instruction_seed stringlengths 18 5.08k | response_seed stringlengths 18 5.08k | reasoning stringlengths 0 99k | deepseek_solution stringlengths 0 4.03k | __original_row_idx int64 0 40k | source stringclasses 1
value | final_reasoning_trace stringlengths 19 102k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
A snail is at the bottom of a 15-foot well. Each day, it climbs up 3 feet, but at night, it slips back 2 feet. How many days will it take for the snail to reach the top of the well? | A snail is at the bottom of a 15-foot well. Each day, it climbs up 3 feet, but at night, it slips back 2 feet. How many days will it take for the snail to reach the top of the well? | Okay, so there's this snail at the bottom of a 15-foot well. Every day, it climbs up 3 feet, but then at night, it slips back 2 feet. The question is, how many days will it take for the snail to reach the top of the well? Hmm, let me try to figure this out.
First, I need to visualize the situation. The well is 15 feet... | The snail is at the bottom of a 15-foot well. Each day, it climbs up 3 feet and slips back 2 feet at night. To determine how many days it will take for the snail to reach the top of the well, we need to consider the net progress each day and the final climb where the snail does not slip back.
1. **Net Progress Calcula... | 9,977 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this snail at the bottom of a 15-foot well. Every day, it climbs up 3 feet, but then at night, it slips back 2 feet. The question is, how many days will it take for the snail to reach the top of the well? Hmm, let me try to figure this out.
First, I need to visualize the situation. The well is... | [
{
"from": "human",
"value": "A snail is at the bottom of a 15-foot well. Each day, it climbs up 3 feet, but at night, it slips back 2 feet. How many days will it take for the snail to reach the top of the well?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so there's this snail at the bottom of ... |
Compute the sum of the geometric series $-1 + 2 - 4 + 8 - \cdots + 512$. | Compute the sum of the geometric series $-1 + 2 - 4 + 8 - \cdots + 512$. | Okay, so I need to compute the sum of this geometric series: -1 + 2 - 4 + 8 - ... + 512. Let me think about how to approach this.
First, I remember that a geometric series is one where each term is multiplied by a constant ratio to get the next term. Let me verify if this series is geometric. Starting with the first ... | The given geometric series is \(-1 + 2 - 4 + 8 - \cdots + 512\).
First, we identify the parameters of the geometric series:
- The first term \(a = -1\)
- The common ratio \(r = -2\)
- The number of terms \(n\) needs to be determined.
The \(n\)-th term of a geometric series is given by \(a_n = a \cdot r^{n-1}\). We s... | 25,384 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to compute the sum of this geometric series: -1 + 2 - 4 + 8 - ... + 512. Let me think about how to approach this.
First, I remember that a geometric series is one where each term is multiplied by a constant ratio to get the next term. Let me verify if this series is geometric. Starting with th... | [
{
"from": "human",
"value": "Compute the sum of the geometric series $-1 + 2 - 4 + 8 - \\cdots + 512$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to compute the sum of this geometric series: -1 + 2 - 4 + 8 - ... + 512. Let me think about how to approach this. \n\nFirst, I remember t... |
In convex quadrilateral $ABCD$, $\angle A = \angle C$, $AB=CD=180$, and $AD \ne BC$. The perimeter of $ABCD$ is 640. Find $\cos A$.
A similar problem:
In convex quadrilateral $ABCD$, $\angle A = \angle C$, $AB=CD=180$, and $AD \ne BC$. The perimeter of $ABCD$ is 640. Find $\cos B$.
In this problem, we are asked to... | In convex quadrilateral $ABCD$, $\angle A = \angle C$, $AB=CD=180$, and $AD \ne BC$. The perimeter of $ABCD$ is 640. Find $\cos A$.
A similar problem:
In convex quadrilateral $ABCD$, $\angle A = \angle C$, $AB=CD=180$, and $AD \ne BC$. The perimeter of $ABCD$ is 640. Find $\cos B$.
In this problem, we are asked to... | Okay, so I have this problem about a convex quadrilateral ABCD where angle A equals angle C, sides AB and CD are both 180 units long, and the perimeter is 640. Also, it's given that AD is not equal to BC, and I need to find cos B. Hmm, interesting. Let me think about how to approach this.
First, let's note down the gi... | Given a convex quadrilateral \(ABCD\) with \(\angle A = \angle C\), \(AB = CD = 180\), and a perimeter of 640, we need to find \(\cos B\).
First, we note that the perimeter gives us \(AB + BC + CD + DA = 640\). Since \(AB = CD = 180\), we have:
\[
180 + BC + 180 + DA = 640 \implies BC + DA = 280
\]
Given \(AD \ne BC\... | 5,105 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem about a convex quadrilateral ABCD where angle A equals angle C, sides AB and CD are both 180 units long, and the perimeter is 640. Also, it's given that AD is not equal to BC, and I need to find cos B. Hmm, interesting. Let me think about how to approach this.
First, let's note dow... | [
{
"from": "human",
"value": "In convex quadrilateral $ABCD$, $\\angle A = \\angle C$, $AB=CD=180$, and $AD \\ne BC$. The perimeter of $ABCD$ is 640. Find $\\cos A$.\n\nA similar problem:\n\nIn convex quadrilateral $ABCD$, $\\angle A = \\angle C$, $AB=CD=180$, and $AD \\ne BC$. The perimeter of $ABCD$ is 6... |
Sides $\overline{AH}$ and $\overline{CD}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$? | Sides $\overline{AH}$ and $\overline{CD}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$? | Okay, so I need to figure out the degree measure of angle P in this regular octagon problem. Let me try to visualize the octagon first. A regular octagon has all sides equal and all internal angles equal. Each internal angle in a regular octagon is 135 degrees because the formula for internal angles is ((n-2)*180)/n, w... | To solve the problem of finding the degree measure of angle \( P \) formed by extending sides \(\overline{AH}\) and \(\overline{CD}\) of regular octagon \(ABCDEFGH\) to meet at point \(P\), we start by noting that a regular octagon has internal angles of \(135^\circ\) and each central angle is \(45^\circ\).
We place t... | 21,627 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the degree measure of angle P in this regular octagon problem. Let me try to visualize the octagon first. A regular octagon has all sides equal and all internal angles equal. Each internal angle in a regular octagon is 135 degrees because the formula for internal angles is ((n-2)*1... | [
{
"from": "human",
"value": "Sides $\\overline{AH}$ and $\\overline{CD}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the degree measure of angle P in this regular o... |
Let $z = e^{i\theta}$, where $\theta$ is a real number. Find the minimum value of $|z^{16} - z^{-16}|$. | Let $z = e^{i\theta}$, where $\theta$ is a real number. Find the minimum value of $|z^{16} - z^{-16}|$. | Okay, so I need to find the minimum value of |z¹⁶ - z⁻¹⁶| where z = e^(iθ) and θ is a real number. Hmm, let's start by understanding what this expression represents. Since z is on the unit circle in the complex plane, raising it to any power will just rotate it around the circle.
First, let me recall that for any com... | Given \( z = e^{i\theta} \) where \( \theta \) is a real number, we need to find the minimum value of \( |z^{16} - z^{-16}| \).
First, we express \( z^{16} \) and \( z^{-16} \) using Euler's formula:
\[ z^{16} = e^{i16\theta} \]
\[ z^{-16} = e^{-i16\theta} \]
The difference \( z^{16} - z^{-16} \) can be written as:
\... | 18,484 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the minimum value of |z¹⁶ - z⁻¹⁶| where z = e^(iθ) and θ is a real number. Hmm, let's start by understanding what this expression represents. Since z is on the unit circle in the complex plane, raising it to any power will just rotate it around the circle.
First, let me recall that for... | [
{
"from": "human",
"value": "Let $z = e^{i\\theta}$, where $\\theta$ is a real number. Find the minimum value of $|z^{16} - z^{-16}|$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the minimum value of |z¹⁶ - z⁻¹⁶| where z = e^(iθ) and θ is a real number. Hmm, let's start by un... |
A cylindrical tank with a height of 12 meters and a radius of 4 meters is filled with water to a height of 8 meters. A sphere with a radius of 2 meters is submerged in the water. What is the volume of the water displaced by the sphere, in cubic meters? | A cylindrical tank with a height of 12 meters and a radius of 4 meters is filled with water to a height of 8 meters. A sphere with a radius of 2 meters is submerged in the water. What is the volume of the water displaced by the sphere, in cubic meters? | Okay, so there's this cylindrical tank, right? It has a height of 12 meters and a radius of 4 meters. They filled it with water up to 8 meters high. Then they submerged a sphere with a radius of 2 meters into the water. The question is asking for the volume of the water displaced by the sphere. Hmm, displacement... I r... | To determine the volume of water displaced by the sphere, we need to calculate the volume of the sphere itself, as the displaced volume is equal to the volume of the submerged object when it is fully submerged.
The formula for the volume of a sphere is:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the ... | 13,936 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this cylindrical tank, right? It has a height of 12 meters and a radius of 4 meters. They filled it with water up to 8 meters high. Then they submerged a sphere with a radius of 2 meters into the water. The question is asking for the volume of the water displaced by the sphere. Hmm, displacemen... | [
{
"from": "human",
"value": "A cylindrical tank with a height of 12 meters and a radius of 4 meters is filled with water to a height of 8 meters. A sphere with a radius of 2 meters is submerged in the water. What is the volume of the water displaced by the sphere, in cubic meters?"
},
{
"from": "gpt... |
There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers. | There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers. | Okay, so I need to find the four positive integers that are divisors of each number in the list: 36, 72, -12, 114, 96. Then, I have to find the sum of these four positive integers. Hmm, let's start by understanding what the problem is asking for.
First, when the problem says "divisors of each number in the list," it ... | To find the four positive integers that are divisors of each number in the list \(36, 72, -12, 114, 96\), we need to determine the greatest common divisor (GCD) of these numbers. The common divisors of all the numbers will be the divisors of this GCD.
First, we compute the GCD step by step:
1. \( \gcd(36, 72) = 36 \)
... | 22,092 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the four positive integers that are divisors of each number in the list: 36, 72, -12, 114, 96. Then, I have to find the sum of these four positive integers. Hmm, let's start by understanding what the problem is asking for.
First, when the problem says "divisors of each number in the li... | [
{
"from": "human",
"value": "There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the four positive integers that are divisors of each n... |
When an integer is divided by 21, the remainder is 10. Find the difference between the remainders when the same integer is divided by 7 and by 3. | When an integer is divided by 21, the remainder is 10. Find the difference between the remainders when the same integer is divided by 7 and by 3. | Okay, so I have this problem here: When an integer is divided by 21, the remainder is 10. I need to find the difference between the remainders when the same integer is divided by 7 and by 3. Hmm, let's see. I need to figure out what the remainders are when this integer is divided by 7 and by 3 separately, then subtract... | Given an integer \( N \) which, when divided by 21, leaves a remainder of 10, we can express \( N \) as \( N = 21k + 10 \) for some integer \( k \).
To find the remainder when \( N \) is divided by 7:
- Since \( 21k \) is a multiple of 7, the remainder comes from \( 10 \div 7 \).
- \( 10 \div 7 \) gives a quotient of ... | 25,684 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: When an integer is divided by 21, the remainder is 10. I need to find the difference between the remainders when the same integer is divided by 7 and by 3. Hmm, let's see. I need to figure out what the remainders are when this integer is divided by 7 and by 3 separately, then ... | [
{
"from": "human",
"value": "When an integer is divided by 21, the remainder is 10. Find the difference between the remainders when the same integer is divided by 7 and by 3."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: When an integer is divided by 21, the remainde... |
If $x$ is to $y$ as $13$ is to $28$, and $y$ is to $z$ as $5$ is to $7$, what is the ratio of $x$ to $z$? Express your answer as a common fraction. | If $x$ is to $y$ as $13$ is to $28$, and $y$ is to $z$ as $5$ is to $7$, what is the ratio of $x$ to $z$? Express your answer as a common fraction. | Okay, so I need to find the ratio of x to z, given that x is to y as 13 is to 28, and y is to z as 5 is to 7. Hmm, ratios can sometimes be tricky, but let me break this down step by step.
First, the problem says "x is to y as 13 is to 28." That means the ratio of x to y is 13:28. So, I can write that as a fraction: x/... | Given that \( x \) is to \( y \) as \( 13 \) is to \( 28 \), we can write the ratio as:
\[
\frac{x}{y} = \frac{13}{28} \implies x = \frac{13}{28}y
\]
Next, given that \( y \) is to \( z \) as \( 5 \) is to \( 7 \), we can write the ratio as:
\[
\frac{y}{z} = \frac{5}{7} \implies z = \frac{7}{5}y
\]
To find the ratio ... | 12,846 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the ratio of x to z, given that x is to y as 13 is to 28, and y is to z as 5 is to 7. Hmm, ratios can sometimes be tricky, but let me break this down step by step.
First, the problem says "x is to y as 13 is to 28." That means the ratio of x to y is 13:28. So, I can write that as a frac... | [
{
"from": "human",
"value": "If $x$ is to $y$ as $13$ is to $28$, and $y$ is to $z$ as $5$ is to $7$, what is the ratio of $x$ to $z$? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the ratio of x to z, given that x is to y as 13 is to 2... |
Find the smallest positive integer that can be expressed as the sum of three distinct powers of 3, and determine its representation in base 3. | Find the smallest positive integer that can be expressed as the sum of three distinct powers of 3, and determine its representation in base 3. | Okay, so I need to find the smallest positive integer that can be expressed as the sum of three distinct powers of 3, and then figure out its base 3 representation. Let me start by recalling what powers of 3 are. A power of 3 is any number of the form 3^n where n is a non-negative integer. So examples are 1 (3^0), 3 (3... | To find the smallest positive integer that can be expressed as the sum of three distinct powers of 3, we start by considering the smallest exponents. The powers of 3 are \(3^0 = 1\), \(3^1 = 3\), \(3^2 = 9\), \(3^3 = 27\), etc.
The sum of the three smallest distinct powers of 3 (with exponents 0, 1, and 2) is:
\[
3^0... | 5,851 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest positive integer that can be expressed as the sum of three distinct powers of 3, and then figure out its base 3 representation. Let me start by recalling what powers of 3 are. A power of 3 is any number of the form 3^n where n is a non-negative integer. So examples are 1 (3^... | [
{
"from": "human",
"value": "Find the smallest positive integer that can be expressed as the sum of three distinct powers of 3, and determine its representation in base 3."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest positive integer that can be expressed as the s... |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$.
Note: The generated problem is different and requires another approach to solve it. The original problem is about finding the difference between the roots of a quadratic equation, while the generated problem ... | The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$.
Note: The generated problem is different and requires another approach to solve it. The original problem is about finding the difference between the roots of a quadratic equation, while the generated problem ... | Alright, let's see. I need to find the value of 'a' such that the equation x² - 3a x + a² = 0 has roots x₁ and x₂ where x₁² + x₂² = 1.75. Hmm, okay. I remember from algebra that for a quadratic equation of the form ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. Wait, but here the quadratic is x²... | Given the quadratic equation \(x^2 - 3ax + a^2 = 0\) with roots \(x_1\) and \(x_2\), we need to determine the value of \(a\) such that \(x_1^2 + x_2^2 = 1.75\).
First, we use the relationships for the sum and product of the roots of a quadratic equation. For the given equation \(x^2 - 3ax + a^2 = 0\), the sum of the r... | 11,105 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's see. I need to find the value of 'a' such that the equation x² - 3a x + a² = 0 has roots x₁ and x₂ where x₁² + x₂² = 1.75. Hmm, okay. I remember from algebra that for a quadratic equation of the form ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. Wait, but here the quadrat... | [
{
"from": "human",
"value": "The roots $x_1$ and $x_2$ of the equation\n$$x^2-3ax+a^2=0$$\nare such that\n$x_1^2+x_2^2=1.75$.\nDetermine $a$.\n\n Note: The generated problem is different and requires another approach to solve it. The original problem is about finding the difference between the roots of a q... |
A bookshelf has 5 shelves, and each shelf can hold a maximum of 8 books. If the bookshelf is currently empty, how many different ways can 12 books be placed on the shelves such that each shelf contains at least one book? | A bookshelf has 5 shelves, and each shelf can hold a maximum of 8 books. If the bookshelf is currently empty, how many different ways can 12 books be placed on the shelves such that each shelf contains at least one book? | Okay, so I have this problem where there's a bookshelf with 5 shelves, each can hold up to 8 books. The bookshelf is empty right now, and I need to figure out how many different ways I can place 12 books on the shelves such that each shelf has at least one book. Hmm, let me break this down.
First, the problem is about... | To determine the number of different ways to place 12 books on 5 shelves such that each shelf contains at least one book and no shelf contains more than 8 books, we can use the following reasoning:
1. **Initial Setup**: We need to distribute 12 books into 5 shelves with each shelf containing at least 1 book. This is a... | 22,387 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem where there's a bookshelf with 5 shelves, each can hold up to 8 books. The bookshelf is empty right now, and I need to figure out how many different ways I can place 12 books on the shelves such that each shelf has at least one book. Hmm, let me break this down.
First, the problem ... | [
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a maximum of 8 books. If the bookshelf is currently empty, how many different ways can 12 books be placed on the shelves such that each shelf contains at least one book?"
},
{
"from": "gpt",
"value": "<think>\nOkay, ... |
What is the smallest possible value of $CD$ if $D$ is a point on the side $AC$ of triangle $ABC$, $\angle B = 90^{\circ}$, $BC = 1$, and $\angle ADB = 45^{\circ}$? | What is the smallest possible value of $CD$ if $D$ is a point on the side $AC$ of triangle $ABC$, $\angle B = 90^{\circ}$, $BC = 1$, and $\angle ADB = 45^{\circ}$? | Okay, let's see. I need to find the smallest possible value of CD where D is a point on side AC of a right-angled triangle ABC with angle B = 90°, BC = 1, and angle ADB = 45°. Hmm, right. Let me start by sketching the problem to visualize it better.
So, triangle ABC is right-angled at B, with BC = 1. Let me denote the... | Given a right-angled triangle \(ABC\) with \(\angle B = 90^\circ\), \(BC = 1\), and point \(D\) on \(AC\) such that \(\angle ADB = 45^\circ\), we need to find the smallest possible value of \(CD\).
1. **Coordinate System Setup**:
- Place \(B\) at the origin \((0, 0)\), \(C\) at \((0, 1)\), and \(A\) at \((x, 0)\) w... | 19,043 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the smallest possible value of CD where D is a point on side AC of a right-angled triangle ABC with angle B = 90°, BC = 1, and angle ADB = 45°. Hmm, right. Let me start by sketching the problem to visualize it better.
So, triangle ABC is right-angled at B, with BC = 1. Let me de... | [
{
"from": "human",
"value": "What is the smallest possible value of $CD$ if $D$ is a point on the side $AC$ of triangle $ABC$, $\\angle B = 90^{\\circ}$, $BC = 1$, and $\\angle ADB = 45^{\\circ}$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the smallest possible value... |
Find the minimum value of $3x^2-18x+12$. | Find the minimum value of $3x^2-18x+12$. | Okay, so I need to find the minimum value of the quadratic function 3x² - 18x + 12. Hmm, let me think. I remember that quadratic functions have either a minimum or maximum value depending on the coefficient of the x² term. Since the coefficient here is 3, which is positive, the parabola opens upwards, so it should have... | To find the minimum value of the quadratic function \(3x^2 - 18x + 12\), we can use the vertex formula. For a quadratic function in the form \(ax^2 + bx + c\), the x-coordinate of the vertex is given by \(-\frac{b}{2a}\).
Here, \(a = 3\), \(b = -18\), and \(c = 12\). Plugging these values into the vertex formula, we g... | 14,004 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the minimum value of the quadratic function 3x² - 18x + 12. Hmm, let me think. I remember that quadratic functions have either a minimum or maximum value depending on the coefficient of the x² term. Since the coefficient here is 3, which is positive, the parabola opens upwards, so it sho... | [
{
"from": "human",
"value": "Find the minimum value of $3x^2-18x+12$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the minimum value of the quadratic function 3x² - 18x + 12. Hmm, let me think. I remember that quadratic functions have either a minimum or maximum value dependin... |
A cube with edge length $a$ has a sphere inscribed within it, such that the sphere touches the cube at the center of each face. The volume of the sphere is $\frac{4}{3}\pi r^3$. Find the ratio of the volume of the cube to the volume of the sphere. | A cube with edge length $a$ has a sphere inscribed within it, such that the sphere touches the cube at the center of each face. The volume of the sphere is $\frac{4}{3}\pi r^3$. Find the ratio of the volume of the cube to the volume of the sphere. | Okay, so I need to find the ratio of the volume of a cube to the volume of a sphere that's inscribed inside it. Let me start by recalling some basic formulas. The volume of a cube is given by edge length cubed, right? So if the edge length is \( a \), then the volume \( V_{cube} \) is \( a^3 \).
Now, the sphere is ins... | The volume of a cube with edge length \( a \) is given by \( V_{\text{cube}} = a^3 \).
The sphere inscribed within the cube has a diameter equal to the edge length of the cube, so the radius \( r \) of the sphere is \( \frac{a}{2} \). The volume of the sphere is calculated as follows:
\[
V_{\text{sphere}} = \frac{4}{3... | 4,017 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the ratio of the volume of a cube to the volume of a sphere that's inscribed inside it. Let me start by recalling some basic formulas. The volume of a cube is given by edge length cubed, right? So if the edge length is \( a \), then the volume \( V_{cube} \) is \( a^3 \).
Now, the spher... | [
{
"from": "human",
"value": "A cube with edge length $a$ has a sphere inscribed within it, such that the sphere touches the cube at the center of each face. The volume of the sphere is $\\frac{4}{3}\\pi r^3$. Find the ratio of the volume of the cube to the volume of the sphere."
},
{
"from": "gpt",
... |
Consider the function $f(x) = \lfloor x \rfloor - 2x$.
Determine the range of values of $x$ for which $f(x) \geq 0$. | Consider the function $f(x) = \lfloor x \rfloor - 2x$.
Determine the range of values of $x$ for which $f(x) \geq 0$. | Okay, so I need to figure out for which values of x the function f(x) = floor(x) - 2x is greater than or equal to zero. Let me start by recalling what the floor function does. The floor of x, denoted as ⎣x⎦, is the greatest integer less than or equal to x. So, for example, if x is 2.3, then floor(x) is 2; if x is -1.7,... | To determine the range of values of \( x \) for which \( f(x) = \lfloor x \rfloor - 2x \geq 0 \), we start by analyzing the inequality \( \lfloor x \rfloor \geq 2x \).
1. **Understanding the Floor Function**:
- The floor function \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \).
- Fo... | 5,177 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out for which values of x the function f(x) = floor(x) - 2x is greater than or equal to zero. Let me start by recalling what the floor function does. The floor of x, denoted as ⎣x⎦, is the greatest integer less than or equal to x. So, for example, if x is 2.3, then floor(x) is 2; if x ... | [
{
"from": "human",
"value": "Consider the function $f(x) = \\lfloor x \\rfloor - 2x$.\nDetermine the range of values of $x$ for which $f(x) \\geq 0$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out for which values of x the function f(x) = floor(x) - 2x is greater than or e... |
Mr. Thompson recorded this information from a recent test taken by all of his students. Using the data, what was the average percent score for these 80 students?
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
95&5\\\hline
85&15\\\hline
75&30\\\hline
65&20\\\... | Mr. Thompson recorded this information from a recent test taken by all of his students. Using the data, what was the average percent score for these 80 students?
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
95&5\\\hline
85&15\\\hline
75&30\\\hline
65&20\\\... | Alright, let's tackle this problem step by step. So, Mr. Thompson has given us a table with percentage scores and the number of students who got each score. We need to find the average percent score for all 80 students. Hmm, averages can sometimes be tricky, but let me recall how that works.
First off, the average pe... | To find the average percent score for Mr. Thompson's 80 students, we need to calculate the weighted average of the percentage scores. The formula for the average is:
\[
\text{Average} = \frac{\sum (\% \text{ Score} \times \text{Number of Students})}{\text{Total Number of Students}}
\]
First, we calculate the total pe... | 10,642 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's tackle this problem step by step. So, Mr. Thompson has given us a table with percentage scores and the number of students who got each score. We need to find the average percent score for all 80 students. Hmm, averages can sometimes be tricky, but let me recall how that works.
First off, the av... | [
{
"from": "human",
"value": "Mr. Thompson recorded this information from a recent test taken by all of his students. Using the data, what was the average percent score for these 80 students?\n\n\\begin{tabular}{|c|c|}\n\\multicolumn{2}{c}{}\\\\\\hline\n\\textbf{$\\%$ Score}&\\textbf{Number of Students}\\\\\... |
Let $g(x)$ be a function defined for all positive real numbers satisfying the conditions $g(x) > 0$ for all $x > 0$ and
\[g(x + y) = \frac{g(xy)}{1 + g(xy)}\]for all $x, y > 0.$ Determine $g(2009).$ | Let $g(x)$ be a function defined for all positive real numbers satisfying the conditions $g(x) > 0$ for all $x > 0$ and
\[g(x + y) = \frac{g(xy)}{1 + g(xy)}\]for all $x, y > 0.$ Determine $g(2009).$ | Alright, so I need to figure out what g(2009) is, given that g is defined for all positive real numbers, is always positive, and satisfies this functional equation: g(x + y) = g(xy)/(1 + g(xy)) for all x, y > 0. Hmm, functional equations can be tricky, but maybe I can find a pattern or substitute specific values for x ... | Given the function \( g(x) \) defined for all positive real numbers satisfying \( g(x) > 0 \) for all \( x > 0 \) and the functional equation \( g(x + y) = \frac{g(xy)}{1 + g(xy)} \) for all \( x, y > 0 \), we need to determine \( g(2009) \).
1. **Substitution \( y = 1 \)**:
- This gives the recursive relation \( g... | 19,047 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to figure out what g(2009) is, given that g is defined for all positive real numbers, is always positive, and satisfies this functional equation: g(x + y) = g(xy)/(1 + g(xy)) for all x, y > 0. Hmm, functional equations can be tricky, but maybe I can find a pattern or substitute specific value... | [
{
"from": "human",
"value": "Let $g(x)$ be a function defined for all positive real numbers satisfying the conditions $g(x) > 0$ for all $x > 0$ and\n\\[g(x + y) = \\frac{g(xy)}{1 + g(xy)}\\]for all $x, y > 0.$ Determine $g(2009).$"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to f... |
In a rectangular room with dimensions 15 feet by 20 feet, two spheres are placed on the floor so that they are tangent to the walls of the room. The radii of the spheres are 3 feet and 5 feet. What is the maximum possible distance between the centers of the two spheres? | In a rectangular room with dimensions 15 feet by 20 feet, two spheres are placed on the floor so that they are tangent to the walls of the room. The radii of the spheres are 3 feet and 5 feet. What is the maximum possible distance between the centers of the two spheres? | Okay, so I need to figure out the maximum possible distance between the centers of two spheres in a rectangular room that's 15 feet by 20 feet. The spheres have radii 3 feet and 5 feet. Hmm, let's start by visualizing the room. It's a rectangle, so the floor is a 15x20 rectangle. The spheres are placed on the floor, so... | To determine the maximum possible distance between the centers of two spheres placed on the floor of a rectangular room with dimensions 15 feet by 20 feet, we need to consider their positions and the 3D distance between their centers. The spheres have radii 3 feet and 5 feet and are tangent to the walls of the room.
1... | 20,965 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the maximum possible distance between the centers of two spheres in a rectangular room that's 15 feet by 20 feet. The spheres have radii 3 feet and 5 feet. Hmm, let's start by visualizing the room. It's a rectangle, so the floor is a 15x20 rectangle. The spheres are placed on the f... | [
{
"from": "human",
"value": "In a rectangular room with dimensions 15 feet by 20 feet, two spheres are placed on the floor so that they are tangent to the walls of the room. The radii of the spheres are 3 feet and 5 feet. What is the maximum possible distance between the centers of the two spheres?"
},
... |
In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths 12 and 20, respectively, and the nonparallel sides $BC$ and $DA$ have equal lengths. Points $X$ and $Y$ are the midpoints of $\overline{BC}$ and $\overline{DA}$, respectively. Compute the area of triangle $AXY$. | In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths 12 and 20, respectively, and the nonparallel sides $BC$ and $DA$ have equal lengths. Points $X$ and $Y$ are the midpoints of $\overline{BC}$ and $\overline{DA}$, respectively. Compute the area of triangle $AXY$. | Okay, so I need to find the area of triangle AXY in trapezoid ABCD where AB and CD are the parallel sides with lengths 12 and 20. The nonparallel sides BC and DA are equal in length. X and Y are the midpoints of BC and DA respectively. Let me start by visualizing the trapezoid. Since it's an isosceles trapezoid (becaus... | In trapezoid \(ABCD\), the parallel sides \(AB\) and \(CD\) have lengths 12 and 20, respectively, and the nonparallel sides \(BC\) and \(DA\) have equal lengths. Points \(X\) and \(Y\) are the midpoints of \(\overline{BC}\) and \(\overline{DA}\), respectively. We need to compute the area of triangle \(AXY\).
1. **Coor... | 3,093 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the area of triangle AXY in trapezoid ABCD where AB and CD are the parallel sides with lengths 12 and 20. The nonparallel sides BC and DA are equal in length. X and Y are the midpoints of BC and DA respectively. Let me start by visualizing the trapezoid. Since it's an isosceles trapezoid... | [
{
"from": "human",
"value": "In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths 12 and 20, respectively, and the nonparallel sides $BC$ and $DA$ have equal lengths. Points $X$ and $Y$ are the midpoints of $\\overline{BC}$ and $\\overline{DA}$, respectively. Compute the area of triangle $AXY$... |
A cylindrical container has a capacity of 81.76 cubic centimeters and is partially filled with a liquid that occupies 34.587 cubic centimeters. What is the height of the liquid in the container, rounded to the nearest tenth, if the container's radius is 5 centimeters? | A cylindrical container has a capacity of 81.76 cubic centimeters and is partially filled with a liquid that occupies 34.587 cubic centimeters. What is the height of the liquid in the container, rounded to the nearest tenth, if the container's radius is 5 centimeters? | Okay, so I have this problem here about a cylindrical container. The total capacity is 81.76 cubic centimeters, and it's partially filled with a liquid that takes up 34.587 cubic centimeters. They want to find the height of the liquid in the cylinder, rounded to the nearest tenth, and the radius of the container is 5 c... | Given a cylindrical container with a capacity of 81.76 cubic centimeters and a radius of 5 centimeters, we need to find the height of the liquid when the container is partially filled with 34.587 cubic centimeters of liquid.
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
We need to solv... | 1,448 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here about a cylindrical container. The total capacity is 81.76 cubic centimeters, and it's partially filled with a liquid that takes up 34.587 cubic centimeters. They want to find the height of the liquid in the cylinder, rounded to the nearest tenth, and the radius of the containe... | [
{
"from": "human",
"value": "A cylindrical container has a capacity of 81.76 cubic centimeters and is partially filled with a liquid that occupies 34.587 cubic centimeters. What is the height of the liquid in the container, rounded to the nearest tenth, if the container's radius is 5 centimeters?"
},
{
... |
Tom's sister Laura will pay him $2.5 for every hour he spends walking her dog. Today, Tom walked the dog for 2 hours and 15 minutes. How much money does Laura owe Tom? Express your answer as a decimal. | Tom's sister Laura will pay him $2.5 for every hour he spends walking her dog. Today, Tom walked the dog for 2 hours and 15 minutes. How much money does Laura owe Tom? Express your answer as a decimal. | Okay, let's see. Tom's sister Laura is paying him $2.5 for every hour he walks her dog. Today, Tom walked the dog for 2 hours and 15 minutes. I need to figure out how much money Laura owes Tom. Hmm, the rate is $2.50 per hour, right? So, first, I need to convert the time he walked the dog into hours because the payment... | Tom's sister Laura will pay him $2.50 for every hour he spends walking her dog. Today, Tom walked the dog for 2 hours and 15 minutes. To calculate how much money Laura owes Tom, we need to convert the time he spent walking the dog into hours and then multiply by the hourly rate.
1. Convert 2 hours and 15 minutes into ... | 8,003 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. Tom's sister Laura is paying him $2.5 for every hour he walks her dog. Today, Tom walked the dog for 2 hours and 15 minutes. I need to figure out how much money Laura owes Tom. Hmm, the rate is $2.50 per hour, right? So, first, I need to convert the time he walked the dog into hours because the... | [
{
"from": "human",
"value": "Tom's sister Laura will pay him $2.5 for every hour he spends walking her dog. Today, Tom walked the dog for 2 hours and 15 minutes. How much money does Laura owe Tom? Express your answer as a decimal."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. Tom's si... |
A right pentagonal prism has two cross sections obtained by cutting the prism with planes parallel to the pentagonal base. The areas of the cross sections are $450\sqrt{2}$ square feet and $250\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the base of the prism is the smaller cross section, in ... | A right pentagonal prism has two cross sections obtained by cutting the prism with planes parallel to the pentagonal base. The areas of the cross sections are $450\sqrt{2}$ square feet and $250\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the base of the prism is the smaller cross section, in ... | Okay, so I need to solve this problem about a right pentagonal prism. Let me first make sure I understand what the problem is asking. A right pentagonal prism has two cross sections created by planes parallel to the pentagonal base. The areas of these cross sections are given as 450√2 and 250√2 square feet. The distanc... | Given a right pentagonal prism with two cross sections obtained by cutting the prism with planes parallel to the pentagonal base, we need to determine the distance from the base of the prism to the smaller cross section. The areas of the cross sections are \(450\sqrt{2}\) square feet and \(250\sqrt{2}\) square feet, an... | 27,658 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this problem about a right pentagonal prism. Let me first make sure I understand what the problem is asking. A right pentagonal prism has two cross sections created by planes parallel to the pentagonal base. The areas of these cross sections are given as 450√2 and 250√2 square feet. The... | [
{
"from": "human",
"value": "A right pentagonal prism has two cross sections obtained by cutting the prism with planes parallel to the pentagonal base. The areas of the cross sections are $450\\sqrt{2}$ square feet and $250\\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the base of ... |
Tom and Jerry are painting a fence together. If Tom could paint the entire fence by himself in 5 hours and Jerry could paint the entire fence by himself in 2.5 hours, how many hours would it take for them to paint the entire fence together? | Tom and Jerry are painting a fence together. If Tom could paint the entire fence by himself in 5 hours and Jerry could paint the entire fence by himself in 2.5 hours, how many hours would it take for them to paint the entire fence together? | Okay, so Tom and Jerry are painting a fence together. Hmm, let me see. The problem says Tom can paint the entire fence by himself in 5 hours, and Jerry can do it in 2.5 hours. We need to find out how long it would take them to paint the fence together. Alright, this seems like a work rate problem. I remember that when ... | Tom can paint the entire fence in 5 hours, so his work rate is \(\frac{1}{5}\) of the fence per hour. Jerry can paint the entire fence in 2.5 hours, so his work rate is \(\frac{1}{2.5}\) of the fence per hour. Converting \(\frac{1}{2.5}\) to a fraction, we get \(\frac{2}{5}\).
Adding their work rates together:
\[
\fra... | 16,843 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Tom and Jerry are painting a fence together. Hmm, let me see. The problem says Tom can paint the entire fence by himself in 5 hours, and Jerry can do it in 2.5 hours. We need to find out how long it would take them to paint the fence together. Alright, this seems like a work rate problem. I remember th... | [
{
"from": "human",
"value": "Tom and Jerry are painting a fence together. If Tom could paint the entire fence by himself in 5 hours and Jerry could paint the entire fence by himself in 2.5 hours, how many hours would it take for them to paint the entire fence together?"
},
{
"from": "gpt",
"valu... |
Consider a polynomial $p(x)$ divided by another polynomial $d(x)$, resulting in a quotient $q(x)$ and a remainder $r(x)$, such that $p(x) = d(x) \cdot q(x) + r(x)$. Given that the polynomial $p(x)$ has a degree of $n$, the quotient $q(x)$ has a degree of $m$, and the remainder $r(x)$ has a degree of $k$, find a general... | Consider a polynomial $p(x)$ divided by another polynomial $d(x)$, resulting in a quotient $q(x)$ and a remainder $r(x)$, such that $p(x) = d(x) \cdot q(x) + r(x)$. Given that the polynomial $p(x)$ has a degree of $n$, the quotient $q(x)$ has a degree of $m$, and the remainder $r(x)$ has a degree of $k$, find a general... | Okay, so I need to find the degree of the divisor polynomial d(x) in terms of the degrees of p(x), q(x), and r(x). Let me start by recalling what I know about polynomial division.
When you divide p(x) by d(x), you get a quotient q(x) and a remainder r(x), such that p(x) = d(x) * q(x) + r(x). The key thing here is that... | Given a polynomial \( p(x) \) divided by another polynomial \( d(x) \), resulting in a quotient \( q(x) \) and a remainder \( r(x) \), such that \( p(x) = d(x) \cdot q(x) + r(x) \), we need to find the degree of \( d(x) \) in terms of the degrees of \( p(x) \), \( q(x) \), and \( r(x) \).
1. **Degrees of Polynomials**... | 10,010 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the degree of the divisor polynomial d(x) in terms of the degrees of p(x), q(x), and r(x). Let me start by recalling what I know about polynomial division.
When you divide p(x) by d(x), you get a quotient q(x) and a remainder r(x), such that p(x) = d(x) * q(x) + r(x). The key thing here... | [
{
"from": "human",
"value": "Consider a polynomial $p(x)$ divided by another polynomial $d(x)$, resulting in a quotient $q(x)$ and a remainder $r(x)$, such that $p(x) = d(x) \\cdot q(x) + r(x)$. Given that the polynomial $p(x)$ has a degree of $n$, the quotient $q(x)$ has a degree of $m$, and the remainder ... |
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$ | A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$ | Okay, so I need to solve this problem where a function f is defined on complex numbers by f(z) = (a + bi)z, where a and b are positive numbers. The key property given is that for any complex number z, f(z) is equidistant from both z and the origin. Also, we're told that the magnitude of a + bi is 8, so |a + bi| = 8, an... | Given a function \( f \) defined on the complex numbers by \( f(z) = (a + bi)z \), where \( a \) and \( b \) are positive numbers, we know that for each complex number \( z \), \( f(z) \) is equidistant from both \( z \) and the origin. Additionally, we are given that \( |a + bi| = 8 \). We need to find \( b^2 \).
Fir... | 9,803 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this problem where a function f is defined on complex numbers by f(z) = (a + bi)z, where a and b are positive numbers. The key property given is that for any complex number z, f(z) is equidistant from both z and the origin. Also, we're told that the magnitude of a + bi is 8, so |a + bi|... | [
{
"from": "human",
"value": "A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$"
},
{
"from... |
A regular hexagon has sides 12 units long. A regular hexagon with sides 6 units long is removed from its center, creating a ring-shaped region. What is the ratio of the area of the smaller hexagon to the area of the ring-shaped region? Express your answer as a common fraction. | A regular hexagon has sides 12 units long. A regular hexagon with sides 6 units long is removed from its center, creating a ring-shaped region. What is the ratio of the area of the smaller hexagon to the area of the ring-shaped region? Express your answer as a common fraction. | Okay, let's see. The problem is about a regular hexagon with side length 12 units. Then, a smaller regular hexagon with side length 6 units is removed from the center, forming a ring-shaped region. We need to find the ratio of the area of the smaller hexagon to the area of the ring-shaped region. Hmm, okay.
First, I r... | To find the ratio of the area of the smaller hexagon to the area of the ring-shaped region, we start by calculating the areas of both hexagons using the formula for the area of a regular hexagon: \(\frac{3\sqrt{3}}{2} \times \text{side length}^2\).
1. **Area of the larger hexagon (side length 12 units):**
\[
\te... | 18,371 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is about a regular hexagon with side length 12 units. Then, a smaller regular hexagon with side length 6 units is removed from the center, forming a ring-shaped region. We need to find the ratio of the area of the smaller hexagon to the area of the ring-shaped region. Hmm, okay.
Fi... | [
{
"from": "human",
"value": "A regular hexagon has sides 12 units long. A regular hexagon with sides 6 units long is removed from its center, creating a ring-shaped region. What is the ratio of the area of the smaller hexagon to the area of the ring-shaped region? Express your answer as a common fraction."
... |
In the arithmetic sequence $23, x, y, z, 53$, what is the value of $y$? | In the arithmetic sequence $23, x, y, z, 53$, what is the value of $y$? | Okay, so I have this arithmetic sequence: 23, x, y, z, 53. And I need to find the value of y. Hmm, arithmetic sequence. Let me remember. An arithmetic sequence is one where each term after the first is obtained by adding a constant difference. So the difference between consecutive terms is always the same.
Let me wri... | Given the arithmetic sequence \(23, x, y, z, 53\), we need to find the value of \(y\).
An arithmetic sequence has a constant common difference \(d\). The first term \(a_1\) is 23, and the fifth term \(a_5\) is 53. The fifth term can be expressed as:
\[
a_5 = a_1 + 4d
\]
Substituting the known values:
\[
53 = 23 + 4d
\... | 4,686 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this arithmetic sequence: 23, x, y, z, 53. And I need to find the value of y. Hmm, arithmetic sequence. Let me remember. An arithmetic sequence is one where each term after the first is obtained by adding a constant difference. So the difference between consecutive terms is always the same.
Le... | [
{
"from": "human",
"value": "In the arithmetic sequence $23, x, y, z, 53$, what is the value of $y$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this arithmetic sequence: 23, x, y, z, 53. And I need to find the value of y. Hmm, arithmetic sequence. Let me remember. An arithmetic sequ... |
An online retailer offers a 20% discount on all orders of $200 or more. A customer has two coupons, one for 10% off and another for 25% off, that can be applied to their purchase. However, the coupons cannot be combined, and only one can be used per order. If the customer wants to buy a laptop originally priced at $250... | An online retailer offers a 20% discount on all orders of $200 or more. A customer has two coupons, one for 10% off and another for 25% off, that can be applied to their purchase. However, the coupons cannot be combined, and only one can be used per order. If the customer wants to buy a laptop originally priced at $250... | Okay, let's see. The customer wants to buy a laptop that's originally $250. The retailer offers a 20% discount on orders of $200 or more. So first, I need to figure out if the laptop qualifies for that 20% discount. Since $250 is more than $200, it does qualify. So the retailer's discount of 20% will be applied first. ... | The customer wants to buy a laptop originally priced at $250. The retailer offers a 20% discount on orders of $200 or more. The customer also has two coupons: 10% off and 25% off, which cannot be combined.
1. **Apply the retailer's 20% discount first:**
\[
20\% \text{ of } \$250 = 0.20 \times 250 = \$50
\]
... | 33,527 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The customer wants to buy a laptop that's originally $250. The retailer offers a 20% discount on orders of $200 or more. So first, I need to figure out if the laptop qualifies for that 20% discount. Since $250 is more than $200, it does qualify. So the retailer's discount of 20% will be applied... | [
{
"from": "human",
"value": "An online retailer offers a 20% discount on all orders of $200 or more. A customer has two coupons, one for 10% off and another for 25% off, that can be applied to their purchase. However, the coupons cannot be combined, and only one can be used per order. If the customer wants ... |
A kitten and two rabbits together weigh 30 pounds. The kitten and the larger rabbit together weigh exactly three times as much as the smaller rabbit, and the kitten and the smaller rabbit together weigh exactly twice as much as the larger rabbit. How many pounds does the kitten weigh? | A kitten and two rabbits together weigh 30 pounds. The kitten and the larger rabbit together weigh exactly three times as much as the smaller rabbit, and the kitten and the smaller rabbit together weigh exactly twice as much as the larger rabbit. How many pounds does the kitten weigh? | Okay, let's see. There's a problem about a kitten and two rabbits weighing 30 pounds together. Hmm. Then there are some other relationships between their weights. The kitten and the larger rabbit together weigh three times as much as the smaller rabbit. And the kitten and the smaller rabbit together weigh twice as much... | Let \( K \) be the weight of the kitten, \( L \) be the weight of the larger rabbit, and \( S \) be the weight of the smaller rabbit. We are given the following equations:
1. \( K + L + S = 30 \)
2. \( K + L = 3S \)
3. \( K + S = 2L \)
First, substitute \( K + L \) from equation 2 into equation 1:
\[
3S + S = 30 \imp... | 14,007 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. There's a problem about a kitten and two rabbits weighing 30 pounds together. Hmm. Then there are some other relationships between their weights. The kitten and the larger rabbit together weigh three times as much as the smaller rabbit. And the kitten and the smaller rabbit together weigh twice... | [
{
"from": "human",
"value": "A kitten and two rabbits together weigh 30 pounds. The kitten and the larger rabbit together weigh exactly three times as much as the smaller rabbit, and the kitten and the smaller rabbit together weigh exactly twice as much as the larger rabbit. How many pounds does the kitten ... |
Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7. | Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7. | Alright, so I need to find the units digit of n, given that mn equals 21^6 and that m has a units digit of 7. Let me break this down step by step.
First off, I know that the units digit of a product depends only on the units digits of the numbers being multiplied. So, if m has a units digit of 7, and n is some number... | Given that \( mn = 21^6 \) and \( m \) has a units digit of 7, we need to find the units digit of \( n \).
First, we determine the units digit of \( 21^6 \). Since \( 21 \equiv 1 \mod 10 \), we have:
\[
21^6 \equiv 1^6 \equiv 1 \mod 10
\]
Thus, the units digit of \( 21^6 \) is 1.
Next, we know that \( m \) has a unit... | 23,584 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to find the units digit of n, given that mn equals 21^6 and that m has a units digit of 7. Let me break this down step by step.
First off, I know that the units digit of a product depends only on the units digits of the numbers being multiplied. So, if m has a units digit of 7, and n is som... | [
{
"from": "human",
"value": "Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to find the units digit of n, given that mn equals 21^6 and that m has a units digit of 7. Let me break this down step by ste... |
In a right triangle, the lengths of the legs are 24 inches and 45 inches. What is the area of the triangle? | In a right triangle, the lengths of the legs are 24 inches and 45 inches. What is the area of the triangle? | Okay, so I need to find the area of a right triangle where the legs are 24 inches and 45 inches. Hmm, let me recall how to calculate the area of a triangle. I remember that the formula for the area of any triangle is (base * height) / 2. But since this is a right triangle, the two legs serve as the base and height, rig... | To find the area of a right triangle with legs measuring 24 inches and 45 inches, we use the formula for the area of a triangle, which is \(\frac{1}{2} \times \text{base} \times \text{height}\). In a right triangle, the legs serve as the base and height.
First, we multiply the lengths of the legs:
\[
24 \times 45 = 10... | 586 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the area of a right triangle where the legs are 24 inches and 45 inches. Hmm, let me recall how to calculate the area of a triangle. I remember that the formula for the area of any triangle is (base * height) / 2. But since this is a right triangle, the two legs serve as the base and hei... | [
{
"from": "human",
"value": "In a right triangle, the lengths of the legs are 24 inches and 45 inches. What is the area of the triangle?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the area of a right triangle where the legs are 24 inches and 45 inches. Hmm, let me recall how... |
Let $a \nabla b = \frac{a}{b^2}$ for all positive numbers $a$ and $b$. If $3 \nabla 2 = c \nabla 3$, find $c$. | Let $a \nabla b = \frac{a}{b^2}$ for all positive numbers $a$ and $b$. If $3 \nabla 2 = c \nabla 3$, find $c$. | Okay, so I have this problem here: they define a new operation, right? It's denoted by a nabla symbol, ∇. And the operation is defined as a ∇ b equals a divided by b squared. So, for any positive numbers a and b, a ∇ b is a over b². Then, they tell me that 3 ∇ 2 equals c ∇ 3, and I need to find c. Alright, let's break ... | Given the operation \( a \nabla b = \frac{a}{b^2} \) for all positive numbers \( a \) and \( b \), we need to find \( c \) such that \( 3 \nabla 2 = c \nabla 3 \).
First, compute \( 3 \nabla 2 \):
\[
3 \nabla 2 = \frac{3}{2^2} = \frac{3}{4}
\]
Next, compute \( c \nabla 3 \):
\[
c \nabla 3 = \frac{c}{3^2} = \frac{c}{9... | 6,855 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: they define a new operation, right? It's denoted by a nabla symbol, ∇. And the operation is defined as a ∇ b equals a divided by b squared. So, for any positive numbers a and b, a ∇ b is a over b². Then, they tell me that 3 ∇ 2 equals c ∇ 3, and I need to find c. Alright, let'... | [
{
"from": "human",
"value": "Let $a \\nabla b = \\frac{a}{b^2}$ for all positive numbers $a$ and $b$. If $3 \\nabla 2 = c \\nabla 3$, find $c$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: they define a new operation, right? It's denoted by a nabla symbol, ∇. And th... |
$B$ and $C$ trisect $\overline{AD}$ and $M$ is the midpoint of $\overline{AD}$. $MC = 8$. How many units are in the length of $\overline{AD}$? | $B$ and $C$ trisect $\overline{AD}$ and $M$ is the midpoint of $\overline{AD}$. $MC = 8$. How many units are in the length of $\overline{AD}$? | Okay, so I need to figure out the length of segment AD, given that points B and C trisect AD, M is the midpoint of AD, and MC equals 8 units. Let me try to visualize this first.
Trisecting a segment means dividing it into three equal parts. So if B and C trisect AD, then the segments AB, BC, and CD should all be equal... | Given that points \( B \) and \( C \) trisect \(\overline{AD}\) and \( M \) is the midpoint of \(\overline{AD}\), we need to find the length of \(\overline{AD}\) given that \( MC = 8 \).
1. **Trisection and Midpoint**:
- Since \( B \) and \( C \) trisect \(\overline{AD}\), the segments \( AB \), \( BC \), and \( CD... | 22,515 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the length of segment AD, given that points B and C trisect AD, M is the midpoint of AD, and MC equals 8 units. Let me try to visualize this first.
Trisecting a segment means dividing it into three equal parts. So if B and C trisect AD, then the segments AB, BC, and CD should all ... | [
{
"from": "human",
"value": "$B$ and $C$ trisect $\\overline{AD}$ and $M$ is the midpoint of $\\overline{AD}$. $MC = 8$. How many units are in the length of $\\overline{AD}$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the length of segment AD, given that points B and C... |
Tom Tresize runs along a circular track with a circumference of 10 meters. He starts at a fixed point and travels halfway around the track on his first lap. On each subsequent lap, he travels half the distance he traveled on the previous lap. If he stops after five laps, what is the total distance Tom has traveled? Exp... | Tom Tresize runs along a circular track with a circumference of 10 meters. He starts at a fixed point and travels halfway around the track on his first lap. On each subsequent lap, he travels half the distance he traveled on the previous lap. If he stops after five laps, what is the total distance Tom has traveled? Exp... | Okay, let's try to figure out how far Tom Tresize has traveled after five laps on this circular track. The track has a circumference of 10 meters, so each full lap is 10 meters. But Tom doesn't run full laps; instead, he starts by going halfway around the track on his first lap, and then each subsequent lap he covers h... | Tom Tresize runs along a circular track with a circumference of 10 meters. He starts at a fixed point and travels halfway around the track on his first lap, which is 5 meters. Each subsequent lap, he travels half the distance he traveled on the previous lap. The distances for each of the five laps are as follows:
1st ... | 28,501 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to figure out how far Tom Tresize has traveled after five laps on this circular track. The track has a circumference of 10 meters, so each full lap is 10 meters. But Tom doesn't run full laps; instead, he starts by going halfway around the track on his first lap, and then each subsequent lap he ... | [
{
"from": "human",
"value": "Tom Tresize runs along a circular track with a circumference of 10 meters. He starts at a fixed point and travels halfway around the track on his first lap. On each subsequent lap, he travels half the distance he traveled on the previous lap. If he stops after five laps, what is... |
Antonette gets $70\%$ on a 10-problem test, $80\%$ on a 20-problem test and $90\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is her overall score, rounded to the nearest percent? | Antonette gets $70\%$ on a 10-problem test, $80\%$ on a 20-problem test and $90\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is her overall score, rounded to the nearest percent? | Okay, let's try to figure out Antonette's overall percentage when all three tests are combined. Hmm, so she took three different tests with different numbers of problems and different percentages. The question is asking, if we consider all those problems as one big test with 60 problems total, what percentage did she g... | Antonette's overall score is calculated by combining her correct answers from three tests and converting that into a percentage of the total number of problems.
1. **First test**: 70% on a 10-problem test:
\[
0.70 \times 10 = 7 \text{ correct answers}
\]
2. **Second test**: 80% on a 20-problem test:
\[
... | 16,279 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to figure out Antonette's overall percentage when all three tests are combined. Hmm, so she took three different tests with different numbers of problems and different percentages. The question is asking, if we consider all those problems as one big test with 60 problems total, what percentage d... | [
{
"from": "human",
"value": "Antonette gets $70\\%$ on a 10-problem test, $80\\%$ on a 20-problem test and $90\\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is her overall score, rounded to the nearest percent?"
},
{
"from": "gpt",
"value": "<t... |
A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece? | A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece? | Okay, so I have this problem where there's a 4-foot by 8-foot piece of plywood, and I need to cut it into 4 congruent rectangles. No wood left over, and the cuts don't waste any wood. Then, I have to find the positive difference between the greatest possible perimeter of a single piece and the least possible perimeter.... | The original piece of plywood is 4 feet by 8 feet, giving an area of 32 square feet. We need to cut this into 4 congruent rectangles, each with an area of 8 square feet.
Possible dimensions for each smaller rectangle are factors of 8: (1x8), (2x4), (4x2), and (8x1). We need to check which of these dimensions can fit ... | 37,512 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem where there's a 4-foot by 8-foot piece of plywood, and I need to cut it into 4 congruent rectangles. No wood left over, and the cuts don't waste any wood. Then, I have to find the positive difference between the greatest possible perimeter of a single piece and the least possible pe... | [
{
"from": "human",
"value": "A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perim... |
A game show offers a contestant three prizes, each with a different value. The contestant has a 30% chance of winning the smallest prize, a 25% chance of winning the middle prize, and a 45% chance of winning the largest prize. If the contestant wins the smallest prize, they receive $100. If they win the middle prize, t... | A game show offers a contestant three prizes, each with a different value. The contestant has a 30% chance of winning the smallest prize, a 25% chance of winning the middle prize, and a 45% chance of winning the largest prize. If the contestant wins the smallest prize, they receive $100. If they win the middle prize, t... | Okay, so I need to find the expected value of the prize the contestant will win. Let me start by understanding the problem step by step.
First, there are three prizes: smallest, middle, and largest. Each has a different value. The chances of winning each are 30%, 25%, and 45% respectively. The prizes are structured su... | The contestant has three possible prizes with the following probabilities and values:
- Smallest prize: 30% chance, value is $100.
- Middle prize: 25% chance, value is $100 + $200 = $300.
- Largest prize: 45% chance, value is $300 + $500 = $800.
To find the expected value (EV), we calculate the sum of each prize valu... | 21,801 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the expected value of the prize the contestant will win. Let me start by understanding the problem step by step.
First, there are three prizes: smallest, middle, and largest. Each has a different value. The chances of winning each are 30%, 25%, and 45% respectively. The prizes are struc... | [
{
"from": "human",
"value": "A game show offers a contestant three prizes, each with a different value. The contestant has a 30% chance of winning the smallest prize, a 25% chance of winning the middle prize, and a 45% chance of winning the largest prize. If the contestant wins the smallest prize, they rece... |
Find the number of integers between 100 and 999 that have exactly two digits the same, where these two digits are not adjacent, and the remaining digit is twice the value of the repeated digit. | Find the number of integers between 100 and 999 that have exactly two digits the same, where these two digits are not adjacent, and the remaining digit is twice the value of the repeated digit. | Okay, so I need to find the number of integers between 100 and 999 that have exactly two digits the same, where those two same digits are not adjacent, and the remaining digit is twice the value of the repeated digit. Hmm, let's break this down step by step.
First, since we're dealing with three-digit numbers, each nu... | To find the number of integers between 100 and 999 that have exactly two digits the same, where these two digits are not adjacent, and the remaining digit is twice the value of the repeated digit, we consider the following:
1. A three-digit number can be represented as \(ABC\) where \(A\), \(B\), and \(C\) are the hun... | 13,600 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the number of integers between 100 and 999 that have exactly two digits the same, where those two same digits are not adjacent, and the remaining digit is twice the value of the repeated digit. Hmm, let's break this down step by step.
First, since we're dealing with three-digit numbers,... | [
{
"from": "human",
"value": "Find the number of integers between 100 and 999 that have exactly two digits the same, where these two digits are not adjacent, and the remaining digit is twice the value of the repeated digit."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the numbe... |
How many positive solutions are there to the equation $y^3 = 3y - 2$? | How many positive solutions are there to the equation $y^3 = 3y - 2$? | Okay, so I need to find the number of positive solutions to the equation \( y^3 = 3y - 2 \). Let me think about how to approach this. Hmm, maybe I can start by rearranging the equation to bring all terms to one side. That way, I can analyze it as a standard polynomial equation. Let me try that.
Starting with the origi... | To determine the number of positive solutions to the equation \( y^3 = 3y - 2 \), we start by rearranging the equation:
\[ y^3 - 3y + 2 = 0 \]
We use the Rational Root Theorem to test possible rational roots. Testing \( y = 1 \):
\[ 1^3 - 3(1) + 2 = 0 \]
This works, so \( y = 1 \) is a root. We factor the cubic pol... | 12,334 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the number of positive solutions to the equation \( y^3 = 3y - 2 \). Let me think about how to approach this. Hmm, maybe I can start by rearranging the equation to bring all terms to one side. That way, I can analyze it as a standard polynomial equation. Let me try that.
Starting with t... | [
{
"from": "human",
"value": "How many positive solutions are there to the equation $y^3 = 3y - 2$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the number of positive solutions to the equation \\( y^3 = 3y - 2 \\). Let me think about how to approach this. Hmm, maybe I can star... |
In a major construction project, there are two pipe-laying machines, PipeMaster and FastPipe, each capable of laying pipes at different rates. PipeMaster lays a pipe at a rate of 150 meters per hour, while FastPipe lays a pipe at a rate of 300 meters per hour. The project requires a total of 4500 meters of pipes to be ... | In a major construction project, there are two pipe-laying machines, PipeMaster and FastPipe, each capable of laying pipes at different rates. PipeMaster lays a pipe at a rate of 150 meters per hour, while FastPipe lays a pipe at a rate of 300 meters per hour. The project requires a total of 4500 meters of pipes to be ... | Okay, let's see. So we have two pipe-laying machines, PipeMaster and FastPipe. PipeMaster can lay 150 meters per hour, and FastPipe can lay 300 meters per hour. The total length needed is 4500 meters. They need to work simultaneously but from opposite ends and finish at the same time. The question is, how far from the ... | To determine how far from the starting point PipeMaster should lay pipes so that both machines finish at the same time, we start by noting their laying rates: PipeMaster at 150 meters per hour and FastPipe at 300 meters per hour. The total length of the project is 4500 meters.
Let \( D \) be the distance PipeMaster la... | 4,543 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. So we have two pipe-laying machines, PipeMaster and FastPipe. PipeMaster can lay 150 meters per hour, and FastPipe can lay 300 meters per hour. The total length needed is 4500 meters. They need to work simultaneously but from opposite ends and finish at the same time. The question is, how far f... | [
{
"from": "human",
"value": "In a major construction project, there are two pipe-laying machines, PipeMaster and FastPipe, each capable of laying pipes at different rates. PipeMaster lays a pipe at a rate of 150 meters per hour, while FastPipe lays a pipe at a rate of 300 meters per hour. The project requir... |
The function $f(x) = \sqrt{x^{2}}$ is graphed below.
At how many points $x$ on the interval $[-10, 10]$ does the graph of the function $g(x) = \lfloor f(x) \rfloor$ intersect with the graph of the function $f(x)$ itself? | The function $f(x) = \sqrt{x^{2}}$ is graphed below.
At how many points $x$ on the interval $[-10, 10]$ does the graph of the function $g(x) = \lfloor f(x) \rfloor$ intersect with the graph of the function $f(x)$ itself? | Okay, let's try to figure out this problem step by step. The question is about the functions f(x) = sqrt(x²) and g(x) = floor(f(x)). We need to find how many points in the interval [-10, 10] the graphs of f(x) and g(x) intersect.
First, let me make sure I understand the functions involved. The function f(x) = sqrt(x²... | The function \( f(x) = \sqrt{x^2} \) simplifies to \( |x| \), which is the absolute value function. The function \( g(x) = \lfloor f(x) \rfloor \) is the floor function of \( |x| \).
To find the points where \( f(x) \) and \( g(x) \) intersect, we need to determine when \( \lfloor |x| \rfloor = |x| \). This occurs wh... | 15,048 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to figure out this problem step by step. The question is about the functions f(x) = sqrt(x²) and g(x) = floor(f(x)). We need to find how many points in the interval [-10, 10] the graphs of f(x) and g(x) intersect.
First, let me make sure I understand the functions involved. The function f(x) =... | [
{
"from": "human",
"value": "The function $f(x) = \\sqrt{x^{2}}$ is graphed below.\n\n\n\nAt how many points $x$ on the interval $[-10, 10]$ does the graph of the function $g(x) = \\lfloor f(x) \\rfloor$ intersect with the graph of the function $f(x)$ itself?"
},
{
"from": "gpt",
"value": "<thin... |
A bookshelf has a height of 60 inches. If there are 2.54 centimeters in an inch, what height should be specified, in centimeters, for the bookshelf in a furniture catalog? (You may use a calculator on this problem; answer to the nearest tenth.) | A bookshelf has a height of 60 inches. If there are 2.54 centimeters in an inch, what height should be specified, in centimeters, for the bookshelf in a furniture catalog? (You may use a calculator on this problem; answer to the nearest tenth.) | Okay, so I need to convert the height of a bookshelf from inches to centimeters. The given height is 60 inches, and there are 2.54 centimeters in an inch. Hmm, let's think about how to approach this.
First, I remember that converting inches to centimeters requires multiplying the number of inches by the conversion fa... | To convert the height of the bookshelf from inches to centimeters, we use the conversion factor 2.54 centimeters per inch.
The height in inches is 60. Multiplying this by the conversion factor gives:
\[
60 \, \text{inches} \times 2.54 \, \frac{\text{centimeters}}{\text{inch}} = 152.4 \, \text{centimeters}
\]
Since t... | 26,543 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to convert the height of a bookshelf from inches to centimeters. The given height is 60 inches, and there are 2.54 centimeters in an inch. Hmm, let's think about how to approach this.
First, I remember that converting inches to centimeters requires multiplying the number of inches by the conve... | [
{
"from": "human",
"value": "A bookshelf has a height of 60 inches. If there are 2.54 centimeters in an inch, what height should be specified, in centimeters, for the bookshelf in a furniture catalog? (You may use a calculator on this problem; answer to the nearest tenth.)"
},
{
"from": "gpt",
"... |
Alice and Bob are playing the "Factor Chain Game". Alice says, "My number is 48". Bob replies, "That's a good one, but I think I can do better. Every prime factor of my number is one more than a prime factor of your number".
What is the smallest possible number that Bob could have? | Alice and Bob are playing the "Factor Chain Game". Alice says, "My number is 48". Bob replies, "That's a good one, but I think I can do better. Every prime factor of my number is one more than a prime factor of your number".
What is the smallest possible number that Bob could have? | Okay, so Alice and Bob are playing this Factor Chain Game, and Alice says her number is 48. Then Bob claims he can do better by having a number where every prime factor is one more than a prime factor of Alice's number. Hmm, let me break this down.
First, I need to figure out what the prime factors of Alice's number, ... | Alice's number is 48, which has prime factors 2 and 3. Bob's number must have prime factors that are each one more than a prime factor of Alice's number.
To determine the possible prime factors of Bob's number:
- Adding 1 to each of Alice's prime factors (2 and 3) gives 3 and 4.
- Since 4 is not a prime number, the ... | 683 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Alice and Bob are playing this Factor Chain Game, and Alice says her number is 48. Then Bob claims he can do better by having a number where every prime factor is one more than a prime factor of Alice's number. Hmm, let me break this down.
First, I need to figure out what the prime factors of Alice's ... | [
{
"from": "human",
"value": "Alice and Bob are playing the \"Factor Chain Game\". Alice says, \"My number is 48\". Bob replies, \"That's a good one, but I think I can do better. Every prime factor of my number is one more than a prime factor of your number\".\n\nWhat is the smallest possible number that Bob... |
Let $a$, $b$, and $c$ be positive integers such that
\[1 < \frac{a + c}{b + c} < 2.\]
If $\frac{a}{b}$ is an integer, what is its value? | Let $a$, $b$, and $c$ be positive integers such that
\[1 < \frac{a + c}{b + c} < 2.\]
If $\frac{a}{b}$ is an integer, what is its value? | Okay, let me try to figure out this problem. So, we have three positive integers a, b, and c. The condition given is that (a + c)/(b + c) is between 1 and 2, but not equal to either. Also, a/b is an integer. We need to find the value of that integer. Hmm, let me break it down step by step.
First, the key points:
1. a... | Given positive integers \(a\), \(b\), and \(c\) such that \(1 < \frac{a + c}{b + c} < 2\) and \(\frac{a}{b}\) is an integer, we need to determine the value of this integer.
1. **Substitution and Inequalities**:
- Let \(k = \frac{a}{b}\), where \(k\) is an integer. Then \(a = kb\).
- Substitute \(a = kb\) into th... | 33,867 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, we have three positive integers a, b, and c. The condition given is that (a + c)/(b + c) is between 1 and 2, but not equal to either. Also, a/b is an integer. We need to find the value of that integer. Hmm, let me break it down step by step.
First, the key point... | [
{
"from": "human",
"value": "Let $a$, $b$, and $c$ be positive integers such that\n\\[1 < \\frac{a + c}{b + c} < 2.\\]\nIf $\\frac{a}{b}$ is an integer, what is its value?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. So, we have three positive integers a, ... |
Compute the product of the reciprocals of the roots of the equation \[x^2 \sin x + 3x^3 + 8 = 0.\] | Compute the product of the reciprocals of the roots of the equation \[x^2 \sin x + 3x^3 + 8 = 0.\] | Okay, so I need to compute the product of the reciprocals of the roots of the equation \(x^2 \sin x + 3x^3 + 8 = 0\). Hmm, reciprocals of roots... that sounds like if the roots are \(r_1, r_2, \dots, r_n\), then the product would be \(\frac{1}{r_1} \times \frac{1}{r_2} \times \dots \times \frac{1}{r_n}\). Right? Which ... | To compute the product of the reciprocals of the roots of the equation \(x^2 \sin x + 3x^3 + 8 = 0\), we need to consider the nature of the roots of this transcendental equation.
1. **Transcendental Equation Analysis**:
- The equation \(x^2 \sin x + 3x^3 + 8 = 0\) combines a cubic polynomial with a trigonometric t... | 35,762 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to compute the product of the reciprocals of the roots of the equation \(x^2 \sin x + 3x^3 + 8 = 0\). Hmm, reciprocals of roots... that sounds like if the roots are \(r_1, r_2, \dots, r_n\), then the product would be \(\frac{1}{r_1} \times \frac{1}{r_2} \times \dots \times \frac{1}{r_n}\). Right... | [
{
"from": "human",
"value": "Compute the product of the reciprocals of the roots of the equation \\[x^2 \\sin x + 3x^3 + 8 = 0.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to compute the product of the reciprocals of the roots of the equation \\(x^2 \\sin x + 3x^3 + 8 = 0\\). Hmm,... |
Let the ordered triples $(a,b,c)$ of real numbers that satisfy
\begin{align*}
a^2 + b^2 - c^2 &= 25, \\
a^2 - b^2 + c^2 &= 13, \\
-a^2 + b^2 + c^2 &= 18.
\end{align*}be $(a_1,b_1,c_1),$ $(a_2,b_2,c_2),$ $\dots,$ $(a_n,b_n,c_n).$ Find $a_1^2 + a_2^2 + \dots + a_n^2.$ | Let the ordered triples $(a,b,c)$ of real numbers that satisfy
\begin{align*}
a^2 + b^2 - c^2 &= 25, \\
a^2 - b^2 + c^2 &= 13, \\
-a^2 + b^2 + c^2 &= 18.
\end{align*}be $(a_1,b_1,c_1),$ $(a_2,b_2,c_2),$ $\dots,$ $(a_n,b_n,c_n).$ Find $a_1^2 + a_2^2 + \dots + a_n^2.$ | Okay, let's see. I need to solve this system of equations for real numbers a, b, c. The equations are:
1. a² + b² - c² = 25
2. a² - b² + c² = 13
3. -a² + b² + c² = 18
And then, once I find all the ordered triples (a, b, c) that satisfy these, I have to sum up all the squares of the a-components. So if there are multi... | Given the system of equations:
\[
\begin{align*}
a^2 + b^2 - c^2 &= 25, \\
a^2 - b^2 + c^2 &= 13, \\
-a^2 + b^2 + c^2 &= 18,
\end{align*}
\]
we start by adding all three equations:
\[
(a^2 + b^2 - c^2) + (a^2 - b^2 + c^2) + (-a^2 + b^2 + c^2) = 25 + 13 + 18.
\]
This simplifies to:
\[
a^2 + b^2 + c^2 = 56.
\]
Next... | 19,986 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to solve this system of equations for real numbers a, b, c. The equations are:
1. a² + b² - c² = 25
2. a² - b² + c² = 13
3. -a² + b² + c² = 18
And then, once I find all the ordered triples (a, b, c) that satisfy these, I have to sum up all the squares of the a-components. So if there a... | [
{
"from": "human",
"value": "Let the ordered triples $(a,b,c)$ of real numbers that satisfy\n\\begin{align*}\na^2 + b^2 - c^2 &= 25, \\\\\na^2 - b^2 + c^2 &= 13, \\\\\n-a^2 + b^2 + c^2 &= 18.\n\\end{align*}be $(a_1,b_1,c_1),$ $(a_2,b_2,c_2),$ $\\dots,$ $(a_n,b_n,c_n).$ Find $a_1^2 + a_2^2 + \\dots + a_n^2.... |
What is the largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5? | What is the largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5? | Alright, let's tackle this problem. We need to find the largest value of n less than 100,000 such that the expression 8(n-2)^5 - n^2 +14n -24 is divisible by 5. Hmm, okay. So essentially, we need 8(n-2)^5 - n^2 +14n -24 ≡ 0 mod 5. Let me break this down step by step.
First, since we're dealing with modulo 5, it might ... | To find the largest value of \( n \) less than 100,000 for which the expression \( 8(n-2)^5 - n^2 + 14n - 24 \) is a multiple of 5, we start by simplifying the expression modulo 5.
First, we note the following congruences modulo 5:
- \( 8 \equiv 3 \mod 5 \)
- \( 14 \equiv 4 \mod 5 \)
- \( -24 \equiv 1 \mod 5 \)
Using... | 8,933 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's tackle this problem. We need to find the largest value of n less than 100,000 such that the expression 8(n-2)^5 - n^2 +14n -24 is divisible by 5. Hmm, okay. So essentially, we need 8(n-2)^5 - n^2 +14n -24 ≡ 0 mod 5. Let me break this down step by step.
First, since we're dealing with modulo 5, i... | [
{
"from": "human",
"value": "What is the largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5?"
},
{
"from": "gpt",
"value": "<think>\nAlright, let's tackle this problem. We need to find the largest value of n less than 100,000 such that the ex... |
If the degree measure of the sum of the interior angles of a convex polygon with $m$ sides is 2700, what is the degree measure of the sum of the interior angles of a convex polygon with $m + 4$ sides? | If the degree measure of the sum of the interior angles of a convex polygon with $m$ sides is 2700, what is the degree measure of the sum of the interior angles of a convex polygon with $m + 4$ sides? | Okay, so I have this problem here: If the sum of the interior angles of a convex polygon with m sides is 2700 degrees, what is the sum for a convex polygon with m + 4 sides? Hmm, let's think.
First, I remember there's a formula for the sum of the interior angles of a convex polygon. Is it (n-2)*180, where n is the nu... | To find the degree measure of the sum of the interior angles of a convex polygon with \( m + 4 \) sides, we start with the given information that the sum of the interior angles of a convex polygon with \( m \) sides is 2700 degrees.
The formula for the sum of the interior angles of a convex polygon with \( n \) sides... | 7,612 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: If the sum of the interior angles of a convex polygon with m sides is 2700 degrees, what is the sum for a convex polygon with m + 4 sides? Hmm, let's think.
First, I remember there's a formula for the sum of the interior angles of a convex polygon. Is it (n-2)*180, where n i... | [
{
"from": "human",
"value": "If the degree measure of the sum of the interior angles of a convex polygon with $m$ sides is 2700, what is the degree measure of the sum of the interior angles of a convex polygon with $m + 4$ sides?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this probl... |
How many three-digit positive integers are congruent to 2 (mod 5)? | How many three-digit positive integers are congruent to 2 (mod 5)? | Okay, so I need to figure out how many three-digit positive integers are congruent to 2 modulo 5. Hmm, let's start by recalling what a three-digit number is. A three-digit number ranges from 100 to 999, inclusive. So, I need to count how many numbers in this range satisfy the condition that when divided by 5, the remai... | To determine how many three-digit positive integers are congruent to 2 modulo 5, we start by noting that three-digit numbers range from 100 to 999. A number congruent to 2 modulo 5 can be expressed as \(5k + 2\) for some integer \(k\).
We need to find the range of \(k\) such that \(100 \leq 5k + 2 \leq 999\). Solving ... | 340 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many three-digit positive integers are congruent to 2 modulo 5. Hmm, let's start by recalling what a three-digit number is. A three-digit number ranges from 100 to 999, inclusive. So, I need to count how many numbers in this range satisfy the condition that when divided by 5, t... | [
{
"from": "human",
"value": "How many three-digit positive integers are congruent to 2 (mod 5)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many three-digit positive integers are congruent to 2 modulo 5. Hmm, let's start by recalling what a three-digit number is. A ... |
When a certain quadratic expression is divided by $x - 1$, the remainder is 2. When the same expression is divided by $x + 1$, the remainder is -4. Determine the value of the quadratic expression when $x = 0$. | When a certain quadratic expression is divided by $x - 1$, the remainder is 2. When the same expression is divided by $x + 1$, the remainder is -4. Determine the value of the quadratic expression when $x = 0$. | Okay, so I have this problem here: When a certain quadratic expression is divided by x - 1, the remainder is 2. When the same expression is divided by x + 1, the remainder is -4. I need to determine the value of the quadratic expression when x = 0. Hmm, let's break this down step by step.
First, I remember that when a... | Given a quadratic expression \( f(x) = ax^2 + bx + c \), we know the following:
1. When divided by \( x - 1 \), the remainder is 2. This implies \( f(1) = 2 \).
2. When divided by \( x + 1 \), the remainder is -4. This implies \( f(-1) = -4 \).
We need to determine the value of the quadratic expression when \( x = 0 \... | 11,939 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: When a certain quadratic expression is divided by x - 1, the remainder is 2. When the same expression is divided by x + 1, the remainder is -4. I need to determine the value of the quadratic expression when x = 0. Hmm, let's break this down step by step.
First, I remember tha... | [
{
"from": "human",
"value": "When a certain quadratic expression is divided by $x - 1$, the remainder is 2. When the same expression is divided by $x + 1$, the remainder is -4. Determine the value of the quadratic expression when $x = 0$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have t... |
The function $f(x)$ satisfies
\[f(x + y) = f(x)f(y)\]for all real numbers $x$ and $y,$ and $f(2) = 3.$ Find $f(3).$ | The function $f(x)$ satisfies
\[f(x + y) = f(x)f(y)\]for all real numbers $x$ and $y,$ and $f(2) = 3.$ Find $f(3).$ | Okay, so I have this problem here: the function f(x) satisfies f(x + y) = f(x)f(y) for all real numbers x and y, and f(2) = 3. I need to find f(3). Hmm, let me think.
First, the equation f(x + y) = f(x)f(y) is ringing a bell. I remember that functions which satisfy this property are called exponential functions. Like... | Given the function \( f(x) \) satisfies the functional equation \( f(x + y) = f(x)f(y) \) for all real numbers \( x \) and \( y \), and \( f(2) = 3 \), we need to find \( f(3) \).
First, we note that the functional equation \( f(x + y) = f(x)f(y) \) is characteristic of exponential functions. To determine the specific... | 22,808 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: the function f(x) satisfies f(x + y) = f(x)f(y) for all real numbers x and y, and f(2) = 3. I need to find f(3). Hmm, let me think.
First, the equation f(x + y) = f(x)f(y) is ringing a bell. I remember that functions which satisfy this property are called exponential functio... | [
{
"from": "human",
"value": "The function $f(x)$ satisfies\n\\[f(x + y) = f(x)f(y)\\]for all real numbers $x$ and $y,$ and $f(2) = 3.$ Find $f(3).$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: the function f(x) satisfies f(x + y) = f(x)f(y) for all real numbers x a... |
A sequence of numbers is formed by repeatedly adding 3, multiplying by 2, and subtracting 5, in that order, starting with 10. What is the 12th term of this sequence? | A sequence of numbers is formed by repeatedly adding 3, multiplying by 2, and subtracting 5, in that order, starting with 10. What is the 12th term of this sequence? | Okay, let me try to figure out this sequence problem. So, the question says that a sequence starts with 10, and then each subsequent term is formed by adding 3, multiplying by 2, and subtracting 5, in that order. We need to find the 12th term. Hmm, let me break this down step by step.
First, let's make sure I understa... | The sequence starts with 10, and each subsequent term is formed by adding 3, multiplying by 2, and subtracting 5. We need to find the 12th term.
First, we observe the recursive relation:
\[ a_{n+1} = (a_n + 3) \cdot 2 - 5 \]
Simplifying this, we get:
\[ a_{n+1} = 2a_n + 1 \]
This is a linear recurrence relation. To s... | 13,623 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this sequence problem. So, the question says that a sequence starts with 10, and then each subsequent term is formed by adding 3, multiplying by 2, and subtracting 5, in that order. We need to find the 12th term. Hmm, let me break this down step by step.
First, let's make sure I ... | [
{
"from": "human",
"value": "A sequence of numbers is formed by repeatedly adding 3, multiplying by 2, and subtracting 5, in that order, starting with 10. What is the 12th term of this sequence?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this sequence problem. So, the... |
A bus leaves a station every 15 minutes. If a passenger arrives at the station at a random time, what is the probability that they will have to wait more than 5 minutes for the next bus? | A bus leaves a station every 15 minutes. If a passenger arrives at the station at a random time, what is the probability that they will have to wait more than 5 minutes for the next bus? | Okay, so there's this problem: A bus leaves a station every 15 minutes. If a passenger arrives at the station at a random time, what is the probability that they will have to wait more than 5 minutes for the next bus? Hmm, let me think about how to approach this.
First, I remember that when dealing with probabilities ... | The problem involves determining the probability that a passenger arriving at a random time will have to wait more than 5 minutes for the next bus, given that buses leave the station every 15 minutes.
1. **Understanding the Uniform Distribution**:
- The passenger's arrival time is uniformly distributed between 0 an... | 39,543 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this problem: A bus leaves a station every 15 minutes. If a passenger arrives at the station at a random time, what is the probability that they will have to wait more than 5 minutes for the next bus? Hmm, let me think about how to approach this.
First, I remember that when dealing with probab... | [
{
"from": "human",
"value": "A bus leaves a station every 15 minutes. If a passenger arrives at the station at a random time, what is the probability that they will have to wait more than 5 minutes for the next bus?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so there's this problem: A bus lea... |
The time it takes to fill a tank varies inversely with the square of the diameter of the pipe used. A pipe with a diameter of 4 inches can fill a certain tank in 18 minutes. A pipe with a diameter of 6 inches will take how many minutes to fill the same tank? | The time it takes to fill a tank varies inversely with the square of the diameter of the pipe used. A pipe with a diameter of 4 inches can fill a certain tank in 18 minutes. A pipe with a diameter of 6 inches will take how many minutes to fill the same tank? | Okay, let me try to figure out this problem. So, it's about filling a tank with different pipe diameters and how the time changes based on that. The problem says the time varies inversely with the square of the diameter. Hmm, inverse variation. Let me recall what that means.
If time varies inversely with the square of... | The time it takes to fill a tank varies inversely with the square of the diameter of the pipe used. This relationship can be expressed as \( T = \frac{k}{d^2} \), where \( T \) is the time, \( d \) is the diameter, and \( k \) is a constant.
Given that a pipe with a diameter of 4 inches fills the tank in 18 minutes, w... | 13,572 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, it's about filling a tank with different pipe diameters and how the time changes based on that. The problem says the time varies inversely with the square of the diameter. Hmm, inverse variation. Let me recall what that means.
If time varies inversely with the s... | [
{
"from": "human",
"value": "The time it takes to fill a tank varies inversely with the square of the diameter of the pipe used. A pipe with a diameter of 4 inches can fill a certain tank in 18 minutes. A pipe with a diameter of 6 inches will take how many minutes to fill the same tank?"
},
{
"from"... |
A point $P(x,y)$ moves in the coordinate plane in such a way that its distance from the point $A(1,-2)$ is always three times its distance from the point $B(-1,1)$. Find the locus of $P$ and determine its shape. | A point $P(x,y)$ moves in the coordinate plane in such a way that its distance from the point $A(1,-2)$ is always three times its distance from the point $B(-1,1)$. Find the locus of $P$ and determine its shape. | Okay, so I need to find the locus of a point P(x, y) such that its distance from point A(1, -2) is always three times its distance from point B(-1, 1). Hmm, let's start by recalling the distance formula between two points. The distance between P(x, y) and A(1, -2) should be three times the distance between P(x, y) and ... | To find the locus of the point \( P(x, y) \) such that its distance from \( A(1, -2) \) is always three times its distance from \( B(-1, 1) \), we start by using the distance formula. The given condition translates to the equation:
\[
\sqrt{(x - 1)^2 + (y + 2)^2} = 3 \sqrt{(x + 1)^2 + (y - 1)^2}
\]
Squaring both side... | 11,355 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the locus of a point P(x, y) such that its distance from point A(1, -2) is always three times its distance from point B(-1, 1). Hmm, let's start by recalling the distance formula between two points. The distance between P(x, y) and A(1, -2) should be three times the distance between P(x,... | [
{
"from": "human",
"value": "A point $P(x,y)$ moves in the coordinate plane in such a way that its distance from the point $A(1,-2)$ is always three times its distance from the point $B(-1,1)$. Find the locus of $P$ and determine its shape."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need... |
In the Fibonacci sequence, each number is the sum of the two numbers before it. In which position of the Fibonacci sequence do three consecutive numbers occur such that the sum of the first and third numbers is twice the second number? | In the Fibonacci sequence, each number is the sum of the two numbers before it. In which position of the Fibonacci sequence do three consecutive numbers occur such that the sum of the first and third numbers is twice the second number? | Okay, so I have this Fibonacci sequence problem. The question is asking: In which position of the Fibonacci sequence do three consecutive numbers occur such that the sum of the first and third numbers is twice the second number? Hmm, let me break this down.
First, let me recall the Fibonacci sequence. It starts with 0... | To solve the problem, we need to find the position in the Fibonacci sequence where three consecutive numbers occur such that the sum of the first and third numbers is twice the second number.
1. **Fibonacci Sequence Definition**: The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the... | 6,079 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this Fibonacci sequence problem. The question is asking: In which position of the Fibonacci sequence do three consecutive numbers occur such that the sum of the first and third numbers is twice the second number? Hmm, let me break this down.
First, let me recall the Fibonacci sequence. It start... | [
{
"from": "human",
"value": "In the Fibonacci sequence, each number is the sum of the two numbers before it. In which position of the Fibonacci sequence do three consecutive numbers occur such that the sum of the first and third numbers is twice the second number?"
},
{
"from": "gpt",
"value": "... |
Find the area of the region enclosed by the graph of $|x-60|+|y|=\left|\frac{x}{4}\right|$.
Write another problem inspired by this one:
Find the area of the region enclosed by the graph of $|x^2-4|+|y|=\left|\frac{x}{2}\right|$. | Find the area of the region enclosed by the graph of $|x-60|+|y|=\left|\frac{x}{4}\right|$.
Write another problem inspired by this one:
Find the area of the region enclosed by the graph of $|x^2-4|+|y|=\left|\frac{x}{2}\right|$. | Okay, so I need to find the area of the region enclosed by the graph of |x - 60| + |y| = |x/4|. Hmm, absolute value equations can sometimes be tricky because they can split into different cases depending on the sign of the expressions inside. Let me start by understanding what this equation represents.
First, the equa... | To find the area of the region enclosed by the graph of \(|x - 60| + |y| = \left|\frac{x}{4}\right|\), we start by solving for \(|y|\):
\[
|y| = \left|\frac{x}{4}\right| - |x - 60|
\]
The right-hand side must be non-negative, leading to the inequality \(|x/4| \geq |x - 60|\). Solving this inequality, we find the doma... | 17,925 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the area of the region enclosed by the graph of |x - 60| + |y| = |x/4|. Hmm, absolute value equations can sometimes be tricky because they can split into different cases depending on the sign of the expressions inside. Let me start by understanding what this equation represents.
First, ... | [
{
"from": "human",
"value": "Find the area of the region enclosed by the graph of $|x-60|+|y|=\\left|\\frac{x}{4}\\right|$.\n\nWrite another problem inspired by this one:\n\nFind the area of the region enclosed by the graph of $|x^2-4|+|y|=\\left|\\frac{x}{2}\\right|$."
},
{
"from": "gpt",
"valu... |
A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace? | A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace? | Okay, so I need to find the probability that when I draw two cards from a standard deck with replacement, at least one of them is a diamond or an ace. Hmm, let's break this down.
First, since the card is replaced each time, the two draws are independent events. That means the outcome of the first draw doesn't affect ... | To find the probability that at least one of the cards chosen at random from a standard deck of 52 cards (with replacement) is a diamond or an ace, we use the complement rule.
1. Calculate the number of cards that are neither diamonds nor aces:
- There are 13 diamonds and 4 aces, but the ace of diamonds is counted... | 28,389 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that when I draw two cards from a standard deck with replacement, at least one of them is a diamond or an ace. Hmm, let's break this down.
First, since the card is replaced each time, the two draws are independent events. That means the outcome of the first draw doesn't... | [
{
"from": "human",
"value": "A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the probabi... |
Consider the quadratic equation:
$$bx^2 - 15x - 2 = 0$$
Find the value of $x$ for which this equation has only one solution. | Consider the quadratic equation:
$$bx^2 - 15x - 2 = 0$$
Find the value of $x$ for which this equation has only one solution. | Okay, so I need to find the value of x for which the quadratic equation \( bx^2 - 15x - 2 = 0 \) has only one solution. Hmm, let me think. I remember that a quadratic equation \( ax^2 + bx + c = 0 \) has only one solution when its discriminant is zero. The discriminant is \( b^2 - 4ac \). If that equals zero, then the ... | To determine the value of \( x \) for which the quadratic equation \( bx^2 - 15x - 2 = 0 \) has only one solution, we need to ensure the discriminant is zero. For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is given by \( b^2 - 4ac \).
For the given equation \( bx^2 - 15x - 2 = 0 \), the coefficien... | 36,340 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of x for which the quadratic equation \( bx^2 - 15x - 2 = 0 \) has only one solution. Hmm, let me think. I remember that a quadratic equation \( ax^2 + bx + c = 0 \) has only one solution when its discriminant is zero. The discriminant is \( b^2 - 4ac \). If that equals zero, t... | [
{
"from": "human",
"value": "Consider the quadratic equation:\n$$bx^2 - 15x - 2 = 0$$\nFind the value of $x$ for which this equation has only one solution."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of x for which the quadratic equation \\( bx^2 - 15x - 2 = 0 \\) h... |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2009,0),(2009,2010),$ and $(0,2010)$. What is the probability that $x > 7y$? Express your answer as a common fraction. | Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2009,0),(2009,2010),$ and $(0,2010)$. What is the probability that $x > 7y$? Express your answer as a common fraction. | Okay, so I need to find the probability that a randomly picked point (x, y) from this rectangle satisfies x > 7y. The rectangle has vertices at (0,0), (2009,0), (2009,2010), and (0,2010). Hmm, let me visualize this. The rectangle is wider in the x-direction (2009 units) and taller in the y-direction (2010 units).
Pro... | To find the probability that a randomly picked point \((x, y)\) from the rectangle with vertices \((0,0)\), \((2009,0)\), \((2009,2010)\), and \((0,2010)\) satisfies \(x > 7y\), we start by calculating the total area of the rectangle and the area of the region where \(x > 7y\).
1. **Total Area of the Rectangle**:
\... | 15,951 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that a randomly picked point (x, y) from this rectangle satisfies x > 7y. The rectangle has vertices at (0,0), (2009,0), (2009,2010), and (0,2010). Hmm, let me visualize this. The rectangle is wider in the x-direction (2009 units) and taller in the y-direction (2010 units... | [
{
"from": "human",
"value": "Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2009,0),(2009,2010),$ and $(0,2010)$. What is the probability that $x > 7y$? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to fin... |
Two spheres of radii $4$ and $9$ are centered at $(0,0,0)$ and $(12,0,0)$, respectively. A plane is tangent to both spheres. Find the $z$-intercept of the plane. | Two spheres of radii $4$ and $9$ are centered at $(0,0,0)$ and $(12,0,0)$, respectively. A plane is tangent to both spheres. Find the $z$-intercept of the plane. | Okay, so I need to find the z-intercept of a plane that's tangent to both spheres. The spheres have radii 4 and 9, and their centers are at (0,0,0) and (12,0,0) respectively. Hmm, let me visualize this. Both spheres are along the x-axis, the first one at the origin with radius 4, and the second one at (12,0,0) with rad... | To find the z-intercept of a plane tangent to both spheres with radii 4 and 9, centered at (0,0,0) and (12,0,0) respectively, we start by noting that the distance from each center to the tangent plane must equal their respective radii.
1. The equation of the plane is \(ax + by + cz = d\).
2. The distance from the ori... | 9,663 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the z-intercept of a plane that's tangent to both spheres. The spheres have radii 4 and 9, and their centers are at (0,0,0) and (12,0,0) respectively. Hmm, let me visualize this. Both spheres are along the x-axis, the first one at the origin with radius 4, and the second one at (12,0,0) ... | [
{
"from": "human",
"value": "Two spheres of radii $4$ and $9$ are centered at $(0,0,0)$ and $(12,0,0)$, respectively. A plane is tangent to both spheres. Find the $z$-intercept of the plane."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the z-intercept of a plane that's tangent... |
A convex pentagon has interior angles with measures $x+1$, $2x$, $3x$, $4x$, and $5x-1$ degrees. What is the measure of the largest angle? | A convex pentagon has interior angles with measures $x+1$, $2x$, $3x$, $4x$, and $5x-1$ degrees. What is the measure of the largest angle? | Okay, so I need to find the measure of the largest angle in a convex pentagon where the interior angles are given as x+1, 2x, 3x, 4x, and 5x-1 degrees. Hmm, let's start by recalling some basic facts about convex pentagons.
First, I remember that the sum of the interior angles of a convex polygon can be calculated usin... | A convex pentagon has interior angles with measures \(x+1\), \(2x\), \(3x\), \(4x\), and \(5x-1\) degrees. To find the measure of the largest angle, we start by using the formula for the sum of the interior angles of a pentagon, which is \((5-2) \times 180 = 540\) degrees.
We set up the equation for the sum of the giv... | 14,307 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the measure of the largest angle in a convex pentagon where the interior angles are given as x+1, 2x, 3x, 4x, and 5x-1 degrees. Hmm, let's start by recalling some basic facts about convex pentagons.
First, I remember that the sum of the interior angles of a convex polygon can be calcula... | [
{
"from": "human",
"value": "A convex pentagon has interior angles with measures $x+1$, $2x$, $3x$, $4x$, and $5x-1$ degrees. What is the measure of the largest angle?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the measure of the largest angle in a convex pentagon where the... |
At Lincoln High School, there are 15 players on the soccer team. All 15 players are taking at least one of algebra or geometry. If 9 players are taking algebra and 3 players are taking both subjects, how many players are taking geometry? | At Lincoln High School, there are 15 players on the soccer team. All 15 players are taking at least one of algebra or geometry. If 9 players are taking algebra and 3 players are taking both subjects, how many players are taking geometry? | Okay, let's see. The problem says there are 15 players on the soccer team at Lincoln High School, and all of them are taking at least one of algebra or geometry. So, that means every player is enrolled in algebra, geometry, or both. Then it tells me that 9 players are taking algebra, and 3 players are taking both subje... | Given that all 15 players on the soccer team are taking at least one of algebra or geometry, we use the principle of inclusion-exclusion to find the number of players taking geometry.
We know:
- The total number of players, \( |A \cup G| = 15 \)
- The number of players taking algebra, \( |A| = 9 \)
- The number of pl... | 19,035 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem says there are 15 players on the soccer team at Lincoln High School, and all of them are taking at least one of algebra or geometry. So, that means every player is enrolled in algebra, geometry, or both. Then it tells me that 9 players are taking algebra, and 3 players are taking bo... | [
{
"from": "human",
"value": "At Lincoln High School, there are 15 players on the soccer team. All 15 players are taking at least one of algebra or geometry. If 9 players are taking algebra and 3 players are taking both subjects, how many players are taking geometry?"
},
{
"from": "gpt",
"value":... |
A taxi company charges a base fare of 2 dollars plus an additional 0.20 USD for every mile driven. A tourist has a 50-euro bill and wants to pay for the taxi ride using this bill. The current exchange rate is 1 USD = 0.88 EUR. If the taxi driver has driven x miles, what is the maximum value of x for which the tourist c... | A taxi company charges a base fare of 2 dollars plus an additional 0.20 USD for every mile driven. A tourist has a 50-euro bill and wants to pay for the taxi ride using this bill. The current exchange rate is 1 USD = 0.88 EUR. If the taxi driver has driven x miles, what is the maximum value of x for which the tourist c... | Okay, let's see. I need to figure out the maximum number of miles, x, that the taxi can drive so that the tourist can pay with a 50-euro bill. Hmm. Let's start by understanding all the components here.
First, the taxi fare is a base of $2 plus $0.20 per mile. So, the total fare in dollars would be 2 + 0.20x. The touri... | The taxi fare is calculated as a base fare of $2 plus $0.20 per mile driven. The tourist has a 50-euro bill, and the exchange rate is 1 USD = 0.88 EUR. We need to determine the maximum number of miles \( x \) that the taxi can drive so that the tourist can pay the fare using his 50-euro bill.
1. Convert 50 euros to US... | 20,732 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to figure out the maximum number of miles, x, that the taxi can drive so that the tourist can pay with a 50-euro bill. Hmm. Let's start by understanding all the components here.
First, the taxi fare is a base of $2 plus $0.20 per mile. So, the total fare in dollars would be 2 + 0.20x. T... | [
{
"from": "human",
"value": "A taxi company charges a base fare of 2 dollars plus an additional 0.20 USD for every mile driven. A tourist has a 50-euro bill and wants to pay for the taxi ride using this bill. The current exchange rate is 1 USD = 0.88 EUR. If the taxi driver has driven x miles, what is the m... |
Let $x, y, z$ be positive real numbers such that $x + y + z = xyz = 12$ and $xy + xz + yz = 48$. Find the value of $x^2 + y^2 + z^2$. | Let $x, y, z$ be positive real numbers such that $x + y + z = xyz = 12$ and $xy + xz + yz = 48$. Find the value of $x^2 + y^2 + z^2$. | Okay, so I need to find the value of \( x^2 + y^2 + z^2 \) given that \( x, y, z \) are positive real numbers satisfying \( x + y + z = 12 \), \( xyz = 12 \), and \( xy + xz + yz = 48 \). Hmm, let me think about how to approach this.
First, I remember that there is a formula relating \( x^2 + y^2 + z^2 \) to the sum o... | Given the positive real numbers \( x, y, z \) such that \( x + y + z = 12 \), \( xyz = 12 \), and \( xy + xz + yz = 48 \), we need to find the value of \( x^2 + y^2 + z^2 \).
We use the identity relating the sum of squares of variables to their sum and the sum of their products:
\[
x^2 + y^2 + z^2 = (x + y + z)^2 - 2... | 36,637 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of \( x^2 + y^2 + z^2 \) given that \( x, y, z \) are positive real numbers satisfying \( x + y + z = 12 \), \( xyz = 12 \), and \( xy + xz + yz = 48 \). Hmm, let me think about how to approach this.
First, I remember that there is a formula relating \( x^2 + y^2 + z^2 \) to t... | [
{
"from": "human",
"value": "Let $x, y, z$ be positive real numbers such that $x + y + z = xyz = 12$ and $xy + xz + yz = 48$. Find the value of $x^2 + y^2 + z^2$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of \\( x^2 + y^2 + z^2 \\) given that \\( x, y, z \\) are p... |
Determine the number of solutions to
\[2\sin^3 x - 5 \sin^2 x + 2 \sin x = 0\]in the range $0 \le x \le 2 \pi.$ | Determine the number of solutions to
\[2\sin^3 x - 5 \sin^2 x + 2 \sin x = 0\]in the range $0 \le x \le 2 \pi.$ | Alright, so I have this equation to solve: \(2\sin^3 x - 5 \sin^2 x + 2 \sin x = 0\), and I need to find how many solutions there are between \(0\) and \(2\pi\). Hmm, let's see. I remember that when solving trigonometric equations, factoring is often a good first step. Maybe I can factor out some common terms here.
Lo... | To determine the number of solutions to the equation \(2\sin^3 x - 5 \sin^2 x + 2 \sin x = 0\) in the range \(0 \le x \le 2\pi\), we start by factoring the equation:
\[
2\sin^3 x - 5 \sin^2 x + 2 \sin x = \sin x (2\sin^2 x - 5\sin x + 2) = 0
\]
This gives us two separate equations to solve: \(\sin x = 0\) and \(2\sin... | 17,673 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I have this equation to solve: \(2\sin^3 x - 5 \sin^2 x + 2 \sin x = 0\), and I need to find how many solutions there are between \(0\) and \(2\pi\). Hmm, let's see. I remember that when solving trigonometric equations, factoring is often a good first step. Maybe I can factor out some common terms h... | [
{
"from": "human",
"value": "Determine the number of solutions to\n\\[2\\sin^3 x - 5 \\sin^2 x + 2 \\sin x = 0\\]in the range $0 \\le x \\le 2 \\pi.$"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this equation to solve: \\(2\\sin^3 x - 5 \\sin^2 x + 2 \\sin x = 0\\), and I need to f... |
A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available? | A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available? | Okay, so I need to figure out how many combinations of one flavor and two different toppings are available at this yogurt shop. Let's see... They have four flavors and six different toppings. Hmm, okay, so the problem is asking for the number of ways to choose one flavor and two different toppings.
First, let's break... | To determine the number of combinations of one flavor and two different toppings available at the yogurt shop, we can break the problem into two parts: choosing the flavor and choosing the toppings.
1. **Choosing one flavor**: There are 4 different flavors, so there are 4 ways to choose one flavor.
2. **Choosing two d... | 29,917 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many combinations of one flavor and two different toppings are available at this yogurt shop. Let's see... They have four flavors and six different toppings. Hmm, okay, so the problem is asking for the number of ways to choose one flavor and two different toppings.
First, let... | [
{
"from": "human",
"value": "A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many combinations of one flavor and two ... |
The line $y=-\frac{3}{4}x+9$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is three times the area of $\triangle TOP$, then what is the value of $r+s$? [asy]
// draw axes
draw((-3, 0)--(15, 0), Arrow); draw((0, -3)--(0, 15), Arrow);
label(... | The line $y=-\frac{3}{4}x+9$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is three times the area of $\triangle TOP$, then what is the value of $r+s$? [asy]
// draw axes
draw((-3, 0)--(15, 0), Arrow); draw((0, -3)--(0, 15), Arrow);
label(... | Okay, let's see. The problem is about a line crossing the x-axis and y-axis at points P and Q, respectively. Then there's a point T(r, s) on the line segment PQ. The area of triangle POQ is three times the area of triangle TOP. We need to find r + s.
First, let's recall that the line is given by the equation y = -3/4 ... | The line \( y = -\frac{3}{4}x + 9 \) crosses the \( x \)-axis at point \( P \) and the \( y \)-axis at point \( Q \). To find the coordinates of \( P \) and \( Q \):
- For \( P \) (x-intercept), set \( y = 0 \):
\[
0 = -\frac{3}{4}x + 9 \implies x = 12
\]
Thus, \( P \) is \( (12, 0) \).
- For \( Q \) (y-inter... | 292 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is about a line crossing the x-axis and y-axis at points P and Q, respectively. Then there's a point T(r, s) on the line segment PQ. The area of triangle POQ is three times the area of triangle TOP. We need to find r + s.
First, let's recall that the line is given by the equation y... | [
{
"from": "human",
"value": "The line $y=-\\frac{3}{4}x+9$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on line segment $PQ$. If the area of $\\triangle POQ$ is three times the area of $\\triangle TOP$, then what is the value of $r+s$? [asy]\n\n// draw axes\ndraw((-3, 0)--(15, 0)... |
Determine the greatest integer less than $e + \sqrt{2}$. | Determine the greatest integer less than $e + \sqrt{2}$. | Okay, so I need to find the greatest integer less than e + √2. Hmm, let's start by recalling the approximate values of e and √2.
First, e is approximately 2.71828... Yeah, that's the base of the natural logarithm. And √2 is about 1.41421... That's the diagonal of a square with side length 1. So if I add those two num... | To determine the greatest integer less than \( e + \sqrt{2} \), we start by recalling the approximate values of \( e \) and \( \sqrt{2} \). The value of \( e \) is approximately \( 2.71828 \) and the value of \( \sqrt{2} \) is approximately \( 1.41421 \).
Adding these values together:
\[
2.71828 + 1.41421 = 4.13249
\]... | 33,873 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the greatest integer less than e + √2. Hmm, let's start by recalling the approximate values of e and √2.
First, e is approximately 2.71828... Yeah, that's the base of the natural logarithm. And √2 is about 1.41421... That's the diagonal of a square with side length 1. So if I add those... | [
{
"from": "human",
"value": "Determine the greatest integer less than $e + \\sqrt{2}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the greatest integer less than e + √2. Hmm, let's start by recalling the approximate values of e and √2. \n\nFirst, e is approximately 2.71828...... |
What is the units digit of the sum of the first four positive composite numbers? | What is the units digit of the sum of the first four positive composite numbers? | Alright, let's see. The question is asking for the units digit of the sum of the first four positive composite numbers. Hmm, okay. First, I need to remember what composite numbers are. A composite number is a positive integer that has at least one positive divisor other than 1 and itself. So, numbers like 4, 6, 8, 9, e... | To determine the units digit of the sum of the first four positive composite numbers, we first identify these numbers. Composite numbers are positive integers greater than 1 that have more than two factors. The first four composite numbers are 4, 6, 8, and 9.
Next, we calculate their sum:
\[
4 + 6 + 8 + 9
\]
Adding t... | 13,058 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's see. The question is asking for the units digit of the sum of the first four positive composite numbers. Hmm, okay. First, I need to remember what composite numbers are. A composite number is a positive integer that has at least one positive divisor other than 1 and itself. So, numbers like 4, 6,... | [
{
"from": "human",
"value": "What is the units digit of the sum of the first four positive composite numbers?"
},
{
"from": "gpt",
"value": "<think>\nAlright, let's see. The question is asking for the units digit of the sum of the first four positive composite numbers. Hmm, okay. First, I need t... |
Find the 50th digit after the decimal point in the decimal representation of $\frac{17}{75}$. | Find the 50th digit after the decimal point in the decimal representation of $\frac{17}{75}$. | Okay, so I need to find the 50th digit after the decimal point in the decimal representation of 17/75. Hmm, let's see. First, I remember that to convert a fraction to its decimal form, you can divide the numerator by the denominator. So maybe I should try dividing 17 by 75. Let me try that.
But wait, 17 divided by 75.... | To find the 50th digit after the decimal point in the decimal representation of \(\frac{17}{75}\), we perform the long division of 17 by 75:
1. \(75\) goes into \(170\) twice (2), leaving a remainder of \(20\).
2. Bring down a \(0\) to make \(200\). \(75\) goes into \(200\) twice (2), leaving a remainder of \(50\).
3.... | 1,471 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the 50th digit after the decimal point in the decimal representation of 17/75. Hmm, let's see. First, I remember that to convert a fraction to its decimal form, you can divide the numerator by the denominator. So maybe I should try dividing 17 by 75. Let me try that.
But wait, 17 divide... | [
{
"from": "human",
"value": "Find the 50th digit after the decimal point in the decimal representation of $\\frac{17}{75}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the 50th digit after the decimal point in the decimal representation of 17/75. Hmm, let's see. First, I reme... |
In a large city, there are five major highways that intersect each other at different points. No two highways are parallel to each other. The city has 15 police stations, and each police station is located at an intersection of two or more highways. What is the minimum number of police stations that can cover all inter... | In a large city, there are five major highways that intersect each other at different points. No two highways are parallel to each other. The city has 15 police stations, and each police station is located at an intersection of two or more highways. What is the minimum number of police stations that can cover all inter... | Okay, so I need to figure out the minimum number of police stations required to cover all intersections in a city with five major highways that aren't parallel, meaning they all intersect each other at different points. Each police station is at an intersection and can cover all directly connected intersections via the... | The problem involves determining the minimum number of police stations required to cover all intersections in a city with five major highways, where each police station can cover only the intersections directly connected by highways. The city has 15 police stations located at intersections of two or more highways.
Key... | 22,711 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the minimum number of police stations required to cover all intersections in a city with five major highways that aren't parallel, meaning they all intersect each other at different points. Each police station is at an intersection and can cover all directly connected intersections... | [
{
"from": "human",
"value": "In a large city, there are five major highways that intersect each other at different points. No two highways are parallel to each other. The city has 15 police stations, and each police station is located at an intersection of two or more highways. What is the minimum number of... |
Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$ | Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$ | Okay, let's try to solve this problem. We have two functions, f and g, such that f(g(x)) = x² and g(f(x)) = x³ for all x ≥ 1. Also, we know that g(16) = 16, and we need to find [g(4)]³. Hmm, interesting.
First, let me note down what is given:
1. f(g(x)) = x² for all x ≥ 1.
2. g(f(x)) = x³ for all x ≥ 1.
3. g(16) = 16... | Given the functions \( f(x) \) and \( g(x) \) which satisfy \( f(g(x)) = x^2 \) and \( g(f(x)) = x^3 \) for all \( x \ge 1 \), and knowing that \( g(16) = 16 \), we need to compute \([g(4)]^3\).
First, we derive the functional equations for \( g \) and \( f \):
1. From \( f(g(x)) = x^2 \), applying \( g \) to both si... | 16,097 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to solve this problem. We have two functions, f and g, such that f(g(x)) = x² and g(f(x)) = x³ for all x ≥ 1. Also, we know that g(16) = 16, and we need to find [g(4)]³. Hmm, interesting.
First, let me note down what is given:
1. f(g(x)) = x² for all x ≥ 1.
2. g(f(x)) = x³ for all x ≥ 1.
3. g(... | [
{
"from": "human",
"value": "Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \\ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to solve this problem. We have two functions, f and g... |
Let the roots of the quadratic equation \(x^2 - mx + 5 = 0\) be \(x_1\) and \(x_2\). Given that \(m\) is such that \(x_1^3 + x_2^3 = 65\), find the value of \(m\). | Let the roots of the quadratic equation \(x^2 - mx + 5 = 0\) be \(x_1\) and \(x_2\). Given that \(m\) is such that \(x_1^3 + x_2^3 = 65\), find the value of \(m\). | Okay, so I need to find the value of \( m \) in the quadratic equation \( x^2 - mx + 5 = 0 \) given that the sum of the cubes of its roots \( x_1^3 + x_2^3 = 65 \). Hmm, let me think about how to approach this.
First, I remember that for a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots \( x_1 + x_2 =... | Given the quadratic equation \(x^2 - mx + 5 = 0\) with roots \(x_1\) and \(x_2\), we need to find the value of \(m\) such that \(x_1^3 + x_2^3 = 65\).
First, we use Vieta's formulas to note that the sum of the roots \(x_1 + x_2 = m\) and the product of the roots \(x_1x_2 = 5\).
The formula for the sum of cubes is:
\[... | 287 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of \( m \) in the quadratic equation \( x^2 - mx + 5 = 0 \) given that the sum of the cubes of its roots \( x_1^3 + x_2^3 = 65 \). Hmm, let me think about how to approach this.
First, I remember that for a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots \( x_1... | [
{
"from": "human",
"value": "Let the roots of the quadratic equation \\(x^2 - mx + 5 = 0\\) be \\(x_1\\) and \\(x_2\\). Given that \\(m\\) is such that \\(x_1^3 + x_2^3 = 65\\), find the value of \\(m\\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of \\( m \\) in t... |
In a certain city, there are two public transportation systems: buses and trains. All residents of the city use at least one of these systems to commute to work. If 45% of the residents use buses and 15% use both buses and trains, what percentage of residents use trains?
Can you solve this problem? | In a certain city, there are two public transportation systems: buses and trains. All residents of the city use at least one of these systems to commute to work. If 45% of the residents use buses and 15% use both buses and trains, what percentage of residents use trains?
Can you solve this problem? | Alright, let's see. So the problem is about figuring out the percentage of residents who use trains, given that everyone uses at least one of the two systems: buses or trains. We know that 45% use buses and 15% use both buses and trains. Hmm, okay.
First, I remember something about sets and maybe using a Venn diagram... | Given that all residents of the city use at least one of the two public transportation systems (buses or trains), we need to determine the percentage of residents who use trains.
We know the following:
- 45% of the residents use buses (denoted as \( B \)).
- 15% of the residents use both buses and trains (denoted as ... | 26,833 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's see. So the problem is about figuring out the percentage of residents who use trains, given that everyone uses at least one of the two systems: buses or trains. We know that 45% use buses and 15% use both buses and trains. Hmm, okay.
First, I remember something about sets and maybe using a Venn... | [
{
"from": "human",
"value": "In a certain city, there are two public transportation systems: buses and trains. All residents of the city use at least one of these systems to commute to work. If 45% of the residents use buses and 15% use both buses and trains, what percentage of residents use trains?\n\nCa... |
A square sheet of paper has a side length of 24 cm. Three congruent rectangles are cut from the sheet along the edges, and the remaining sheet is divided into four smaller congruent rectangles. What is the length of the side of the smaller rectangles if it is known that each of the three cut rectangles has an area 2.5 ... | A square sheet of paper has a side length of 24 cm. Three congruent rectangles are cut from the sheet along the edges, and the remaining sheet is divided into four smaller congruent rectangles. What is the length of the side of the smaller rectangles if it is known that each of the three cut rectangles has an area 2.5 ... | Okay, so I have this problem here about a square sheet of paper that's 24 cm on each side. They say that three congruent rectangles are cut from the sheet along the edges, and the remaining sheet is divided into four smaller congruent rectangles. Each of the three cut rectangles has an area that's 2.5 times larger than... | Given a square sheet of paper with a side length of 24 cm, three congruent rectangles are cut from the edges, and the remaining sheet is divided into four smaller congruent rectangles. Each of the three cut rectangles has an area 2.5 times larger than each of the smaller rectangles. We need to find the side lengths of ... | 8,642 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here about a square sheet of paper that's 24 cm on each side. They say that three congruent rectangles are cut from the sheet along the edges, and the remaining sheet is divided into four smaller congruent rectangles. Each of the three cut rectangles has an area that's 2.5 times lar... | [
{
"from": "human",
"value": "A square sheet of paper has a side length of 24 cm. Three congruent rectangles are cut from the sheet along the edges, and the remaining sheet is divided into four smaller congruent rectangles. What is the length of the side of the smaller rectangles if it is known that each of ... |
What is $\dbinom{n}{2}$ for any positive integer $n$? | What is $\dbinom{n}{2}$ for any positive integer $n$? | Okay, so I need to figure out what the binomial coefficient $\dbinom{n}{2}$ is for any positive integer $n$. Hmm, binomial coefficients... right, they represent the number of ways to choose a subset of items from a larger set without considering the order. So $\dbinom{n}{2}$ specifically should give the number of ways ... | To determine the value of the binomial coefficient $\dbinom{n}{2}$ for any positive integer $n$, we start with the combination formula:
\[
\dbinom{n}{k} = \frac{n!}{k!(n - k)!}
\]
Substituting $k = 2$ into the formula, we get:
\[
\dbinom{n}{2} = \frac{n!}{2!(n - 2)!}
\]
We can simplify this expression by recognizin... | 9,760 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out what the binomial coefficient $\dbinom{n}{2}$ is for any positive integer $n$. Hmm, binomial coefficients... right, they represent the number of ways to choose a subset of items from a larger set without considering the order. So $\dbinom{n}{2}$ specifically should give the number ... | [
{
"from": "human",
"value": "What is $\\dbinom{n}{2}$ for any positive integer $n$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out what the binomial coefficient $\\dbinom{n}{2}$ is for any positive integer $n$. Hmm, binomial coefficients... right, they represent the number... |
Let $A,B,C$ be angles of a triangle, where angle $C$ is acute, and \begin{align*}
\cos^2 A + \cos^2 C + 2 \sin A \sin C \cos B &= \frac{11}{7} \text{ and} \\
\cos^2 B + \cos^2 A + 2 \sin B \sin A \cos C &= \frac{13}{10}.
\end{align*}There are positive integers $p$, $q$, $r$, and $s$ for which \[ \cos^2 C + \cos^2 B + 2... | Let $A,B,C$ be angles of a triangle, where angle $C$ is acute, and \begin{align*}
\cos^2 A + \cos^2 C + 2 \sin A \sin C \cos B &= \frac{11}{7} \text{ and} \\
\cos^2 B + \cos^2 A + 2 \sin B \sin A \cos C &= \frac{13}{10}.
\end{align*}There are positive integers $p$, $q$, $r$, and $s$ for which \[ \cos^2 C + \cos^2 B + 2... | Okay, so we have this triangle with angles A, B, C, and angle C is acute. There are two equations given involving cos² terms and some sine and cosine products. The third equation we need to find, which is similar in structure, and then express it in a specific form. The answer will be the sum of the numerators and deno... | Given the angles \(A\), \(B\), and \(C\) of a triangle where \(C\) is acute, we have the equations:
\[
\cos^2 A + \cos^2 C + 2 \sin A \sin C \cos B = \frac{11}{7}
\]
\[
\cos^2 B + \cos^2 A + 2 \sin B \sin A \cos C = \frac{13}{10}
\]
We need to find \(\cos^2 C + \cos^2 B + 2 \sin C \sin B \cos A\) and express it in the... | 3,438 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so we have this triangle with angles A, B, C, and angle C is acute. There are two equations given involving cos² terms and some sine and cosine products. The third equation we need to find, which is similar in structure, and then express it in a specific form. The answer will be the sum of the numerators ... | [
{
"from": "human",
"value": "Let $A,B,C$ be angles of a triangle, where angle $C$ is acute, and \\begin{align*}\n\\cos^2 A + \\cos^2 C + 2 \\sin A \\sin C \\cos B &= \\frac{11}{7} \\text{ and} \\\\\n\\cos^2 B + \\cos^2 A + 2 \\sin B \\sin A \\cos C &= \\frac{13}{10}.\n\\end{align*}There are positive integer... |
What is the product of all the even integers between $-10$ and $20$? | What is the product of all the even integers between $-10$ and $20$? | Okay, let's see. The problem is asking for the product of all the even integers between -10 and 20. Hmm, so first, I need to figure out which numbers are included in that range. Let me start by understanding the boundaries. The numbers should be between -10 and 20, but not including -10 and 20 themselves, right? Wait, ... | The problem asks for the product of all the even integers between \(-10\) and \(20\).
First, we need to determine the range of even integers. The term "between" can be ambiguous, but regardless of whether it is inclusive or exclusive of the endpoints, the product will be zero because zero is an even integer within th... | 31,118 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is asking for the product of all the even integers between -10 and 20. Hmm, so first, I need to figure out which numbers are included in that range. Let me start by understanding the boundaries. The numbers should be between -10 and 20, but not including -10 and 20 themselves, right... | [
{
"from": "human",
"value": "What is the product of all the even integers between $-10$ and $20$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem is asking for the product of all the even integers between -10 and 20. Hmm, so first, I need to figure out which numbers are incl... |
The number $37_{10}$ is written in base 2 as $100101_{2}$. What is the next power of 5 greater than 37 that, when written in base 2, consists only of 1s and 0s, and has exactly four 1s? | The number $37_{10}$ is written in base 2 as $100101_{2}$. What is the next power of 5 greater than 37 that, when written in base 2, consists only of 1s and 0s, and has exactly four 1s? | Okay, so I need to find the next power of 5 greater than 37 that, when written in base 2, is composed only of 1s and 0s and has exactly four 1s. Let me start by understanding the problem step by step.
First, the number 37 in base 10 is given as 100101 in base 2. Let me verify that. To convert 37 to binary, I divide by... | To find the next power of 5 greater than 37 that, when written in base 2, consists only of 1s and 0s and has exactly four 1s, we start by converting powers of 5 to binary and checking the number of 1s.
1. **5^3 = 125** in binary is \(1111101_2\), which has six 1s.
2. **5^4 = 625** in binary is \(1001110001_2\), which ... | 23,712 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the next power of 5 greater than 37 that, when written in base 2, is composed only of 1s and 0s and has exactly four 1s. Let me start by understanding the problem step by step.
First, the number 37 in base 10 is given as 100101 in base 2. Let me verify that. To convert 37 to binary, I d... | [
{
"from": "human",
"value": "The number $37_{10}$ is written in base 2 as $100101_{2}$. What is the next power of 5 greater than 37 that, when written in base 2, consists only of 1s and 0s, and has exactly four 1s?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the next power of... |
How many positive divisors of 180 are not divisible by 3? | How many positive divisors of 180 are not divisible by 3? | Okay, so I need to figure out how many positive divisors of 180 are not divisible by 3. Hmm, let me start by recalling some basic number theory concepts. Divisors of a number can be found by looking at its prime factorization. So first, let me factorize 180 into primes.
Starting with 180. I know that 180 divided by 2 ... | To determine how many positive divisors of 180 are not divisible by 3, we start with its prime factorization:
\[
180 = 2^2 \times 3^2 \times 5^1
\]
The total number of divisors of 180 is calculated by adding 1 to each exponent in the prime factorization and multiplying the results:
\[
(2+1)(2+1)(1+1) = 3 \times 3 \t... | 22,244 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many positive divisors of 180 are not divisible by 3. Hmm, let me start by recalling some basic number theory concepts. Divisors of a number can be found by looking at its prime factorization. So first, let me factorize 180 into primes.
Starting with 180. I know that 180 divid... | [
{
"from": "human",
"value": "How many positive divisors of 180 are not divisible by 3?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many positive divisors of 180 are not divisible by 3. Hmm, let me start by recalling some basic number theory concepts. Divisors of a n... |
Compute the value of $\sin 0^\circ$ using the identity $\sin^2 0^\circ + \cos^2 0^\circ = 1$. | Compute the value of $\sin 0^\circ$ using the identity $\sin^2 0^\circ + \cos^2 0^\circ = 1$. | Okay, so I need to figure out the value of sin 0 degrees using the Pythagorean identity, which is sin²θ + cos²θ = 1. Hmm, let me start by recalling what I know about trigonometric functions at 0 degrees.
First off, 0 degrees is one of those special angles where the values of sine and cosine are pretty straightforward... | To compute the value of \(\sin 0^\circ\) using the identity \(\sin^2 0^\circ + \cos^2 0^\circ = 1\):
1. Start with the given identity:
\[
\sin^2 0^\circ + \cos^2 0^\circ = 1
\]
2. Recall that \(\cos 0^\circ = 1\). This is a known value from the unit circle or right triangle definitions.
3. Substitute \(\cos... | 2,338 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the value of sin 0 degrees using the Pythagorean identity, which is sin²θ + cos²θ = 1. Hmm, let me start by recalling what I know about trigonometric functions at 0 degrees.
First off, 0 degrees is one of those special angles where the values of sine and cosine are pretty straigh... | [
{
"from": "human",
"value": "Compute the value of $\\sin 0^\\circ$ using the identity $\\sin^2 0^\\circ + \\cos^2 0^\\circ = 1$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the value of sin 0 degrees using the Pythagorean identity, which is sin²θ + cos²θ = 1. Hmm, let m... |
What is the value of $x$ for which $(8-x)^2=x^2$? | What is the value of $x$ for which $(8-x)^2=x^2$? | Okay, so I need to find the value of x that makes (8 - x)^2 equal to x^2. Hmm, let's see. I remember that squaring a binomial like (a - b)^2 gives a^2 - 2ab + b^2. So maybe I can expand the left side and then solve for x?
Alright, let me start by expanding (8 - x)^2. Using the formula (a - b)^2 = a^2 - 2ab + b^2, wher... | To find the value of \( x \) for which \( (8 - x)^2 = x^2 \), we start by expanding the left side:
\[
(8 - x)^2 = 8^2 - 2 \cdot 8 \cdot x + x^2 = 64 - 16x + x^2
\]
Setting this equal to \( x^2 \):
\[
64 - 16x + x^2 = x^2
\]
Subtracting \( x^2 \) from both sides:
\[
64 - 16x = 0
\]
Solving for \( x \):
\[
64 = 16... | 11,655 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of x that makes (8 - x)^2 equal to x^2. Hmm, let's see. I remember that squaring a binomial like (a - b)^2 gives a^2 - 2ab + b^2. So maybe I can expand the left side and then solve for x?
Alright, let me start by expanding (8 - x)^2. Using the formula (a - b)^2 = a^2 - 2ab + b... | [
{
"from": "human",
"value": "What is the value of $x$ for which $(8-x)^2=x^2$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of x that makes (8 - x)^2 equal to x^2. Hmm, let's see. I remember that squaring a binomial like (a - b)^2 gives a^2 - 2ab + b^2. So maybe I ca... |
Which of the following, when rounded to the nearest tenth, does not round to 42.7?
A) 42.749
B) 42.75
C) 42.699
D) 42.7001
E) 42.79 | Which of the following, when rounded to the nearest tenth, does not round to 42.7?
A) 42.749
B) 42.75
C) 42.699
D) 42.7001
E) 42.79 | Okay, let's see. The question is asking which of the options, when rounded to the nearest tenth, does not round to 42.7. The options are A through E. I need to check each one and see if when rounded to the nearest tenth, they become 42.7 or not. The one that doesn't round to 42.7 is the answer.
First, let me remember ... | To determine which number, when rounded to the nearest tenth, does not round to 42.7, we need to check each option:
**Option A: 42.749**
- Tenths place: 7, Hundredths place: 4 (less than 5), so rounds to 42.7.
**Option B: 42.75**
- Tenths place: 7, Hundredths place: 5 (equal to 5), so rounds up to 42.8.
**Option C: ... | 29,411 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The question is asking which of the options, when rounded to the nearest tenth, does not round to 42.7. The options are A through E. I need to check each one and see if when rounded to the nearest tenth, they become 42.7 or not. The one that doesn't round to 42.7 is the answer.
First, let me r... | [
{
"from": "human",
"value": "Which of the following, when rounded to the nearest tenth, does not round to 42.7?\n\nA) 42.749\n\nB) 42.75\n\nC) 42.699\n\nD) 42.7001\n\nE) 42.79"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The question is asking which of the options, when rounded to th... |
If $a\circ b = \frac{a}{b} + \frac{b}{a} - 2$, what is the value of $4\circ\frac{1}{2}$? | If $a\circ b = \frac{a}{b} + \frac{b}{a} - 2$, what is the value of $4\circ\frac{1}{2}$? | Okay, let me try to figure out this problem. So, we have this operation defined as a circle between two numbers, right? And the operation is given by a∘b = (a/b) + (b/a) - 2. The question is asking for the value of 4∘(1/2). Alright, let's break this down step by step.
First, I need to remember how these operations wor... | To find the value of \(4 \circ \frac{1}{2}\) given the operation \(a \circ b = \frac{a}{b} + \frac{b}{a} - 2\):
1. Substitute \(a = 4\) and \(b = \frac{1}{2}\) into the operation:
\[
4 \circ \frac{1}{2} = \frac{4}{\frac{1}{2}} + \frac{\frac{1}{2}}{4} - 2
\]
2. Calculate each term separately:
- \(\frac{4}{... | 7,385 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, we have this operation defined as a circle between two numbers, right? And the operation is given by a∘b = (a/b) + (b/a) - 2. The question is asking for the value of 4∘(1/2). Alright, let's break this down step by step.
First, I need to remember how these operat... | [
{
"from": "human",
"value": "If $a\\circ b = \\frac{a}{b} + \\frac{b}{a} - 2$, what is the value of $4\\circ\\frac{1}{2}$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. So, we have this operation defined as a circle between two numbers, right? And the opera... |
A fair coin is flipped 20 times. What is the probability that the sequence of heads and tails obtained contains exactly three "runs" of heads (a "run" being a sequence of one or more consecutive heads)? Express your answer as a decimal rounded to the nearest thousandth. | A fair coin is flipped 20 times. What is the probability that the sequence of heads and tails obtained contains exactly three "runs" of heads (a "run" being a sequence of one or more consecutive heads)? Express your answer as a decimal rounded to the nearest thousandth. | Okay, so I need to find the probability that when flipping a fair coin 20 times, there are exactly three runs of heads. A run is defined as a sequence of one or more consecutive heads. Hmm. Let me try to understand this.
First, let me recall what a run is. If I have a sequence like H T T H H H T H, then there are thre... | To find the probability that a sequence of 20 coin flips contains exactly three runs of heads, we need to consider the structure of such sequences. A run of heads is a sequence of one or more consecutive heads, separated by at least one tail.
The number of such sequences can be calculated using combinatorial methods.... | 13,316 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that when flipping a fair coin 20 times, there are exactly three runs of heads. A run is defined as a sequence of one or more consecutive heads. Hmm. Let me try to understand this.
First, let me recall what a run is. If I have a sequence like H T T H H H T H, then there ... | [
{
"from": "human",
"value": "A fair coin is flipped 20 times. What is the probability that the sequence of heads and tails obtained contains exactly three \"runs\" of heads (a \"run\" being a sequence of one or more consecutive heads)? Express your answer as a decimal rounded to the nearest thousandth."
}... |
In a modified version of the SHORT BINGO game, a $4\times4$ card is used instead, and the middle square is marked as FREE SPACE. The remaining 15 squares are filled with 15 distinct numbers from the set $1-20$, with each row containing numbers from a different set of 5 consecutive integers (e.g., the first row contains... | In a modified version of the SHORT BINGO game, a $4\times4$ card is used instead, and the middle square is marked as FREE SPACE. The remaining 15 squares are filled with 15 distinct numbers from the set $1-20$, with each row containing numbers from a different set of 5 consecutive integers (e.g., the first row contains... | Okay, so I need to figure out how many distinct possibilities there are for the values in the first row of this modified SHORT BINGO card. The card is 4x4, with the middle square being a FREE SPACE. The remaining 15 squares are filled with 15 distinct numbers from 1-20. Each row must contain numbers from a different se... | To determine the number of distinct possibilities for the values in the first row of a modified SHORT BINGO card, we need to consider the following:
1. The card is a 4x4 grid with the middle square marked as FREE SPACE, leaving 15 squares to be filled with distinct numbers from the set 1-20.
2. Each row must contain n... | 12,777 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many distinct possibilities there are for the values in the first row of this modified SHORT BINGO card. The card is 4x4, with the middle square being a FREE SPACE. The remaining 15 squares are filled with 15 distinct numbers from 1-20. Each row must contain numbers from a diff... | [
{
"from": "human",
"value": "In a modified version of the SHORT BINGO game, a $4\\times4$ card is used instead, and the middle square is marked as FREE SPACE. The remaining 15 squares are filled with 15 distinct numbers from the set $1-20$, with each row containing numbers from a different set of 5 consecut... |
Find the product of the reciprocals of the roots of $x^2-13x+4=0$. | Find the product of the reciprocals of the roots of $x^2-13x+4=0$. | Okay, so I need to find the product of the reciprocals of the roots of the quadratic equation x² - 13x + 4 = 0. Hmm, let's see. Let me recall some algebra concepts here.
First, when dealing with quadratic equations, I remember there's something called Vieta's formulas that relates the coefficients of the equation to ... | To find the product of the reciprocals of the roots of the quadratic equation \(x^2 - 13x + 4 = 0\), we can use Vieta's formulas. For a quadratic equation \(ax^2 + bx + c = 0\), the product of the roots \(r_1\) and \(r_2\) is given by \(\frac{c}{a}\).
For the given equation \(x^2 - 13x + 4 = 0\), the coefficients are... | 37,137 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the product of the reciprocals of the roots of the quadratic equation x² - 13x + 4 = 0. Hmm, let's see. Let me recall some algebra concepts here.
First, when dealing with quadratic equations, I remember there's something called Vieta's formulas that relates the coefficients of the equa... | [
{
"from": "human",
"value": "Find the product of the reciprocals of the roots of $x^2-13x+4=0$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the product of the reciprocals of the roots of the quadratic equation x² - 13x + 4 = 0. Hmm, let's see. Let me recall some algebra conce... |
Consider the sequence $T = \{t_1, t_2, t_3, \ldots\}$ defined by $t_1 = 11$ and $t_n = 11^{t_{n-1}}$ for each integer $n > 1$. What is the remainder when $t_{75}$ is divided by $9$? | Consider the sequence $T = \{t_1, t_2, t_3, \ldots\}$ defined by $t_1 = 11$ and $t_n = 11^{t_{n-1}}$ for each integer $n > 1$. What is the remainder when $t_{75}$ is divided by $9$? | Okay, so I need to find the remainder when t₇₅ is divided by 9. The sequence is defined with t₁ = 11 and each subsequent term is 11 raised to the power of the previous term. That means t₂ = 11¹¹, t₃ = 11^(11¹¹), and so on. Since t₇₅ is going to be an enormous number, directly computing it isn't feasible. Therefore, I n... | To find the remainder when \( t_{75} \) is divided by 9, we start with the sequence \( T = \{t_1, t_2, t_3, \ldots\} \) defined by \( t_1 = 11 \) and \( t_n = 11^{t_{n-1}} \) for \( n > 1 \).
1. **Modular Arithmetic with Euler's Theorem**:
- Since 11 and 9 are coprime, Euler's theorem tells us \( 11^{\phi(9)} \equi... | 1,655 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the remainder when t₇₅ is divided by 9. The sequence is defined with t₁ = 11 and each subsequent term is 11 raised to the power of the previous term. That means t₂ = 11¹¹, t₃ = 11^(11¹¹), and so on. Since t₇₅ is going to be an enormous number, directly computing it isn't feasible. Theref... | [
{
"from": "human",
"value": "Consider the sequence $T = \\{t_1, t_2, t_3, \\ldots\\}$ defined by $t_1 = 11$ and $t_n = 11^{t_{n-1}}$ for each integer $n > 1$. What is the remainder when $t_{75}$ is divided by $9$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the remainder when... |
Five people are sitting at a round table. What is the probability that no two people wearing the same color shirt are sitting next to each other, given that three people are wearing white shirts and two people are wearing black shirts? | Five people are sitting at a round table. What is the probability that no two people wearing the same color shirt are sitting next to each other, given that three people are wearing white shirts and two people are wearing black shirts? | Okay, so there's this problem about five people sitting around a round table. Three are wearing white shirts, and two are wearing black shirts. We need to find the probability that no two people wearing the same color shirt are sitting next to each other. Hmm, okay. Let me try to figure this out step by step.
First, p... | To determine the probability that no two people wearing the same color shirt are sitting next to each other at a round table with three white shirts (W) and two black shirts (B), we need to consider the following:
1. **Circular Permutations with Identical Items**: The total number of distinct circular arrangements of ... | 26,861 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this problem about five people sitting around a round table. Three are wearing white shirts, and two are wearing black shirts. We need to find the probability that no two people wearing the same color shirt are sitting next to each other. Hmm, okay. Let me try to figure this out step by step.
... | [
{
"from": "human",
"value": "Five people are sitting at a round table. What is the probability that no two people wearing the same color shirt are sitting next to each other, given that three people are wearing white shirts and two people are wearing black shirts?"
},
{
"from": "gpt",
"value": "... |
What is the value of $\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\left(\frac{4}{5}\right)\left(\frac{5}{6}\right)$? Express your answer as a common fraction. | What is the value of $\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\left(\frac{4}{5}\right)\left(\frac{5}{6}\right)$? Express your answer as a common fraction. | Okay, so I need to figure out the value of this product: (2/3)(3/4)(4/5)(5/6). Hmm, let's see. At first glance, it looks like a series of fractions multiplied together. Each numerator and denominator seems to be consecutive numbers. Let me write them out one by one to see if there's a pattern or something that cancels ... | To find the value of \(\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\left(\frac{4}{5}\right)\left(\frac{5}{6}\right)\), we start by observing the pattern in the fractions. Each numerator after the first fraction cancels with the denominator of the previous fraction.
Writing the product as a single fraction, we hav... | 873 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the value of this product: (2/3)(3/4)(4/5)(5/6). Hmm, let's see. At first glance, it looks like a series of fractions multiplied together. Each numerator and denominator seems to be consecutive numbers. Let me write them out one by one to see if there's a pattern or something that ... | [
{
"from": "human",
"value": "What is the value of $\\left(\\frac{2}{3}\\right)\\left(\\frac{3}{4}\\right)\\left(\\frac{4}{5}\\right)\\left(\\frac{5}{6}\\right)$? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the value of this prod... |
What is the sum of the last two digits of this portion of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!$? | What is the sum of the last two digits of this portion of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!$? | Okay, let's see. I need to find the sum of the last two digits of this Fibonacci Factorial Series: 1! + 1! + 2! + 3! + 5! + 8! + 13! + 21! + 34! + 55! + 89!. Hmm, so first, maybe I should write down each term, compute their factorials, find the last two digits of each factorial, add them all together, and then take the... | To find the sum of the last two digits of the Fibonacci Factorial Series \(1! + 1! + 2! + 3! + 5! + 8! + 13! + 21! + 34! + 55! + 89!\), we need to compute the last two digits of each factorial and then sum them up.
1. Calculate the last two digits of each factorial:
- \(1! = 1\) → last two digits: 01
- \(1! = 1\... | 687 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the sum of the last two digits of this Fibonacci Factorial Series: 1! + 1! + 2! + 3! + 5! + 8! + 13! + 21! + 34! + 55! + 89!. Hmm, so first, maybe I should write down each term, compute their factorials, find the last two digits of each factorial, add them all together, and then ... | [
{
"from": "human",
"value": "What is the sum of the last two digits of this portion of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the sum of the last two digits of this Fibonacci Factorial Serie... |
In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 5$, and $\angle ABO = \text{arc } CD = 60^\circ$. Find the length of $BC$.
[asy]
import graph;
unitsize(2 cm);
pair O, A, B, C, D;
O = (0,0);
A = dir(30);
C = dir(160);
B = (2*C + A)/3;
D = -A;
draw(Circle(O,1));
draw(C--A--D);
draw(B--O);
... | In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 5$, and $\angle ABO = \text{arc } CD = 60^\circ$. Find the length of $BC$.
[asy]
import graph;
unitsize(2 cm);
pair O, A, B, C, D;
O = (0,0);
A = dir(30);
C = dir(160);
B = (2*C + A)/3;
D = -A;
draw(Circle(O,1));
draw(C--A--D);
draw(B--O);
... | Okay, let me try to figure out how to solve this geometry problem. So, we have a circle with center O. AD is a diameter, ABC is a chord, BO = 5, angle ABO is 60 degrees, and the arc CD is also 60 degrees. We need to find the length of BC.
First, let me visualize the diagram. AD is a diameter, so points A and D are on ... | Given a circle with center \( O \), where \( AD \) is a diameter, \( ABC \) is a chord, \( BO = 5 \), and \( \angle ABO = \text{arc } CD = 60^\circ \). We need to find the length of \( BC \).
1. **Identify Key Information**:
- \( AD \) is a diameter, so \( OA = OD = 5 \) (radius of the circle).
- \( \angle ABO =... | 14,877 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out how to solve this geometry problem. So, we have a circle with center O. AD is a diameter, ABC is a chord, BO = 5, angle ABO is 60 degrees, and the arc CD is also 60 degrees. We need to find the length of BC.
First, let me visualize the diagram. AD is a diameter, so points A and D... | [
{
"from": "human",
"value": "In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 5$, and $\\angle ABO = \\text{arc } CD = 60^\\circ$. Find the length of $BC$.\n\n[asy]\nimport graph;\n\nunitsize(2 cm);\n\npair O, A, B, C, D;\n\nO = (0,0);\nA = dir(30);\nC = dir(160);\nB = (2*C + A)/3;\... |
Simplify 3x + 6x + 9x + 12x + 15x + 20. | Simplify 3x + 6x + 9x + 12x + 15x + 20. | Okay, so I need to simplify this expression: 3x + 6x + 9x + 12x + 15x + 20. Let me think about how to approach this. Hmm, simplifying expressions usually involves combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, all the terms with x are like terms, and the c... | To simplify the expression \(3x + 6x + 9x + 12x + 15x + 20\), we need to combine the like terms. The like terms are the ones with the variable \(x\) and the constant term.
First, we add the coefficients of the \(x\) terms:
\[
3 + 6 + 9 + 12 + 15
\]
Adding these step by step:
\[
3 + 6 = 9
\]
\[
9 + 9 = 18
\]
\[
18 + 1... | 19,569 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to simplify this expression: 3x + 6x + 9x + 12x + 15x + 20. Let me think about how to approach this. Hmm, simplifying expressions usually involves combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, all the terms with x are like terms, a... | [
{
"from": "human",
"value": "Simplify 3x + 6x + 9x + 12x + 15x + 20."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to simplify this expression: 3x + 6x + 9x + 12x + 15x + 20. Let me think about how to approach this. Hmm, simplifying expressions usually involves combining like terms. Li... |
Problem:
Given that $a$ is an odd multiple of $7767$, find the greatest common divisor of $6a^2+49a+108$ and $2a+9$.
Write another problem inspired by this one:
Let $a$ be an integer and $b$ a positive integer. Find the greatest common divisor of $4a^2+8a+9$ and $a^2+2a+1$ if it is known that $\frac{b}{a^2+2a+1}$ is a... | Problem:
Given that $a$ is an odd multiple of $7767$, find the greatest common divisor of $6a^2+49a+108$ and $2a+9$.
Write another problem inspired by this one:
Let $a$ be an integer and $b$ a positive integer. Find the greatest common divisor of $4a^2+8a+9$ and $a^2+2a+1$ if it is known that $\frac{b}{a^2+2a+1}$ is a... | Okay, so I have this problem here where I need to find the greatest common divisor (GCD) of two expressions: 6a² + 49a + 108 and 2a + 9, given that a is an odd multiple of 7767. Alright, let's break this down step by step.
First, I remember that the GCD of two numbers also divides any linear combination of them. So ma... | To find the greatest common divisor (GCD) of \(6a^2 + 49a + 108\) and \(2a + 9\) where \(a\) is an odd multiple of \(7767\):
1. **Apply the Euclidean Algorithm**:
- Divide \(6a^2 + 49a + 108\) by \(2a + 9\):
- The quotient is \(3a\) and the remainder is \(22a + 108\).
- Next, divide \(22a + 108\) by \(2a ... | 1,973 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here where I need to find the greatest common divisor (GCD) of two expressions: 6a² + 49a + 108 and 2a + 9, given that a is an odd multiple of 7767. Alright, let's break this down step by step.
First, I remember that the GCD of two numbers also divides any linear combination of the... | [
{
"from": "human",
"value": "Problem:\nGiven that $a$ is an odd multiple of $7767$, find the greatest common divisor of $6a^2+49a+108$ and $2a+9$.\n\nWrite another problem inspired by this one:\nLet $a$ be an integer and $b$ a positive integer. Find the greatest common divisor of $4a^2+8a+9$ and $a^2+2a+1$ ... |
The function $f(x)$ takes positive real numbers to real numbers, such that
\[xf(y) - yf(x) = f \left( \frac{x}{y} \right)\]for all positive real numbers $x$ and $y.$ Find all possible values of $f(100).$ Enter all possible values, separated by commas. | The function $f(x)$ takes positive real numbers to real numbers, such that
\[xf(y) - yf(x) = f \left( \frac{x}{y} \right)\]for all positive real numbers $x$ and $y.$ Find all possible values of $f(100).$ Enter all possible values, separated by commas. | Okay, so I need to solve this functional equation: for all positive real numbers x and y, the equation x*f(y) - y*f(x) = f(x/y) holds. And I need to find all possible values of f(100). Hmm, functional equations can be tricky, but let's see if I can figure this out step by step.
First, functional equations often requir... | Given the functional equation \( xf(y) - yf(x) = f \left( \frac{x}{y} \right) \) for all positive real numbers \( x \) and \( y \), we need to find all possible values of \( f(100) \).
First, we set \( x = y \). This simplifies the equation to:
\[
x f(x) - x f(x) = f(1) \implies 0 = f(1)
\]
Thus, \( f(1) = 0 \).
Next... | 4,994 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this functional equation: for all positive real numbers x and y, the equation x*f(y) - y*f(x) = f(x/y) holds. And I need to find all possible values of f(100). Hmm, functional equations can be tricky, but let's see if I can figure this out step by step.
First, functional equations ofte... | [
{
"from": "human",
"value": "The function $f(x)$ takes positive real numbers to real numbers, such that\n\\[xf(y) - yf(x) = f \\left( \\frac{x}{y} \\right)\\]for all positive real numbers $x$ and $y.$ Find all possible values of $f(100).$ Enter all possible values, separated by commas."
},
{
"from... |
Circle $A$ has its center at $(3, -4)$ and a radius of $8$ units. What is the sum of the $x$-coordinates of the two points on circle $A$ that are also on the $x$-axis? | Circle $A$ has its center at $(3, -4)$ and a radius of $8$ units. What is the sum of the $x$-coordinates of the two points on circle $A$ that are also on the $x$-axis? | Okay, let's see. I need to find the sum of the x-coordinates of the two points where circle A intersects the x-axis. The circle is centered at (3, -4) with a radius of 8 units. Hmm, how do I approach this?
First, I remember that the equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. S... | To find the sum of the x-coordinates of the two points where circle \( A \) intersects the x-axis, we start with the equation of the circle. The circle is centered at \((3, -4)\) with a radius of 8 units, so its equation is:
\[
(x - 3)^2 + (y + 4)^2 = 64
\]
To determine the points of intersection with the x-axis, we ... | 15,079 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the sum of the x-coordinates of the two points where circle A intersects the x-axis. The circle is centered at (3, -4) with a radius of 8 units. Hmm, how do I approach this?
First, I remember that the equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 ... | [
{
"from": "human",
"value": "Circle $A$ has its center at $(3, -4)$ and a radius of $8$ units. What is the sum of the $x$-coordinates of the two points on circle $A$ that are also on the $x$-axis?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the sum of the x-coordinate... |
Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$. | Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$. | Okay, let me try to figure out this problem. So, we have two functions f and g, and we're told that the inverse of f composed with g of x is equal to 5x + 3. And we need to find the inverse of g composed with f at -7. That is, we need to find g⁻¹(f(-7)). Hmm, let's break this down step by step.
First, let me recall wh... | Given that \( f^{-1}(g(x)) = 5x + 3 \), we need to find \( g^{-1}(f(-7)) \).
First, we start by applying \( f \) to both sides of the given equation:
\[
f(f^{-1}(g(x))) = f(5x + 3)
\]
This simplifies to:
\[
g(x) = f(5x + 3)
\]
Next, we need to find \( g^{-1}(f(-7)) \). Let \( y = g^{-1}(f(-7)) \). By the definition o... | 15,808 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, we have two functions f and g, and we're told that the inverse of f composed with g of x is equal to 5x + 3. And we need to find the inverse of g composed with f at -7. That is, we need to find g⁻¹(f(-7)). Hmm, let's break this down step by step.
First, let me r... | [
{
"from": "human",
"value": "Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. So, we have two functions f and g, and we're told that the inverse of f composed with g of... |
A semicircle is randomly chosen on a sphere. What is the probability that it is the boundary of a spherical triangle, which contains the sphere's center?
One should note that points on the semicircle do not necessarily have to form the vertices of a spherical triangle.
The only condition is that the semicircle must b... | A semicircle is randomly chosen on a sphere. What is the probability that it is the boundary of a spherical triangle, which contains the sphere's center?
One should note that points on the semicircle do not necessarily have to form the vertices of a spherical triangle.
The only condition is that the semicircle must b... | Okay, so I need to find the probability that a randomly chosen semicircle on a sphere is the boundary of a spherical triangle that contains the sphere's center. Hmm, let me try to unpack this step by step.
First, let me recall what a spherical triangle is. A spherical triangle is a figure formed on the surface of a sp... | To determine the probability that a randomly chosen semicircle on a sphere is the boundary of a spherical triangle that contains the sphere's center, we need to consider the following key points:
1. **Spherical Triangle and Center Containment**: A spherical triangle contains the center of the sphere if and only if it ... | 23,888 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that a randomly chosen semicircle on a sphere is the boundary of a spherical triangle that contains the sphere's center. Hmm, let me try to unpack this step by step.
First, let me recall what a spherical triangle is. A spherical triangle is a figure formed on the surface... | [
{
"from": "human",
"value": "A semicircle is randomly chosen on a sphere. What is the probability that it is the boundary of a spherical triangle, which contains the sphere's center?\n\nOne should note that points on the semicircle do not necessarily have to form the vertices of a spherical triangle.\nThe ... |
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