of side $AC$. | Let's denote the sides of triangle \( ABC \) as follows: let \( AC = b \), \( BC = a \), and \( AB = c = 10 \text{ cm} \).
Given that \( \angle B = 120^\circ \) and the angle bisector \( AD \) divides side \( BC \) in the ratio \( BD:DC = 2:5 \), we can denote \( BD = 2x \) and \( DC = 5x \). Thus, the length of \( B... | 10,257 | [
{
"from": "human",
"value": "In triangle $ABC$, $\\angle B = 120^\\circ$, $AB = 10$ cm, and the bisector of $\\angle A$ intersects side $BC$ at point $D$. If $BD:DC = 2:5$, find the length of side $AC$."
},
{
"from": "gpt",
"value": "Let's denote the sides of triangle \\( ABC \\) as follows: let... |
A rumor is spreading in a high school such that each day, every student who has heard the rumor tells it to all their classmates who have not yet heard it. The school has 1000 students, and initially only 5 students have heard the rumor. If the students are evenly distributed into 20 classes of 50 students each, and th... | A rumor is spreading in a high school such that each day, every student who has heard the rumor tells it to all their classmates who have not yet heard it. The school has 1000 students, and initially only 5 students have heard the rumor. If the students are evenly distributed into 20 classes of 50 students each, and th... | To solve this problem, we need to consider how the rumor spreads within the constraints of the classroom distribution.
The school has 1000 students distributed evenly into 20 classes of 50 students each. The rumor starts with 5 students, and these students can only tell their classmates within their own class.
### D... | 10,258 | [
{
"from": "human",
"value": "A rumor is spreading in a high school such that each day, every student who has heard the rumor tells it to all their classmates who have not yet heard it. The school has 1000 students, and initially only 5 students have heard the rumor. If the students are evenly distributed in... |
Evaluate the sum \[\frac{1}{1^2} - \frac{2}{2^2} + \frac{3}{3^2} - \cdots + \frac{k}{k^2} + \cdots \] | Evaluate the sum \[\frac{1}{1^2} - \frac{2}{2^2} + \frac{3}{3^2} - \cdots + \frac{k}{k^2} + \cdots \] | To evaluate the sum
\[
S = \frac{1}{1^2} - \frac{2}{2^2} + \frac{3}{3^2} - \frac{4}{4^2} + \cdots + \frac{k}{k^2} + \cdots
\]
we can express \( S \) in a more convenient form:
\[
S = \sum_{k=1}^{\infty} \frac{(-1)^{k+1} k}{k^2} = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}
\]
Now, observe that the term \( \frac{k}{k^... | 10,259 | [
{
"from": "human",
"value": "Evaluate the sum \\[\\frac{1}{1^2} - \\frac{2}{2^2} + \\frac{3}{3^2} - \\cdots + \\frac{k}{k^2} + \\cdots \\]"
},
{
"from": "gpt",
"value": "To evaluate the sum \n\n\\[\nS = \\frac{1}{1^2} - \\frac{2}{2^2} + \\frac{3}{3^2} - \\frac{4}{4^2} + \\cdots + \\frac{k}{k^2} ... |
How many integers from 1 to 1000 have exactly 5 positive divisors? | How many integers from 1 to 1000 have exactly 5 positive divisors? | To determine how many integers from 1 to 1000 have exactly 5 positive divisors, we need to analyze the structure of a number's divisors based on its prime factorization.
If a positive integer \( n \) can be expressed as:
\[
n = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k}
\]
where \( p_1, p_2, \ldots, p_k \) are dist... | 10,260 | [
{
"from": "human",
"value": "How many integers from 1 to 1000 have exactly 5 positive divisors?"
},
{
"from": "gpt",
"value": "To determine how many integers from 1 to 1000 have exactly 5 positive divisors, we need to analyze the structure of a number's divisors based on its prime factorization.... |
Consider the expression $e^{-t^2} \cdot t^n$. For what value of $n$ will the maximum value of this expression occur when $t = 2$? | Consider the expression $e^{-t^2} \cdot t^n$. For what value of $n$ will the maximum value of this expression occur when $t = 2$? | To find the value of \( n \) such that the maximum of the expression \( e^{-t^2} \cdot t^n \) occurs at \( t = 2 \), we start by defining the function:
\[
f(t) = e^{-t^2} \cdot t^n
\]
To locate the maximum of \( f(t) \), we compute its derivative and set it to zero. Using the product rule:
\[
f'(t) = \frac{d}{dt}(e^... | 10,261 | [
{
"from": "human",
"value": "Consider the expression $e^{-t^2} \\cdot t^n$. For what value of $n$ will the maximum value of this expression occur when $t = 2$?"
},
{
"from": "gpt",
"value": "To find the value of \\( n \\) such that the maximum of the expression \\( e^{-t^2} \\cdot t^n \\) occurs... |
Determine all real numbers x for which the function f(x) = 1/(1 + |x|) is increasing. | Determine all real numbers x for which the function f(x) = 1/(1 + |x|) is increasing. | To determine where the function \( f(x) = \frac{1}{1 + |x|} \) is increasing, we start by analyzing the function's behavior with respect to its derivative.
First, we need to consider the absolute value function \( |x| \) and what it means for \( f(x) \):
- For \( x \geq 0 \), we have \( |x| = x \), hence:
\[
f(x)... | 10,262 | [
{
"from": "human",
"value": "Determine all real numbers x for which the function f(x) = 1/(1 + |x|) is increasing."
},
{
"from": "gpt",
"value": "To determine where the function \\( f(x) = \\frac{1}{1 + |x|} \\) is increasing, we start by analyzing the function's behavior with respect to its der... |
The product of the second and fourth terms of an arithmetic sequence is 96. If the common difference is 2, what is the value of the third term? | The product of the second and fourth terms of an arithmetic sequence is 96. If the common difference is 2, what is the value of the third term? | Let the first term of the arithmetic sequence be denoted by \( a \), and the common difference be denoted by \( d \). In this case, we are given that \( d = 2 \).
The terms of the arithmetic sequence can be represented as follows:
- The first term: \( a \)
- The second term: \( a + d = a + 2 \)
- The third term: \( a ... | 10,263 | [
{
"from": "human",
"value": "The product of the second and fourth terms of an arithmetic sequence is 96. If the common difference is 2, what is the value of the third term?"
},
{
"from": "gpt",
"value": "Let the first term of the arithmetic sequence be denoted by \\( a \\), and the common differ... |
In a triangle $ABC$, the altitudes from $A$ and $B$ intersect at a point $H$. Given that $\tan A = \frac{2}{3}$ and $\cot B = \frac{3}{4}$, find $\tan C$. | In a triangle $ABC$, the altitudes from $A$ and $B$ intersect at a point $H$. Given that $\tan A = \frac{2}{3}$ and $\cot B = \frac{3}{4}$, find $\tan C$. | To find \(\tan C\), we start by using the given values of \(\tan A\) and \(\cot B\).
Given:
\[
\tan A = \frac{2}{3}
\]
For \(\cot B\):
\[
\cot B = \frac{3}{4} \implies \tan B = \frac{1}{\cot B} = \frac{4}{3}
\]
Now we can utilize the identity that relates the tangent of three angles in a triangle:
\[
\tan A + \ta... | 10,264 | [
{
"from": "human",
"value": "In a triangle $ABC$, the altitudes from $A$ and $B$ intersect at a point $H$. Given that $\\tan A = \\frac{2}{3}$ and $\\cot B = \\frac{3}{4}$, find $\\tan C$."
},
{
"from": "gpt",
"value": "To find \\(\\tan C\\), we start by using the given values of \\(\\tan A\\) a... |
How many ways can you distribute five identical marbles among three distinct jars if the marbles are indistinguishable and each jar can contain zero or more marbles? | How many ways can you distribute five identical marbles among three distinct jars if the marbles are indistinguishable and each jar can contain zero or more marbles? | To find the number of ways to distribute five identical marbles among three distinct jars, we can approach the problem using the "stars and bars" theorem, which is a common combinatorial method for distributing indistinguishable objects into distinct bins.
In this scenario, we want to count the number of solutions to ... | 10,265 | [
{
"from": "human",
"value": "How many ways can you distribute five identical marbles among three distinct jars if the marbles are indistinguishable and each jar can contain zero or more marbles?"
},
{
"from": "gpt",
"value": "To find the number of ways to distribute five identical marbles among ... |
Find the product of all values of $t$ such that $t^2 = 36$. | Find the product of all values of $t$ such that $t^2 = 36$. | To solve the equation \( t^2 = 36 \), we start by taking the square root of both sides. Recall that taking the square root of a number yields both a positive and a negative value:
\[
t = \sqrt{36}
\]
\[
t = 6 \quad \text{or} \quad t = -6
\]
Thus, the solutions to the equation \( t^2 = 36 \) are \( t = 6 \) and \( t =... | 10,266 | [
{
"from": "human",
"value": "Find the product of all values of $t$ such that $t^2 = 36$."
},
{
"from": "gpt",
"value": "To solve the equation \\( t^2 = 36 \\), we start by taking the square root of both sides. Recall that taking the square root of a number yields both a positive and a negative v... |
In the equation
$$4x^2-kx+2=0$$
determine $k$ such that the difference of the roots be equal to unity. | In the equation
$$4x^2-kx+2=0$$
determine $k$ such that the difference of the roots be equal to unity. | Given the quadratic equation
\[
4x^2 - kx + 2 = 0,
\]
we denote the roots by \(r_1\) and \(r_2\). According to Vieta's formulas, the sum and the product of the roots can be expressed as:
\[
r_1 + r_2 = \frac{k}{4}
\]
and
\[
r_1 r_2 = \frac{2}{4} = \frac{1}{2}.
\]
We want the difference of the roots \( |r_1 - r_2... | 10,267 | [
{
"from": "human",
"value": "In the equation\n$$4x^2-kx+2=0$$\ndetermine $k$ such that the difference of the roots be equal to unity."
},
{
"from": "gpt",
"value": "Given the quadratic equation \n\n\\[\n4x^2 - kx + 2 = 0,\n\\]\n\nwe denote the roots by \\(r_1\\) and \\(r_2\\). According to Vieta... |
Let $f(x) = x^3 - x + c$, where $c$ is a real number. Suppose that $r$, $s$, and $t$ are distinct real zeros of $f$, and that $f(r) = r$, $f(s) = s$, and $f(t) = t$. Find the value of $c$. | Let $f(x) = x^3 - x + c$, where $c$ is a real number. Suppose that $r$, $s$, and $t$ are distinct real zeros of $f$, and that $f(r) = r$, $f(s) = s$, and $f(t) = t$. Find the value of $c$. | Given the function \( f(x) = x^3 - x + c \), we need to find the value of \( c \) under the conditions that \( r \), \( s \), and \( t \) are distinct real zeros of \( f \) such that \( f(r) = r \), \( f(s) = s \), and \( f(t) = t \).
First, we rewrite the conditions \( f(r) = r \), \( f(s) = s \), and \( f(t) = t \) ... | 10,268 | [
{
"from": "human",
"value": "Let $f(x) = x^3 - x + c$, where $c$ is a real number. Suppose that $r$, $s$, and $t$ are distinct real zeros of $f$, and that $f(r) = r$, $f(s) = s$, and $f(t) = t$. Find the value of $c$."
},
{
"from": "gpt",
"value": "Given the function \\( f(x) = x^3 - x + c \\)... |
A box contains a certain number of balls, and when divided into groups of 2, 3, 4, 5, or 6, there is always one ball left over. If the box is not empty, what is the minimum number of balls it can contain? | A box contains a certain number of balls, and when divided into groups of 2, 3, 4, 5, or 6, there is always one ball left over. If the box is not empty, what is the minimum number of balls it can contain? | Let the number of balls in the box be \( N \). According to the problem, when \( N \) is divided by 2, 3, 4, 5, or 6, there is always a remainder of 1. This can be expressed mathematically as:
\[
N \equiv 1 \mod{2}
\]
\[
N \equiv 1 \mod{3}
\]
\[
N \equiv 1 \mod{4}
\]
\[
N \equiv 1 \mod{5}
\]
\[
N \equiv 1 \mod{6}
\]
... | 10,269 | [
{
"from": "human",
"value": "A box contains a certain number of balls, and when divided into groups of 2, 3, 4, 5, or 6, there is always one ball left over. If the box is not empty, what is the minimum number of balls it can contain?"
},
{
"from": "gpt",
"value": "Let the number of balls in the ... |
If the graph of the line $y = 2x + c$ passes through the points $(3,7)$ and $(6,16)$, what is $2c - 3$? | If the graph of the line $y = 2x + c$ passes through the points $(3,7)$ and $(6,16)$, what is $2c - 3$? | To find the value of \( c \) in the equation of the line \( y = 2x + c \), we start by substituting the coordinates of the points that the line passes through.
**Step 1: Using the point (3, 7)**
Substituting \( x = 3 \) and \( y = 7 \) into the line equation:
\[
7 = 2(3) + c
\]
Calculating \( 2(3) \):
\[
7 = 6 + c... | 10,270 | [
{
"from": "human",
"value": "If the graph of the line $y = 2x + c$ passes through the points $(3,7)$ and $(6,16)$, what is $2c - 3$?"
},
{
"from": "gpt",
"value": "To find the value of \\( c \\) in the equation of the line \\( y = 2x + c \\), we start by substituting the coordinates of the point... |
Let $f(x) = x^2$ be a function, and $A(x_1, y_1)$ and $B(x_2, y_2)$ be two points on its graph such that $0 < x_1 < x_2$. The points $C$ and $D$ divide $\overline{AB}$ into four equal parts, with $AC < CB$. A vertical line is drawn through $C$ to intersect the graph of $f(x)$ at point $E(x_3, y_3)$. If $x_1 = 1$ and $x... | Let $f(x) = x^2$ be a function, and $A(x_1, y_1)$ and $B(x_2, y_2)$ be two points on its graph such that $0 < x_1 < x_2$. The points $C$ and $D$ divide $\overline{AB}$ into four equal parts, with $AC < CB$. A vertical line is drawn through $C$ to intersect the graph of $f(x)$ at point $E(x_3, y_3)$. If $x_1 = 1$ and $x... | The points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) on the graph of the function \( f(x) = x^2 \) are given, with \( x_1 = 1 \) and \( x_2 = 4 \). We start by calculating the corresponding \( y \)-coordinates for these points:
\[
y_1 = f(1) = 1^2 = 1,
\]
\[
y_2 = f(4) = 4^2 = 16.
\]
Thus, the points are:
\[
A(1, 1) \... | 10,271 | [
{
"from": "human",
"value": "Let $f(x) = x^2$ be a function, and $A(x_1, y_1)$ and $B(x_2, y_2)$ be two points on its graph such that $0 < x_1 < x_2$. The points $C$ and $D$ divide $\\overline{AB}$ into four equal parts, with $AC < CB$. A vertical line is drawn through $C$ to intersect the graph of $f(x)$ a... |
Let $f(x)$ be the function defined by
$$f(x) = x^2 + 2022 \left( 1 - |\sin x| \right).$$
Find the number of distinct real roots of the equation $f(x) = 2023.$ | Let $f(x)$ be the function defined by
$$f(x) = x^2 + 2022 \left( 1 - |\sin x| \right).$$
Find the number of distinct real roots of the equation $f(x) = 2023.$ | To find the distinct real roots of the equation \( f(x) = 2023 \), we first rewrite the function:
\[
f(x) = x^2 + 2022(1 - |\sin x|).
\]
Setting \( f(x) = 2023 \), we get
\[
x^2 + 2022(1 - |\sin x|) = 2023.
\]
Rearranging this, we find
\[
x^2 + 2022 - 2022|\sin x| = 2023.
\]
This simplifies to
\[
x^2 - 2022|\sin... | 10,272 | [
{
"from": "human",
"value": "Let $f(x)$ be the function defined by\n\n$$f(x) = x^2 + 2022 \\left( 1 - |\\sin x| \\right).$$\n\nFind the number of distinct real roots of the equation $f(x) = 2023.$"
},
{
"from": "gpt",
"value": "To find the distinct real roots of the equation \\( f(x) = 2023 \\),... |
What is the largest positive integer $m$ for which $m^2+36$ is divisible by $m+6$? | What is the largest positive integer $m$ for which $m^2+36$ is divisible by $m+6$? | To determine the largest positive integer \( m \) such that \( m^2 + 36 \) is divisible by \( m + 6 \), we can start by expressing the divisibility condition in terms of polynomial division.
We need to check if \( m^2 + 36 \equiv 0 \mod (m + 6) \). This is equivalent to evaluating \( m^2 + 36 \) at \( m = -6 \):
\[
f... | 10,273 | [
{
"from": "human",
"value": "What is the largest positive integer $m$ for which $m^2+36$ is divisible by $m+6$?"
},
{
"from": "gpt",
"value": "To determine the largest positive integer \\( m \\) such that \\( m^2 + 36 \\) is divisible by \\( m + 6 \\), we can start by expressing the divisibility... |
Let f(x) be a function that satisfies f(f(x)) = x for all real numbers x.
Determine the possible values of f(10) if it is given that f(f(f(10))) = 20. | Let f(x) be a function that satisfies f(f(x)) = x for all real numbers x.
Determine the possible values of f(10) if it is given that f(f(f(10))) = 20. | Given the function \( f \) satisfies the property:
\[
f(f(x)) = x \quad \text{for all } x,
\]
we can deduce that \( f \) is an involution, meaning applying \( f \) twice returns the original input.
We are given the condition:
\[
f(f(f(10))) = 20.
\]
Let's denote \( a = f(10) \). Then we can replace \( f(10) \) wit... | 10,274 | [
{
"from": "human",
"value": "Let f(x) be a function that satisfies f(f(x)) = x for all real numbers x.\n\nDetermine the possible values of f(10) if it is given that f(f(f(10))) = 20."
},
{
"from": "gpt",
"value": "Given the function \\( f \\) satisfies the property:\n\n\\[\nf(f(x)) = x \\quad \\... |
Find the median of five consecutive integers whose sum is -15. | Find the median of five consecutive integers whose sum is -15. | To find the median of five consecutive integers that sum to -15, let's denote the first integer as \( n \). The five consecutive integers can then be represented as:
\[
n, \quad n+1, \quad n+2, \quad n+3, \quad n+4
\]
The sum of these integers is:
\[
n + (n + 1) + (n + 2) + (n + 3) + (n + 4) = 5n + 10
\]
We know th... | 10,275 | [
{
"from": "human",
"value": "Find the median of five consecutive integers whose sum is -15."
},
{
"from": "gpt",
"value": "To find the median of five consecutive integers that sum to -15, let's denote the first integer as \\( n \\). The five consecutive integers can then be represented as:\n\n\\... |
Find the number of integers n with the property that the product of the digits of n equals 1995 and n itself is divisible by 1995. | Find the number of integers n with the property that the product of the digits of n equals 1995 and n itself is divisible by 1995. | To solve this problem, we start by factorizing the number 1995. We can do this as follows:
\[
1995 = 3 \times 5 \times 7 \times 19
\]
Next, we observe that for an integer \( n \) to satisfy the property that its product of digits equals 1995, we first need to find the digits \( d_1, d_2, \ldots, d_k \) of \( n \) suc... | 10,276 | [
{
"from": "human",
"value": "Find the number of integers n with the property that the product of the digits of n equals 1995 and n itself is divisible by 1995."
},
{
"from": "gpt",
"value": "To solve this problem, we start by factorizing the number 1995. We can do this as follows:\n\n\\[\n1995 =... |
The shaded region in the following figure consists of 16 congruent squares. If the area of the entire figure is $1200\pi$ square meters, what is the side length of one of the squares? Express your answer in terms of $\pi$. | The shaded region in the following figure consists of 16 congruent squares. If the area of the entire figure is $1200\pi$ square meters, what is the side length of one of the squares? Express your answer in terms of $\pi$. | To solve the problem, we start by observing the given information. The total area of the entire figure is \( 1200\pi \) square meters, and it consists of 16 congruent squares.
Let the side length of one of the squares be denoted as \( s \). Then, the area of one square is given by:
\[
s^2
\]
Since there are 16 cong... | 10,277 | [
{
"from": "human",
"value": "The shaded region in the following figure consists of 16 congruent squares. If the area of the entire figure is $1200\\pi$ square meters, what is the side length of one of the squares? Express your answer in terms of $\\pi$."
},
{
"from": "gpt",
"value": "To solve th... |
Four boxes are marked A and three other boxes are marked B. The seven boxes are randomly arranged in a row. What is the probability that the arrangement reads ABABABA? | Four boxes are marked A and three other boxes are marked B. The seven boxes are randomly arranged in a row. What is the probability that the arrangement reads ABABABA? | To find the probability that the boxes arranged in a row read ABABABA, we can follow these steps:
1. **Total Arrangements**: First, we need to determine the total number of arrangements of 4 boxes marked A and 3 boxes marked B. The total number of boxes is 7.
The formula for the number of arrangements of \( n \) obje... | 10,278 | [
{
"from": "human",
"value": "Four boxes are marked A and three other boxes are marked B. The seven boxes are randomly arranged in a row. What is the probability that the arrangement reads ABABABA?"
},
{
"from": "gpt",
"value": "To find the probability that the boxes arranged in a row read ABABAB... |
A bicycle is sold for $198 after a 10% discount. How much did the bicycle originally cost? | A bicycle is sold for $198 after a 10% discount. How much did the bicycle originally cost? | To find the original cost of the bicycle before the 10% discount, we can represent the original price as \( P \).
The bicycle is sold for $198 after applying a 10% discount. This means that the selling price is 90% of the original price, since a 10% discount leaves you with 90% of the cost.
We can express this relati... | 10,279 | [
{
"from": "human",
"value": "A bicycle is sold for $198 after a 10% discount. How much did the bicycle originally cost?"
},
{
"from": "gpt",
"value": "To find the original cost of the bicycle before the 10% discount, we can represent the original price as \\( P \\).\n\nThe bicycle is sold for $1... |
Expand $(x+y)(x+2y)$. | Expand $(x+y)(x+2y)$. | To expand the expression \((x+y)(x+2y)\), we will use the distributive property, also known as the FOIL method for binomials.
1. First, distribute \(x\) from the first binomial to each term in the second binomial:
\[
x \cdot (x + 2y) = x^2 + 2xy
\]
2. Next, distribute \(y\) from the first binomial to each te... | 10,280 | [
{
"from": "human",
"value": "Expand $(x+y)(x+2y)$."
},
{
"from": "gpt",
"value": "To expand the expression \\((x+y)(x+2y)\\), we will use the distributive property, also known as the FOIL method for binomials.\n\n1. First, distribute \\(x\\) from the first binomial to each term in the second bin... |
The population of a certain city is growing at a rate of 5% per year. If the population in 2010 was approximately 2.5 million people, what will be the estimated population in 2025? | The population of a certain city is growing at a rate of 5% per year. If the population in 2010 was approximately 2.5 million people, what will be the estimated population in 2025? | To calculate the estimated population of the city in 2025, we can use the formula for exponential growth:
\[
P(t) = P_0 \times (1 + r)^t
\]
where:
- \(P(t)\) is the population at time \(t\),
- \(P_0\) is the initial population,
- \(r\) is the growth rate (as a decimal),
- \(t\) is the number of years since the initia... | 10,281 | [
{
"from": "human",
"value": "The population of a certain city is growing at a rate of 5% per year. If the population in 2010 was approximately 2.5 million people, what will be the estimated population in 2025?"
},
{
"from": "gpt",
"value": "To calculate the estimated population of the city in 20... |
The mean of $3,5$ and $15$ is equal to the mean of $8$ and $x$. What is the value of $x$? | The mean of $3,5$ and $15$ is equal to the mean of $8$ and $x$. What is the value of $x$? | To find the value of \( x \), we start by calculating the mean of the numbers \( 3, 5, \) and \( 15 \).
The mean is calculated as follows:
\[
\text{Mean} = \frac{\text{Sum of the numbers}}{\text{Number of values}}
\]
Calculating the sum of \( 3, 5, \) and \( 15 \):
\[
3 + 5 + 15 = 23
\]
The number of values is \( ... | 10,282 | [
{
"from": "human",
"value": "The mean of $3,5$ and $15$ is equal to the mean of $8$ and $x$. What is the value of $x$?"
},
{
"from": "gpt",
"value": "To find the value of \\( x \\), we start by calculating the mean of the numbers \\( 3, 5, \\) and \\( 15 \\).\n\nThe mean is calculated as follows... |
The diagonal of a rectangle is 3 inches longer than twice the length of the shorter side. The perimeter of the rectangle is 160 inches. What is the length of the longer side, in inches? | The diagonal of a rectangle is 3 inches longer than twice the length of the shorter side. The perimeter of the rectangle is 160 inches. What is the length of the longer side, in inches? | Let \( l \) be the length of the longer side and \( w \) be the length of the shorter side of the rectangle.
From the problem statement, we have the following relationships:
1. The perimeter of the rectangle is given by the formula:
\[
P = 2(l + w) = 160
\]
Dividing both sides by 2,
\[
l + w = 80
... | 10,283 | [
{
"from": "human",
"value": "The diagonal of a rectangle is 3 inches longer than twice the length of the shorter side. The perimeter of the rectangle is 160 inches. What is the length of the longer side, in inches?"
},
{
"from": "gpt",
"value": "Let \\( l \\) be the length of the longer side and... |
Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive intege... | Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive intege... | To solve the problem, we start by defining the variables involved. Let \( j = 25 \) (Jane's current age) and \( d \) (Dick's current age) where \( d > 25 \). After \( n \) years, Jane's age will be \( 25 + n \) and Dick's age will be \( d + n \).
We require that both ages are two-digit numbers and that Jane's age can ... | 10,284 | [
{
"from": "human",
"value": "Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. ... |
Consider the cube in 3-dimensional space with vertices $(0,0,0)$, $(2,0,0)$, $(0,3,0)$, $(0,0,5)$, $(2,3,0)$, $(2,0,5)$, $(0,3,5)$, and $(2,3,5)$. Find the area of the triangle with vertices $(1,0,1)$, $(0,2,3)$, and $(1,3,0)$. | Consider the cube in 3-dimensional space with vertices $(0,0,0)$, $(2,0,0)$, $(0,3,0)$, $(0,0,5)$, $(2,3,0)$, $(2,0,5)$, $(0,3,5)$, and $(2,3,5)$. Find the area of the triangle with vertices $(1,0,1)$, $(0,2,3)$, and $(1,3,0)$. | To find the area of the triangle with vertices \( A = (1, 0, 1) \), \( B = (0, 2, 3) \), and \( C = (1, 3, 0) \), we can use the formula for the area of a triangle defined by three points in three-dimensional space:
\[
\text{Area} = \frac{1}{2} \left\| \vec{AB} \times \vec{AC} \right\|
\]
First, we need to calculate ... | 10,285 | [
{
"from": "human",
"value": "Consider the cube in 3-dimensional space with vertices $(0,0,0)$, $(2,0,0)$, $(0,3,0)$, $(0,0,5)$, $(2,3,0)$, $(2,0,5)$, $(0,3,5)$, and $(2,3,5)$. Find the area of the triangle with vertices $(1,0,1)$, $(0,2,3)$, and $(1,3,0)$."
},
{
"from": "gpt",
"value": "To find... |
A bakery sells a special bread that requires a specific ratio of flour to sugar. If a batch of bread made with 12 cups of flour and 3 cups of sugar yields 20 loaves, how many loaves can be made with 15 cups of flour and an unknown amount of sugar, given that the ratio of flour to sugar remains the same? | A bakery sells a special bread that requires a specific ratio of flour to sugar. If a batch of bread made with 12 cups of flour and 3 cups of sugar yields 20 loaves, how many loaves can be made with 15 cups of flour and an unknown amount of sugar, given that the ratio of flour to sugar remains the same? | To determine how many loaves can be made with 15 cups of flour while maintaining the same flour-to-sugar ratio, we start by finding the original flour-to-sugar ratio in the initial batch.
The initial batch uses:
- Flour = 12 cups
- Sugar = 3 cups
First, we calculate the flour-to-sugar ratio:
\[
\text{Ratio of flour ... | 10,286 | [
{
"from": "human",
"value": "A bakery sells a special bread that requires a specific ratio of flour to sugar. If a batch of bread made with 12 cups of flour and 3 cups of sugar yields 20 loaves, how many loaves can be made with 15 cups of flour and an unknown amount of sugar, given that the ratio of flour t... |
A cylindrical tank with a height of 10 meters and a radius of 4 meters is partially filled with water. A sphere with a radius of 2 meters is submerged in the water. The water level rises by 1 meter. Find the volume of the sphere that is not submerged in the water. Express your answer in terms of π. | A cylindrical tank with a height of 10 meters and a radius of 4 meters is partially filled with water. A sphere with a radius of 2 meters is submerged in the water. The water level rises by 1 meter. Find the volume of the sphere that is not submerged in the water. Express your answer in terms of π. | To solve the problem, we need to find the volume of the sphere that is not submerged in the water after it has been placed into the tank.
First, we can determine the volume of the cylindrical tank that corresponds to the increase in water level due to the submerged sphere. The cylinder has a radius \( r = 4 \) meters... | 10,287 | [
{
"from": "human",
"value": "A cylindrical tank with a height of 10 meters and a radius of 4 meters is partially filled with water. A sphere with a radius of 2 meters is submerged in the water. The water level rises by 1 meter. Find the volume of the sphere that is not submerged in the water. Express your a... |
If $x$, $y$, $z$, and $w$ are real numbers such that $\sin x + \cos y = 3$, $\cos y + \tan z = 4$, $\tan z + \ln w = 5$, and $\ln w + \sin x = 6$, find the value of $\sin x + \cos y + \tan z + \ln w$. | If $x$, $y$, $z$, and $w$ are real numbers such that $\sin x + \cos y = 3$, $\cos y + \tan z = 4$, $\tan z + \ln w = 5$, and $\ln w + \sin x = 6$, find the value of $\sin x + \cos y + \tan z + \ln w$. | Given the equations:
\[
\sin x + \cos y = 3 \tag{1}
\]
\[
\cos y + \tan z = 4 \tag{2}
\]
\[
\tan z + \ln w = 5 \tag{3}
\]
\[
\ln w + \sin x = 6 \tag{4}
\]
We need to find the value of \( S = \sin x + \cos y + \tan z + \ln w \).
From equation (1), we express \(\cos y\) in terms of \(\sin x\):
\[
\cos y = 3 - \sin x ... | 10,288 | [
{
"from": "human",
"value": "If $x$, $y$, $z$, and $w$ are real numbers such that $\\sin x + \\cos y = 3$, $\\cos y + \\tan z = 4$, $\\tan z + \\ln w = 5$, and $\\ln w + \\sin x = 6$, find the value of $\\sin x + \\cos y + \\tan z + \\ln w$."
},
{
"from": "gpt",
"value": "Given the equations:\n\... |
Consider the four numbers:
\[2 + \sqrt{3}, \; 2 - \sqrt{3}, \; -1 + \sqrt{5}, \; -1 - \sqrt{5}\]
as roots of the same nonzero polynomial with rational coefficients.
What is the smallest possible degree of the polynomial? | Consider the four numbers:
\[2 + \sqrt{3}, \; 2 - \sqrt{3}, \; -1 + \sqrt{5}, \; -1 - \sqrt{5}\]
as roots of the same nonzero polynomial with rational coefficients.
What is the smallest possible degree of the polynomial? | To find the smallest degree of a polynomial with rational coefficients that has the given numbers as roots, we need to check if they are all rational or if they involve irrational numbers. If they involve irrationals, we must also take the conjugates of any irrational roots to ensure the polynomial has rational coeffic... | 10,289 | [
{
"from": "human",
"value": "Consider the four numbers:\n\\[2 + \\sqrt{3}, \\; 2 - \\sqrt{3}, \\; -1 + \\sqrt{5}, \\; -1 - \\sqrt{5}\\]\nas roots of the same nonzero polynomial with rational coefficients.\nWhat is the smallest possible degree of the polynomial?"
},
{
"from": "gpt",
"value": "To ... |
Given that $2^{10x} \cdot 8^{15x} \cdot 16^{5x} = 4^{50x}$, find the value of $x$. | Given that $2^{10x} \cdot 8^{15x} \cdot 16^{5x} = 4^{50x}$, find the value of $x$. | To begin, we will express all the terms in the equation \(2^{10x} \cdot 8^{15x} \cdot 16^{5x} = 4^{50x}\) in terms of base \(2\).
First, we convert \(8\) and \(16\):
- \(8 = 2^3\), therefore \(8^{15x} = (2^3)^{15x} = 2^{45x}\).
- \(16 = 2^4\), hence \(16^{5x} = (2^4)^{5x} = 2^{20x}\).
- \(4 = 2^2\), so \(4^{50x} = (2^... | 10,290 | [
{
"from": "human",
"value": "Given that $2^{10x} \\cdot 8^{15x} \\cdot 16^{5x} = 4^{50x}$, find the value of $x$."
},
{
"from": "gpt",
"value": "To begin, we will express all the terms in the equation \\(2^{10x} \\cdot 8^{15x} \\cdot 16^{5x} = 4^{50x}\\) in terms of base \\(2\\).\n\nFirst, we co... |
Determine the range of $x$ that satisfies the inequality $\frac{2x-1}{x^2-x-2}>0$.
Express your answer in interval notation. | Determine the range of $x$ that satisfies the inequality $\frac{2x-1}{x^2-x-2}>0$.
Express your answer in interval notation. | To solve the inequality \(\frac{2x-1}{x^2-x-2} > 0\), we begin with the critical points of the function by determining where the numerator and denominator are zero.
**Step 1: Find critical points.**
1. **Numerator**: Set \(2x - 1 = 0\).
\[
2x = 1 \implies x = \frac{1}{2}
\]
2. **Denominator**: Factor \(x^2 ... | 10,291 | [
{
"from": "human",
"value": "Determine the range of $x$ that satisfies the inequality $\\frac{2x-1}{x^2-x-2}>0$.\n\nExpress your answer in interval notation."
},
{
"from": "gpt",
"value": "To solve the inequality \\(\\frac{2x-1}{x^2-x-2} > 0\\), we begin with the critical points of the function ... |
The sum of the squares of three consecutive odd positive integers is equal to 265. What is the middle integer? | The sum of the squares of three consecutive odd positive integers is equal to 265. What is the middle integer? | Let the three consecutive odd positive integers be represented as \( x - 2 \), \( x \), and \( x + 2 \). Here, \( x \) is the middle integer.
The sum of their squares can be expressed as:
\[
(x - 2)^2 + x^2 + (x + 2)^2
\]
We can expand each square:
\[
(x - 2)^2 = x^2 - 4x + 4
\]
\[
x^2 = x^2
\]
\[
(x + 2)^2 = x^2 +... | 10,292 | [
{
"from": "human",
"value": "The sum of the squares of three consecutive odd positive integers is equal to 265. What is the middle integer?"
},
{
"from": "gpt",
"value": "Let the three consecutive odd positive integers be represented as \\( x - 2 \\), \\( x \\), and \\( x + 2 \\). Here, \\( x \\... |
What multiple of 24 is closest to 3257? | What multiple of 24 is closest to 3257? | To find the multiple of 24 that is closest to 3257, we first divide 3257 by 24.
\[
\frac{3257}{24} \approx 135.7083
\]
Next, we round this quotient to the nearest whole number. The closest whole numbers are 135 and 136.
Now we can calculate the multiples of 24 for these two whole numbers:
For 135:
\[
135 \times 24... | 10,293 | [
{
"from": "human",
"value": "What multiple of 24 is closest to 3257?"
},
{
"from": "gpt",
"value": "To find the multiple of 24 that is closest to 3257, we first divide 3257 by 24.\n\n\\[\n\\frac{3257}{24} \\approx 135.7083\n\\]\n\nNext, we round this quotient to the nearest whole number. The clo... |
In a certain high school, there are three different clubs: debate, drama, and music. Each student is a member of at most one club, and no student is a member of all three clubs. If there are 5 students in debate, 7 students in drama, and 3 students in music, and there are 2 students who are members of both debate and d... | In a certain high school, there are three different clubs: debate, drama, and music. Each student is a member of at most one club, and no student is a member of all three clubs. If there are 5 students in debate, 7 students in drama, and 3 students in music, and there are 2 students who are members of both debate and d... | To find the total number of students in the high school, we can use the principle of inclusion-exclusion to calculate the number of distinct students in the different clubs.
Let's define the sets:
- Let \( D \) be the set of students in the debate club.
- Let \( R \) be the set of students in the drama club.
- Let \(... | 10,294 | [
{
"from": "human",
"value": "In a certain high school, there are three different clubs: debate, drama, and music. Each student is a member of at most one club, and no student is a member of all three clubs. If there are 5 students in debate, 7 students in drama, and 3 students in music, and there are 2 stud... |
Find all points where the parabolas $y=3x^2-12x-4$ and $y=2x^2-16x+24$ intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | Find all points where the parabolas $y=3x^2-12x-4$ and $y=2x^2-16x+24$ intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | To find the points where the parabolas \( y = 3x^2 - 12x - 4 \) and \( y = 2x^2 - 16x + 24 \) intersect, we need to set the equations equal to each other:
\[
3x^2 - 12x - 4 = 2x^2 - 16x + 24.
\]
Next, we'll move all terms to one side of the equation:
\[
3x^2 - 12x - 4 - 2x^2 + 16x - 24 = 0.
\]
This simplifies to:
... | 10,295 | [
{
"from": "human",
"value": "Find all points where the parabolas $y=3x^2-12x-4$ and $y=2x^2-16x+24$ intersect. List the points in order of increasing $x$-coordinate, separated by semicolons."
},
{
"from": "gpt",
"value": "To find the points where the parabolas \\( y = 3x^2 - 12x - 4 \\) and \\( ... |
What is the smallest five-digit multiple of 23 that is also a multiple of 7? | What is the smallest five-digit multiple of 23 that is also a multiple of 7? | To find the smallest five-digit multiple of \( 23 \) that is also a multiple of \( 7 \), we can express a multiple of both numbers as a multiple of their least common multiple (LCM).
First, we find the LCM of \( 23 \) and \( 7 \). Since \( 23 \) and \( 7 \) are both prime, their LCM is given by the product:
\[
\text... | 10,296 | [
{
"from": "human",
"value": "What is the smallest five-digit multiple of 23 that is also a multiple of 7?"
},
{
"from": "gpt",
"value": "To find the smallest five-digit multiple of \\( 23 \\) that is also a multiple of \\( 7 \\), we can express a multiple of both numbers as a multiple of their l... |
A water tank can be filled by two pipes, A and B, in 24 and 36 minutes, respectively. If both pipes are opened simultaneously, how long will it take to fill the tank? | A water tank can be filled by two pipes, A and B, in 24 and 36 minutes, respectively. If both pipes are opened simultaneously, how long will it take to fill the tank? | To determine how long it will take to fill the water tank when pipes A and B are opened simultaneously, we first need to determine the rate at which each pipe fills the tank.
1. **Calculate the rates of each pipe:**
- Pipe A can fill the tank in 24 minutes, so its rate is:
\[
\text{Rate of A} = \frac{1 \t... | 10,297 | [
{
"from": "human",
"value": "A water tank can be filled by two pipes, A and B, in 24 and 36 minutes, respectively. If both pipes are opened simultaneously, how long will it take to fill the tank?"
},
{
"from": "gpt",
"value": "To determine how long it will take to fill the water tank when pipes ... |
The three-digit number $xxy$ is divisible by 7, and the four-digit number $xyyx$ is divisible by 9.
Determine $x+y$. | The three-digit number $xxy$ is divisible by 7, and the four-digit number $xyyx$ is divisible by 9.
Determine $x+y$. | Let the three-digit number \( xxy \) be represented as \( 100x + 10x + y = 110x + y \). Given that \( xxy \) is divisible by 7, we need to find \( 110x + y \equiv 0 \pmod{7} \).
Calculating \( 110 \mod 7 \):
\[
110 \div 7 = 15 \quad \text{(Quotient)}
\]
\[
110 - 15 \times 7 = 110 - 105 = 5 \quad \Rightarrow \quad 110 ... | 10,298 | [
{
"from": "human",
"value": "The three-digit number $xxy$ is divisible by 7, and the four-digit number $xyyx$ is divisible by 9.\nDetermine $x+y$."
},
{
"from": "gpt",
"value": "Let the three-digit number \\( xxy \\) be represented as \\( 100x + 10x + y = 110x + y \\). Given that \\( xxy \\) is ... |
Find the largest root of the polynomial $$P(x) = (x^2 - 1)(x^3 - 2)(x^4 - 3) \dotsm (x^{13} - 12).$$ | Find the largest root of the polynomial $$P(x) = (x^2 - 1)(x^3 - 2)(x^4 - 3) \dotsm (x^{13} - 12).$$ | To find the largest root of the polynomial
\[
P(x) = (x^2 - 1)(x^3 - 2)(x^4 - 3) \cdots (x^{13} - 12),
\]
we first note that \(P(x)\) is a product of polynomial factors \(x^n - (n-1)\) for \(n = 2\) to \(13\). To investigate the roots, we need to look at each factor \(x^n - (n-1)\):
1. The roots of \(x^2 - 1 = 0\) ... | 10,299 | [
{
"from": "human",
"value": "Find the largest root of the polynomial $$P(x) = (x^2 - 1)(x^3 - 2)(x^4 - 3) \\dotsm (x^{13} - 12).$$"
},
{
"from": "gpt",
"value": "To find the largest root of the polynomial \n\n\\[\nP(x) = (x^2 - 1)(x^3 - 2)(x^4 - 3) \\cdots (x^{13} - 12),\n\\]\n\nwe first note th... |
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