instruction_seed stringlengths 15 5.67k | response_seed stringlengths 15 5.67k | reasoning stringlengths 1.28k 65.9k | deepseek_solution stringlengths 7 2.46k | __original_row_idx int64 3 40k | source stringclasses 1
value | final_reasoning_trace stringlengths 1.51k 67.5k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
The ratio of measures of two complementary angles is 4 to 5. The smallest measure is increased by $10\%$. By what percent must the larger measure be decreased so that the two angles remain complementary? | The ratio of measures of two complementary angles is 4 to 5. The smallest measure is increased by $10\%$. By what percent must the larger measure be decreased so that the two angles remain complementary? | Okay, so I need to solve this problem about complementary angles. Let me read it again carefully. The ratio of the measures of two complementary angles is 4 to 5. The smallest measure is increased by 10%. By what percent must the larger measure be decreased so that the two angles remain complementary?
Alright, complem... | The ratio of the measures of two complementary angles is 4 to 5. Let the measures of the angles be \(4x\) and \(5x\). Since complementary angles add up to 90 degrees, we have:
\[
4x + 5x = 90 \implies 9x = 90 \implies x = 10
\]
Thus, the measures of the angles are \(40^\circ\) and \(50^\circ\).
The smallest angle (4... | 23,262 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this problem about complementary angles. Let me read it again carefully. The ratio of the measures of two complementary angles is 4 to 5. The smallest measure is increased by 10%. By what percent must the larger measure be decreased so that the two angles remain complementary?
Alright,... | [
{
"from": "human",
"value": "The ratio of measures of two complementary angles is 4 to 5. The smallest measure is increased by $10\\%$. By what percent must the larger measure be decreased so that the two angles remain complementary?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to sol... |
A five-digit number is divisible by 9. If the sum of the first three digits is equal to the sum of the last two digits, find the remainder when the square of the number is divided by 9. | A five-digit number is divisible by 9. If the sum of the first three digits is equal to the sum of the last two digits, find the remainder when the square of the number is divided by 9. | Okay, let's tackle this problem step by step. So, we have a five-digit number that's divisible by 9. The condition given is that the sum of the first three digits is equal to the sum of the last two digits. We need to find the remainder when the square of this number is divided by 9. Alright, let's start breaking this ... | Given a five-digit number divisible by 9, where the sum of the first three digits is equal to the sum of the last two digits, we need to find the remainder when the square of the number is divided by 9.
1. **Divisibility by 9**: A number is divisible by 9 if the sum of its digits is a multiple of 9. Let the five-digit... | 5,545 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's tackle this problem step by step. So, we have a five-digit number that's divisible by 9. The condition given is that the sum of the first three digits is equal to the sum of the last two digits. We need to find the remainder when the square of this number is divided by 9. Alright, let's start breaki... | [
{
"from": "human",
"value": "A five-digit number is divisible by 9. If the sum of the first three digits is equal to the sum of the last two digits, find the remainder when the square of the number is divided by 9."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's tackle this problem step by s... |
A rectangular prism with integer edge lengths is cut into eight non-overlapping cubes. The volume of the prism is 2160 cubic units, and the largest cube has an edge length of 10 units. Find the surface area of the prism. | A rectangular prism with integer edge lengths is cut into eight non-overlapping cubes. The volume of the prism is 2160 cubic units, and the largest cube has an edge length of 10 units. Find the surface area of the prism. | Alright, so I have this problem here about a rectangular prism with integer edge lengths that's been cut into eight non-overlapping cubes. The volume is 2160 cubic units, and the largest cube has an edge length of 10 units. I need to find the surface area of the prism. Hmm, okay, let's break this down step by step.
Fi... | Given a rectangular prism with integer edge lengths, volume 2160 cubic units, and divided into eight non-overlapping cubes with the largest cube having an edge length of 10 units, we need to find the surface area of the prism.
1. **Volume Calculation**:
- The volume of the largest cube (10x10x10) is 1000 cubic unit... | 33,554 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I have this problem here about a rectangular prism with integer edge lengths that's been cut into eight non-overlapping cubes. The volume is 2160 cubic units, and the largest cube has an edge length of 10 units. I need to find the surface area of the prism. Hmm, okay, let's break this down step by s... | [
{
"from": "human",
"value": "A rectangular prism with integer edge lengths is cut into eight non-overlapping cubes. The volume of the prism is 2160 cubic units, and the largest cube has an edge length of 10 units. Find the surface area of the prism."
},
{
"from": "gpt",
"value": "<think>\nAlrigh... |
In a regular tetrahedron, each edge measures 8 units. What is the total area, in square units, of the four triangular faces of this tetrahedron? | In a regular tetrahedron, each edge measures 8 units. What is the total area, in square units, of the four triangular faces of this tetrahedron? | Okay, so I need to find the total area of the four triangular faces of a regular tetrahedron where each edge is 8 units long. Hmm, let me start by recalling what a regular tetrahedron is. A regular tetrahedron is a three-dimensional shape with four triangular faces, and all edges are equal in length. That means every e... | To find the total area of the four triangular faces of a regular tetrahedron with each edge measuring 8 units, we start by calculating the area of one equilateral triangular face.
The formula for the area of an equilateral triangle with side length \( a \) is:
\[
\text{Area} = \frac{\sqrt{3}}{4} a^2
\]
Substituting \... | 24,459 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the total area of the four triangular faces of a regular tetrahedron where each edge is 8 units long. Hmm, let me start by recalling what a regular tetrahedron is. A regular tetrahedron is a three-dimensional shape with four triangular faces, and all edges are equal in length. That means... | [
{
"from": "human",
"value": "In a regular tetrahedron, each edge measures 8 units. What is the total area, in square units, of the four triangular faces of this tetrahedron?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the total area of the four triangular faces of a regular t... |
Find the sum of the coefficients in the polynomial $-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)$. | Find the sum of the coefficients in the polynomial $-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)$. | Okay, so I need to find the sum of the coefficients in this polynomial: -2(x⁷ - x⁴ + 3x² - 5) + 4(x³ + 2x) - 3(x⁵ - 4). Hmm, let me remember. The sum of the coefficients in a polynomial is found by plugging in x = 1, right? Because when you substitute 1 for x, each term becomes just the coefficient times 1 to some powe... | To find the sum of the coefficients in the polynomial \(-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)\), we can substitute \(x = 1\) into the polynomial and simplify.
Substituting \(x = 1\):
\[
-2(1^7 - 1^4 + 3 \cdot 1^2 - 5) + 4(1^3 + 2 \cdot 1) - 3(1^5 - 4)
\]
Simplifying each part:
\[
-2(1 - 1 + 3 - 5) + 4... | 18,855 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the sum of the coefficients in this polynomial: -2(x⁷ - x⁴ + 3x² - 5) + 4(x³ + 2x) - 3(x⁵ - 4). Hmm, let me remember. The sum of the coefficients in a polynomial is found by plugging in x = 1, right? Because when you substitute 1 for x, each term becomes just the coefficient times 1 to s... | [
{
"from": "human",
"value": "Find the sum of the coefficients in the polynomial $-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of the coefficients in this polynomial: -2(x⁷ - x⁴ + 3x² - 5) + 4(x³ + 2x) - 3(x⁵ - 4). H... |
A certain organization has 18 members who each take turns serving as officers for 2 weeks at a time. In order to facilitate scheduling, they break up into smaller groups to cover the 52 weeks of the year. If 6 of the members decide to leave the organization before the schedules are made, how many different ways can the... | A certain organization has 18 members who each take turns serving as officers for 2 weeks at a time. In order to facilitate scheduling, they break up into smaller groups to cover the 52 weeks of the year. If 6 of the members decide to leave the organization before the schedules are made, how many different ways can the... | Okay, let me try to figure out this problem. So, there's an organization with 18 members, and each member serves as an officer for 2 weeks. They need to split into smaller groups to cover all 52 weeks of the year. But before the schedules are made, 6 members leave. Now, we need to find out how many different ways the r... | The organization originally has 18 members, but 6 leave, leaving 12 members. We need to find the number of ways to divide these 12 members into 3 groups of 4 each.
Since the groups are indistinct, we use the formula for dividing \( n \) items into groups of equal size, which is given by the multinomial coefficient di... | 16,037 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, there's an organization with 18 members, and each member serves as an officer for 2 weeks. They need to split into smaller groups to cover all 52 weeks of the year. But before the schedules are made, 6 members leave. Now, we need to find out how many different wa... | [
{
"from": "human",
"value": "A certain organization has 18 members who each take turns serving as officers for 2 weeks at a time. In order to facilitate scheduling, they break up into smaller groups to cover the 52 weeks of the year. If 6 of the members decide to leave the organization before the schedules ... |
The hypotenuse of a right triangle whose legs are consecutive multiples of 3 is 40 units. What is the sum of the lengths of the two legs? | The hypotenuse of a right triangle whose legs are consecutive multiples of 3 is 40 units. What is the sum of the lengths of the two legs? | Okay, so I have this problem here: The hypotenuse of a right triangle whose legs are consecutive multiples of 3 is 40 units. I need to find the sum of the lengths of the two legs. Hmm, let me try to figure this out step by step.
First, let me parse the problem. It's a right triangle, so the Pythagorean theorem applies... | Given a right triangle with legs that are consecutive multiples of 3 and a hypotenuse of 40 units, we need to find the sum of the lengths of the two legs.
1. **Define the legs**: Let the smaller leg be \(3n\) and the next consecutive multiple be \(3(n + 1)\). Therefore, the legs are \(3n\) and \(3(n + 1)\).
2. **Apply... | 38,501 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: The hypotenuse of a right triangle whose legs are consecutive multiples of 3 is 40 units. I need to find the sum of the lengths of the two legs. Hmm, let me try to figure this out step by step.
First, let me parse the problem. It's a right triangle, so the Pythagorean theorem... | [
{
"from": "human",
"value": "The hypotenuse of a right triangle whose legs are consecutive multiples of 3 is 40 units. What is the sum of the lengths of the two legs?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: The hypotenuse of a right triangle whose legs are cons... |
A prime number is called "unusual" if it is not divisible by the product of its digits. Find the smallest unusual prime number. | A prime number is called "unusual" if it is not divisible by the product of its digits. Find the smallest unusual prime number. | Okay, so I need to find the smallest unusual prime number. Let me start by understanding the problem properly. An unusual prime is defined as a prime number that is not divisible by the product of its digits. Hmm, wait, actually, the problem says a prime is called "unusual" if it is **not** divisible by the product of ... | To determine the smallest unusual prime number, we need to check each prime number in ascending order and verify if it is not divisible by the product of its digits. An unusual prime is defined as a prime number that is not divisible by the product of its digits.
1. **Prime number 2**: Product of digits is 2. \(2 \div... | 7,063 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest unusual prime number. Let me start by understanding the problem properly. An unusual prime is defined as a prime number that is not divisible by the product of its digits. Hmm, wait, actually, the problem says a prime is called "unusual" if it is **not** divisible by the pro... | [
{
"from": "human",
"value": "A prime number is called \"unusual\" if it is not divisible by the product of its digits. Find the smallest unusual prime number."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest unusual prime number. Let me start by understanding the prob... |
Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Find the radius of the circle.
[asy]
unitsize(1 cm);
real func (real x) {
return (x^2 + 3/4);
}
path parab = graph(func,-1.5,1.5);
for(int i = 0;... | Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Find the radius of the circle.
[asy]
unitsize(1 cm);
real func (real x) {
return (x^2 + 3/4);
}
path parab = graph(func,-1.5,1.5);
for(int i = 0;... | Alright, so I need to find the radius of the circle that's tangent to six congruent parabolas arranged around it, each touching their neighbors. The given parabola is y = x², but they're probably rotated and positioned around the circle. Let me try to visualize this. The Asymptote code shows that each parabola is rotat... | To find the radius of the circle tangent to six congruent copies of the parabola \( y = x^2 \) arranged around the circle, we need to consider the geometric configuration and tangency conditions.
1. **Arrangement of Parabolas**: Each parabola is congruent to \( y = x^2 \) and arranged such that their vertices are tang... | 15,372 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to find the radius of the circle that's tangent to six congruent parabolas arranged around it, each touching their neighbors. The given parabola is y = x², but they're probably rotated and positioned around the circle. Let me try to visualize this. The Asymptote code shows that each parabola ... | [
{
"from": "human",
"value": "Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Find the radius of the circle.\n\n[asy]\nunitsize(1 cm);\n\nreal func (real x) {\n return (x^2 + 3/4);\n}\n\n... |
Let $O$ be the origin. There exists a scalar $m$ so that for any points $P,$ $Q,$ $R,$ and $S$ such that
\[2 \overrightarrow{OP} + 4 \overrightarrow{OQ} - 3 \overrightarrow{OR} + m \overrightarrow{OS} = \mathbf{0},\]the four points $P,$ $Q,$ $R,$ and $S$ are coplanar. Find $m.$ | Let $O$ be the origin. There exists a scalar $m$ so that for any points $P,$ $Q,$ $R,$ and $S$ such that
\[2 \overrightarrow{OP} + 4 \overrightarrow{OQ} - 3 \overrightarrow{OR} + m \overrightarrow{OS} = \mathbf{0},\]the four points $P,$ $Q,$ $R,$ and $S$ are coplanar. Find $m.$ | Okay, so I need to find the scalar \( m \) such that for any points \( P, Q, R, S \) satisfying the equation \( 2 \overrightarrow{OP} + 4 \overrightarrow{OQ} - 3 \overrightarrow{OR} + m \overrightarrow{OS} = \mathbf{0} \), the four points are coplanar. Hmm, let's break this down step by step.
First, the equation given... | To find the scalar \( m \) such that for any points \( P, Q, R, \) and \( S \) satisfying the equation
\[ 2 \overrightarrow{OP} + 4 \overrightarrow{OQ} - 3 \overrightarrow{OR} + m \overrightarrow{OS} = \mathbf{0}, \]
the four points are coplanar, we need to ensure that the coefficients of the linear combination sum to ... | 38,663 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the scalar \( m \) such that for any points \( P, Q, R, S \) satisfying the equation \( 2 \overrightarrow{OP} + 4 \overrightarrow{OQ} - 3 \overrightarrow{OR} + m \overrightarrow{OS} = \mathbf{0} \), the four points are coplanar. Hmm, let's break this down step by step.
First, the equati... | [
{
"from": "human",
"value": "Let $O$ be the origin. There exists a scalar $m$ so that for any points $P,$ $Q,$ $R,$ and $S$ such that\n\\[2 \\overrightarrow{OP} + 4 \\overrightarrow{OQ} - 3 \\overrightarrow{OR} + m \\overrightarrow{OS} = \\mathbf{0},\\]the four points $P,$ $Q,$ $R,$ and $S$ are coplanar. ... |
Let $g_0(x) = \frac{x}{|x-1|}$, and for $n \geq 1$, let $g_n(x) = \frac{1}{g_{n-1}(x)} + 1$. For how many values of $x$ is $g_{10}(x) = 1$? | Let $g_0(x) = \frac{x}{|x-1|}$, and for $n \geq 1$, let $g_n(x) = \frac{1}{g_{n-1}(x)} + 1$. For how many values of $x$ is $g_{10}(x) = 1$? | Okay, so I need to figure out how many values of x satisfy g_{10}(x) = 1. The functions g_n are defined recursively. Let me try to understand the problem step by step.
First, the initial function is g_0(x) = x / |x - 1|. Then, for each n ≥ 1, g_n(x) = 1 / g_{n-1}(x) + 1. So each subsequent function is defined in terms... | To determine how many values of \( x \) satisfy \( g_{10}(x) = 1 \), we start with the given functions \( g_0(x) = \frac{x}{|x - 1|} \) and \( g_n(x) = \frac{1}{g_{n-1}(x)} + 1 \) for \( n \geq 1 \).
We need to reverse-engineer the recursion to find \( x \) such that \( g_{10}(x) = 1 \). Starting from \( g_{10}(x) = 1... | 20,618 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many values of x satisfy g_{10}(x) = 1. The functions g_n are defined recursively. Let me try to understand the problem step by step.
First, the initial function is g_0(x) = x / |x - 1|. Then, for each n ≥ 1, g_n(x) = 1 / g_{n-1}(x) + 1. So each subsequent function is defined ... | [
{
"from": "human",
"value": "Let $g_0(x) = \\frac{x}{|x-1|}$, and for $n \\geq 1$, let $g_n(x) = \\frac{1}{g_{n-1}(x)} + 1$. For how many values of $x$ is $g_{10}(x) = 1$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many values of x satisfy g_{10}(x) = 1. The functi... |
Find the product of the roots of the equation \[x^2\sqrt{x} - 3x^2 + 2\sqrt{x} - 5 = 0,\]given that all of the roots are real and nonnegative. | Find the product of the roots of the equation \[x^2\sqrt{x} - 3x^2 + 2\sqrt{x} - 5 = 0,\]given that all of the roots are real and nonnegative. | Okay, so I have this equation to solve: \(x^2\sqrt{x} - 3x^2 + 2\sqrt{x} - 5 = 0\). The question asks for the product of the roots, and it mentions that all roots are real and nonnegative. Hmm, let me think about how to approach this.
First off, the equation involves both \(x^2\) and \(\sqrt{x}\), which makes it a bit... | Given the equation \(x^2\sqrt{x} - 3x^2 + 2\sqrt{x} - 5 = 0\), we need to find the product of the roots, given that all roots are real and nonnegative.
First, we substitute \(t = \sqrt{x}\), which implies \(x = t^2\). Substituting \(t\) into the equation, we get:
\[
(t^2)^2 \cdot t - 3(t^2)^2 + 2t - 5 = 0
\]
Simplifyi... | 30,323 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this equation to solve: \(x^2\sqrt{x} - 3x^2 + 2\sqrt{x} - 5 = 0\). The question asks for the product of the roots, and it mentions that all roots are real and nonnegative. Hmm, let me think about how to approach this.
First off, the equation involves both \(x^2\) and \(\sqrt{x}\), which makes ... | [
{
"from": "human",
"value": "Find the product of the roots of the equation \\[x^2\\sqrt{x} - 3x^2 + 2\\sqrt{x} - 5 = 0,\\]given that all of the roots are real and nonnegative."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this equation to solve: \\(x^2\\sqrt{x} - 3x^2 + 2\\sqrt{x} - 5 ... |
A data set consists of six numbers: 2, 3, 4, 5, x, and y, where x and y are unknown positive integers. The mean of the data set is 4.5, and the mode is 2. If the data set is arranged in increasing order, the median is 4. What is the value of x + y? | A data set consists of six numbers: 2, 3, 4, 5, x, and y, where x and y are unknown positive integers. The mean of the data set is 4.5, and the mode is 2. If the data set is arranged in increasing order, the median is 4. What is the value of x + y? | Alright, so I have this problem here where there's a dataset with six numbers: 2, 3, 4, 5, x, and y. Both x and y are positive integers, and I need to find their sum. The given information includes the mean being 4.5, the mode being 2, and the median being 4 when the data is arranged in increasing order. Hmm, okay, let... | Given a dataset consisting of six numbers: 2, 3, 4, 5, \(x\), and \(y\), where \(x\) and \(y\) are unknown positive integers, we need to find the value of \(x + y\) given the following conditions:
1. The mean of the dataset is 4.5.
2. The mode is 2.
3. The median is 4 when the dataset is arranged in increasing order.
... | 26,700 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I have this problem here where there's a dataset with six numbers: 2, 3, 4, 5, x, and y. Both x and y are positive integers, and I need to find their sum. The given information includes the mean being 4.5, the mode being 2, and the median being 4 when the data is arranged in increasing order. Hmm, o... | [
{
"from": "human",
"value": "A data set consists of six numbers: 2, 3, 4, 5, x, and y, where x and y are unknown positive integers. The mean of the data set is 4.5, and the mode is 2. If the data set is arranged in increasing order, the median is 4. What is the value of x + y?"
},
{
"from": "gpt",
... |
What is the area of triangle $ABC$ if the side length of the square is 7 inches instead of 6 inches? | What is the area of triangle $ABC$ if the side length of the square is 7 inches instead of 6 inches? | Okay, so I need to find the area of triangle ABC where the square has a side length of 7 inches instead of 6. Hmm, let me visualize this. If there's a square, and triangle ABC is somehow related to that square, maybe inscribed or part of it. The problem doesn't provide a diagram, so I have to assume based on standard p... | To find the area of triangle \(ABC\) when the side length of the square is 7 inches, we consider the most common configuration where triangle \(ABC\) is a right triangle formed by two adjacent sides and the diagonal of the square.
1. The side length of the square is 7 inches.
2. A right triangle formed by two adjacen... | 38,260 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the area of triangle ABC where the square has a side length of 7 inches instead of 6. Hmm, let me visualize this. If there's a square, and triangle ABC is somehow related to that square, maybe inscribed or part of it. The problem doesn't provide a diagram, so I have to assume based on st... | [
{
"from": "human",
"value": "What is the area of triangle $ABC$ if the side length of the square is 7 inches instead of 6 inches?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the area of triangle ABC where the square has a side length of 7 inches instead of 6. Hmm, let me visu... |
Determine $a$ and $b$ such that the expression $(x+a)(x+b)$ has the same value for any $x$. | Determine $a$ and $b$ such that the expression $(x+a)(x+b)$ has the same value for any $x$. | Okay, let me try to figure out how to determine the values of a and b such that the expression (x + a)(x + b) is the same for any x. Hmm, so the problem is asking for constants a and b where, no matter what value of x you plug in, the product (x + a)(x + b) remains constant. That means the expression doesn't actually d... | To determine \(a\) and \(b\) such that the expression \((x + a)(x + b)\) has the same value for any \(x\), we start by expanding the expression:
\[
(x + a)(x + b) = x^2 + (a + b)x + ab
\]
For this quadratic expression to be constant for all \(x\), the coefficients of \(x^2\) and \(x\) must be zero, and the constant t... | 4,582 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out how to determine the values of a and b such that the expression (x + a)(x + b) is the same for any x. Hmm, so the problem is asking for constants a and b where, no matter what value of x you plug in, the product (x + a)(x + b) remains constant. That means the expression doesn't ac... | [
{
"from": "human",
"value": "Determine $a$ and $b$ such that the expression $(x+a)(x+b)$ has the same value for any $x$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out how to determine the values of a and b such that the expression (x + a)(x + b) is the same for any x. Hm... |
In a conference, there are 5 speakers who need to address 4 different topics. Each speaker will address only one topic, and each topic will be addressed by only one speaker. Before the conference, each speaker meets individually with each of the other speakers who will be addressing a different topic, to discuss their ... | In a conference, there are 5 speakers who need to address 4 different topics. Each speaker will address only one topic, and each topic will be addressed by only one speaker. Before the conference, each speaker meets individually with each of the other speakers who will be addressing a different topic, to discuss their ... | Alright, so I have this problem here about 5 speakers addressing 4 different topics. Each speaker is assigned to one topic, and each topic has only one speaker. Before the conference, each speaker meets individually with every other speaker who's addressing a different topic. I need to find out the total number of meet... | The problem involves 5 speakers addressing 4 different topics, with each speaker addressing only one topic and each topic being addressed by only one speaker. However, this setup seems contradictory because 5 speakers cannot be assigned to 4 topics if each topic must be addressed by exactly one speaker.
Reinterpretin... | 18,742 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I have this problem here about 5 speakers addressing 4 different topics. Each speaker is assigned to one topic, and each topic has only one speaker. Before the conference, each speaker meets individually with every other speaker who's addressing a different topic. I need to find out the total number... | [
{
"from": "human",
"value": "In a conference, there are 5 speakers who need to address 4 different topics. Each speaker will address only one topic, and each topic will be addressed by only one speaker. Before the conference, each speaker meets individually with each of the other speakers who will be addres... |
There are 5 people waiting to take a bus: 2 girls and 3 boys. The bus can take only 3 passengers. How many ways are there to choose the passengers if there must be at least one girl among them? | There are 5 people waiting to take a bus: 2 girls and 3 boys. The bus can take only 3 passengers. How many ways are there to choose the passengers if there must be at least one girl among them? | Okay, so there's this problem here: there are 5 people waiting for a bus—2 girls and 3 boys. The bus can only take 3 passengers. We need to figure out how many ways there are to choose the passengers if there must be at least one girl among them. Hmm, okay, let me break this down.
First off, the total number of people... | To determine the number of ways to choose 3 passengers from 5 people (2 girls and 3 boys) such that there is at least one girl among them, we can use two methods: subtracting the unwanted cases from the total or adding up the valid cases.
**Method 1: Subtracting Unwanted Cases**
1. Calculate the total number of ways ... | 14,767 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this problem here: there are 5 people waiting for a bus—2 girls and 3 boys. The bus can only take 3 passengers. We need to figure out how many ways there are to choose the passengers if there must be at least one girl among them. Hmm, okay, let me break this down.
First off, the total number o... | [
{
"from": "human",
"value": "There are 5 people waiting to take a bus: 2 girls and 3 boys. The bus can take only 3 passengers. How many ways are there to choose the passengers if there must be at least one girl among them?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so there's this problem her... |
If a triangle has two sides of lengths 5 and 7 units, then how many different integer lengths can the third side be? | If a triangle has two sides of lengths 5 and 7 units, then how many different integer lengths can the third side be? | Okay, so I need to figure out how many different integer lengths the third side of a triangle can be if two of the sides are 5 and 7 units long. Hmm, triangles have those rules about the sum of the lengths of any two sides being greater than the length of the remaining side. Let me recall exactly what the triangle ineq... | To determine how many different integer lengths the third side of a triangle can be if two sides are 5 and 7 units long, we use the triangle inequality theorem. The third side, denoted as \( x \), must satisfy the following inequalities:
1. \( 5 + 7 > x \) which simplifies to \( x < 12 \)
2. \( 5 + x > 7 \) which simp... | 6,183 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many different integer lengths the third side of a triangle can be if two of the sides are 5 and 7 units long. Hmm, triangles have those rules about the sum of the lengths of any two sides being greater than the length of the remaining side. Let me recall exactly what the trian... | [
{
"from": "human",
"value": "If a triangle has two sides of lengths 5 and 7 units, then how many different integer lengths can the third side be?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many different integer lengths the third side of a triangle can be if two of... |
Solve for $n$, if $8^n\cdot8^n\cdot8^n=64^3$. | Solve for $n$, if $8^n\cdot8^n\cdot8^n=64^3$. | Okay, so I need to solve for n in the equation 8^n multiplied by itself three times equals 64 cubed. Let me write that down first to make sure I have it right: 8^n · 8^n · 8^n = 64^3. Hmm, so that's three instances of 8^n multiplied together. Maybe I can simplify the left side first.
Since all the bases are the same a... | To solve for \( n \) in the equation \( 8^n \cdot 8^n \cdot 8^n = 64^3 \):
1. Simplify the left side using the exponent rule \( a^m \cdot a^n = a^{m+n} \):
\[
8^n \cdot 8^n \cdot 8^n = 8^{n + n + n} = 8^{3n}
\]
2. Express both sides as powers of 2:
- \( 8 = 2^3 \), so \( 8^{3n} = (2^3)^{3n} = 2^{9n} \)
... | 18,249 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve for n in the equation 8^n multiplied by itself three times equals 64 cubed. Let me write that down first to make sure I have it right: 8^n · 8^n · 8^n = 64^3. Hmm, so that's three instances of 8^n multiplied together. Maybe I can simplify the left side first.
Since all the bases are th... | [
{
"from": "human",
"value": "Solve for $n$, if $8^n\\cdot8^n\\cdot8^n=64^3$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve for n in the equation 8^n multiplied by itself three times equals 64 cubed. Let me write that down first to make sure I have it right: 8^n · 8^n · 8^n = 6... |
Find the sum of $2010_9$ and $777_9$ in base $9$. | Find the sum of $2010_9$ and $777_9$ in base $9$. | Okay, so I need to add two numbers that are in base 9, right? The numbers are 2010_9 and 777_9. Hmm, base 9. Let me remember how addition works in different bases. I think it's similar to base 10, but once the sum reaches 9, we carry over to the next place. Let me confirm that. Yes, in base 9, each digit can go from 0 ... | To find the sum of \(2010_9\) and \(777_9\) in base 9, we need to align the numbers by their place values and add each column starting from the rightmost digit (units place). Here are the key steps:
1. **Align the numbers vertically by their place values:**
```
2 0 1 0
+ 0 7 7 7
-----------
```
2. **... | 20,339 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to add two numbers that are in base 9, right? The numbers are 2010_9 and 777_9. Hmm, base 9. Let me remember how addition works in different bases. I think it's similar to base 10, but once the sum reaches 9, we carry over to the next place. Let me confirm that. Yes, in base 9, each digit can go... | [
{
"from": "human",
"value": "Find the sum of $2010_9$ and $777_9$ in base $9$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to add two numbers that are in base 9, right? The numbers are 2010_9 and 777_9. Hmm, base 9. Let me remember how addition works in different bases. I think it's ... |
A circle of radius 1 is surrounded by 4 circles of radius $r$ as shown. What is $r$?
[asy]
unitsize(0.6cm);
for(int i=0; i<2; ++i){
for(int j=0; j<2; ++j){
draw(Circle((-2.4+4.8i,-2.4+4.8j),2.4),linewidth(0.7));
draw((-2.4+4.8i,-2.4+4.8j)--(-0.7+4.8i,-0.7+4.8j));
label("$r$",(-1.5+4.8i,-1.5+4.8j),SE);
};
}
draw(Circle... | A circle of radius 1 is surrounded by 4 circles of radius $r$ as shown. What is $r$?
[asy]
unitsize(0.6cm);
for(int i=0; i<2; ++i){
for(int j=0; j<2; ++j){
draw(Circle((-2.4+4.8i,-2.4+4.8j),2.4),linewidth(0.7));
draw((-2.4+4.8i,-2.4+4.8j)--(-0.7+4.8i,-0.7+4.8j));
label("$r$",(-1.5+4.8i,-1.5+4.8j),SE);
};
}
draw(Circle... | Okay, so I need to find the radius \( r \) of the four surrounding circles that are arranged around a central circle of radius 1. Hmm, let me visualize the Asymptote figure they provided. There's a central circle with radius 1, and four circles around it, each touching the central circle and also touching their adjacen... | To find the radius \( r \) of the surrounding circles, we start by noting that the distance between the center of the central circle (radius 1) and the center of any surrounding circle (radius \( r \)) is \( 1 + r \). The centers of the four surrounding circles form a square, with each side of the square being the dist... | 2,077 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the radius \( r \) of the four surrounding circles that are arranged around a central circle of radius 1. Hmm, let me visualize the Asymptote figure they provided. There's a central circle with radius 1, and four circles around it, each touching the central circle and also touching their... | [
{
"from": "human",
"value": "A circle of radius 1 is surrounded by 4 circles of radius $r$ as shown. What is $r$?\n\n[asy]\nunitsize(0.6cm);\nfor(int i=0; i<2; ++i){\nfor(int j=0; j<2; ++j){\ndraw(Circle((-2.4+4.8i,-2.4+4.8j),2.4),linewidth(0.7));\ndraw((-2.4+4.8i,-2.4+4.8j)--(-0.7+4.8i,-0.7+4.8j));\nlabel(... |
The $24^\text{th}$ day of a particular month is a Saturday. On what day of the week was the first day of that month? | The $24^\text{th}$ day of a particular month is a Saturday. On what day of the week was the first day of that month? | Okay, so the problem is: The 24th day of a particular month is a Saturday. On what day of the week was the first day of that month?
Hmm, let's see. I need to figure out what day the 1st of the month was, given that the 24th is a Saturday. Days of the week repeat every 7 days, right? So if I can find out how many days ... | Given that the 24th day of a particular month is a Saturday, we need to determine the day of the week for the first day of that month.
1. **Calculate the number of days between the 1st and the 24th**:
- The difference is \(24 - 1 = 23\) days.
2. **Determine the number of weeks and extra days**:
- Dividing 23 by... | 12,054 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so the problem is: The 24th day of a particular month is a Saturday. On what day of the week was the first day of that month?
Hmm, let's see. I need to figure out what day the 1st of the month was, given that the 24th is a Saturday. Days of the week repeat every 7 days, right? So if I can find out how ma... | [
{
"from": "human",
"value": "The $24^\\text{th}$ day of a particular month is a Saturday. On what day of the week was the first day of that month?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem is: The 24th day of a particular month is a Saturday. On what day of the week was the fi... |
A set of five numbers has a median equal to 10 and a mode equal to 12. If the sum of the numbers is 54, what is the smallest possible value of the largest number in the set? | A set of five numbers has a median equal to 10 and a mode equal to 12. If the sum of the numbers is 54, what is the smallest possible value of the largest number in the set? | Okay, so I need to figure out the smallest possible value of the largest number in a set of five numbers. The given information is that the median is 10, the mode is 12, and the sum of the numbers is 54. Let me break this down step by step.
First, let's recall what median, mode, and sum mean. The median is the middle ... | Given a set of five numbers with a median of 10, a mode of 12, and a sum of 54, we need to find the smallest possible value of the largest number in the set.
1. **Median and Mode Constraints**:
- The median of 10 means the third number in the ordered set is 10.
- The mode of 12 means 12 must appear more frequent... | 3,484 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the smallest possible value of the largest number in a set of five numbers. The given information is that the median is 10, the mode is 12, and the sum of the numbers is 54. Let me break this down step by step.
First, let's recall what median, mode, and sum mean. The median is the... | [
{
"from": "human",
"value": "A set of five numbers has a median equal to 10 and a mode equal to 12. If the sum of the numbers is 54, what is the smallest possible value of the largest number in the set?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the smallest possible v... |
A group of friends want to take a road trip from their hometown to a nearby city, which is 210 miles away. If they drive at an average speed of 35 miles per hour for the first 120 miles, and then increase their speed to 50 miles per hour for the remaining distance, how many hours will the entire trip take? | A group of friends want to take a road trip from their hometown to a nearby city, which is 210 miles away. If they drive at an average speed of 35 miles per hour for the first 120 miles, and then increase their speed to 50 miles per hour for the remaining distance, how many hours will the entire trip take? | Okay, so I need to figure out how long this road trip will take. Let me start by understanding the problem again. The friends are driving 210 miles total. They drive the first 120 miles at 35 mph, and then the remaining distance at 50 mph. I need to find the total time taken for the entire trip.
Alright, time is equa... | To determine the total time taken for the entire trip, we need to calculate the time for each segment of the trip and then sum them together.
1. **First segment: 120 miles at 35 mph**
\[
\text{Time} = \frac{120 \text{ miles}}{35 \text{ mph}} = \frac{24}{7} \text{ hours}
\]
2. **Second segment: 90 miles at 50... | 18,324 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how long this road trip will take. Let me start by understanding the problem again. The friends are driving 210 miles total. They drive the first 120 miles at 35 mph, and then the remaining distance at 50 mph. I need to find the total time taken for the entire trip.
Alright, time... | [
{
"from": "human",
"value": "A group of friends want to take a road trip from their hometown to a nearby city, which is 210 miles away. If they drive at an average speed of 35 miles per hour for the first 120 miles, and then increase their speed to 50 miles per hour for the remaining distance, how many hour... |
The number of bacteria in a culture grows according to a geometric progression. On the third day, the number of bacteria is 50% more than the initial amount, and on the fifth day, it is three times the initial amount. What was the initial number of bacteria? | The number of bacteria in a culture grows according to a geometric progression. On the third day, the number of bacteria is 50% more than the initial amount, and on the fifth day, it is three times the initial amount. What was the initial number of bacteria? | Okay, let's see. The problem says that the number of bacteria in a culture grows according to a geometric progression. On the third day, it's 50% more than the initial amount, and on the fifth day, it's three times the initial amount. We need to find the initial number of bacteria.
Hmm, geometric progression. So that ... | The problem states that the number of bacteria in a culture grows according to a geometric progression. On the third day, the number of bacteria is 50% more than the initial amount, and on the fifth day, it is three times the initial amount. We need to find the initial number of bacteria.
1. **Setting up the equations... | 608 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem says that the number of bacteria in a culture grows according to a geometric progression. On the third day, it's 50% more than the initial amount, and on the fifth day, it's three times the initial amount. We need to find the initial number of bacteria.
Hmm, geometric progression. ... | [
{
"from": "human",
"value": "The number of bacteria in a culture grows according to a geometric progression. On the third day, the number of bacteria is 50% more than the initial amount, and on the fifth day, it is three times the initial amount. What was the initial number of bacteria?"
},
{
"from"... |
Let $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ be three vectors such that $\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}$, $\|\mathbf{u}\| = 3,$ $\|\mathbf{v}\| = 4,$ and $\|\mathbf{u} \times \mathbf{v}\| = 6.$ Find $\mathbf{w} \cdot (\mathbf{u} \times \mathbf{v}).$ | Let $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ be three vectors such that $\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}$, $\|\mathbf{u}\| = 3,$ $\|\mathbf{v}\| = 4,$ and $\|\mathbf{u} \times \mathbf{v}\| = 6.$ Find $\mathbf{w} \cdot (\mathbf{u} \times \mathbf{v}).$ | Okay, let's see. The problem gives three vectors u, v, and w such that their sum is zero. We also know the magnitudes of u and v, and the magnitude of the cross product of u and v. We need to find the dot product of w with the cross product of u and v. Hmm. Let's break this down step by step.
First, since u + v + w = ... | Given vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) such that \(\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}\), with \(\|\mathbf{u}\| = 3\), \(\|\mathbf{v}\| = 4\), and \(\|\mathbf{u} \times \mathbf{v}\| = 6\), we need to find \(\mathbf{w} \cdot (\mathbf{u} \times \mathbf{v})\).
Since \(\mathbf{u} + ... | 29,815 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem gives three vectors u, v, and w such that their sum is zero. We also know the magnitudes of u and v, and the magnitude of the cross product of u and v. We need to find the dot product of w with the cross product of u and v. Hmm. Let's break this down step by step.
First, since u + ... | [
{
"from": "human",
"value": "Let $\\mathbf{u}$, $\\mathbf{v}$, and $\\mathbf{w}$ be three vectors such that $\\mathbf{u} + \\mathbf{v} + \\mathbf{w} = \\mathbf{0}$, $\\|\\mathbf{u}\\| = 3,$ $\\|\\mathbf{v}\\| = 4,$ and $\\|\\mathbf{u} \\times \\mathbf{v}\\| = 6.$ Find $\\mathbf{w} \\cdot (\\mathbf{u} \\tim... |
Determine the base $b$ such that the equation $\log_b bx = \frac{1}{2}$ has exactly two solutions. | Determine the base $b$ such that the equation $\log_b bx = \frac{1}{2}$ has exactly two solutions. | Okay, let's see. I need to find the base \( b \) such that the equation \( \log_b (bx) = \frac{1}{2} \) has exactly two solutions. Hmm, alright. Let me start by recalling some logarithm properties.
First, the equation is \( \log_b (bx) = \frac{1}{2} \). Maybe I should convert this logarithmic equation into its expone... | To determine the base \( b \) such that the equation \(\log_b (bx) = \frac{1}{2}\) has exactly two solutions, we start by converting the logarithmic equation to its exponential form.
Given the equation:
\[
\log_b (bx) = \frac{1}{2}
\]
Using the definition of logarithms, we rewrite it as:
\[
b^{\frac{1}{2}} = bx
\]
... | 22,057 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the base \( b \) such that the equation \( \log_b (bx) = \frac{1}{2} \) has exactly two solutions. Hmm, alright. Let me start by recalling some logarithm properties.
First, the equation is \( \log_b (bx) = \frac{1}{2} \). Maybe I should convert this logarithmic equation into it... | [
{
"from": "human",
"value": "Determine the base $b$ such that the equation $\\log_b bx = \\frac{1}{2}$ has exactly two solutions."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the base \\( b \\) such that the equation \\( \\log_b (bx) = \\frac{1}{2} \\) has exactly two ... |
An ant is placed at the origin of a three-dimensional grid. It can move in any of the six directions (up, down, left, right, forward, backward) with equal probability. What is the probability that the ant reaches the point (3,3,3) in exactly 15 steps? | An ant is placed at the origin of a three-dimensional grid. It can move in any of the six directions (up, down, left, right, forward, backward) with equal probability. What is the probability that the ant reaches the point (3,3,3) in exactly 15 steps? | Okay, so I need to find the probability that an ant starting at the origin (0,0,0) in a 3D grid reaches the point (3,3,3) in exactly 15 steps. Each step, the ant can move in one of the six directions—up, down, left, right, forward, or backward—each with equal probability. Hmm, let me try to break this down.
First, let... | The ant starts at the origin (0,0,0) in a 3D grid and can move in any of the six directions with equal probability. We need to find the probability that the ant reaches the point (3,3,3) in exactly 15 steps.
1. **Displacement Requirements**:
- For each coordinate (x, y, z), the net displacement must be +3.
- Let... | 33,129 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that an ant starting at the origin (0,0,0) in a 3D grid reaches the point (3,3,3) in exactly 15 steps. Each step, the ant can move in one of the six directions—up, down, left, right, forward, or backward—each with equal probability. Hmm, let me try to break this down.
Fi... | [
{
"from": "human",
"value": "An ant is placed at the origin of a three-dimensional grid. It can move in any of the six directions (up, down, left, right, forward, backward) with equal probability. What is the probability that the ant reaches the point (3,3,3) in exactly 15 steps?"
},
{
"from": "gpt"... |
A box contains 10 blue balls and 15 red balls. We draw 4 balls at random without replacement. What is the probability that exactly 3 of the drawn balls are blue? | A box contains 10 blue balls and 15 red balls. We draw 4 balls at random without replacement. What is the probability that exactly 3 of the drawn balls are blue? | Okay, so I have this probability problem here: there's a box with 10 blue balls and 15 red balls. We need to draw 4 balls at random without replacement, and find the probability that exactly 3 of them are blue. Hmm, let's see. I think this is a hypergeometric distribution problem because we're dealing with successes an... | To find the probability that exactly 3 out of 4 drawn balls are blue from a box containing 10 blue balls and 15 red balls, we use the hypergeometric distribution formula:
\[
P(X = k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}}
\]
where:
- \(N = 25\) (total number of balls),
- \(K = 10\) (number of blue b... | 26,313 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this probability problem here: there's a box with 10 blue balls and 15 red balls. We need to draw 4 balls at random without replacement, and find the probability that exactly 3 of them are blue. Hmm, let's see. I think this is a hypergeometric distribution problem because we're dealing with succ... | [
{
"from": "human",
"value": "A box contains 10 blue balls and 15 red balls. We draw 4 balls at random without replacement. What is the probability that exactly 3 of the drawn balls are blue?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this probability problem here: there's a box with... |
A number $x$ has the property that when multiplied by $0.\overline{2}$, the result is $0.\overline{8}$. Determine the value of $x$. | A number $x$ has the property that when multiplied by $0.\overline{2}$, the result is $0.\overline{8}$. Determine the value of $x$. | Okay, so I need to find a number x such that when I multiply it by 0.2222... (repeating), the result is 0.8888... (repeating). Hmm, let's see.
First, I remember that repeating decimals can be converted into fractions by using some algebra. Maybe that will help here. Let me start by converting both 0.\overline{2} and ... | To determine the value of \( x \) such that \( x \times 0.\overline{2} = 0.\overline{8} \), we start by converting the repeating decimals into fractions.
First, convert \( 0.\overline{2} \) to a fraction:
Let \( a = 0.\overline{2} \). Then,
\[
10a = 2.\overline{2}
\]
Subtracting the original equation from this,
\[
10a... | 12,561 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find a number x such that when I multiply it by 0.2222... (repeating), the result is 0.8888... (repeating). Hmm, let's see.
First, I remember that repeating decimals can be converted into fractions by using some algebra. Maybe that will help here. Let me start by converting both 0.\overline... | [
{
"from": "human",
"value": "A number $x$ has the property that when multiplied by $0.\\overline{2}$, the result is $0.\\overline{8}$. Determine the value of $x$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find a number x such that when I multiply it by 0.2222... (repeating), the... |
If
$$\log_ba=\log_ca^2\qquad\mbox{and}\qquad\log_ca=\log_ba^3$$
determine $a$ in terms of $b$ and $c$. | If
$$\log_ba=\log_ca^2\qquad\mbox{and}\qquad\log_ca=\log_ba^3$$
determine $a$ in terms of $b$ and $c$. | Okay, let's see. I need to find a in terms of b and c using these two logarithmic equations. Hmm. The equations given are:
First, log base b of a equals log base c of a squared. So, in symbols, that would be:
$$\log_b a = \log_c a^2$$
And the second equation is log base c of a equals log base b of a cubed:
$$\log_c... | Given the equations:
$$
\log_b a = \log_c a^2
$$
and
$$
\log_c a = \log_b a^3,
$$
we need to determine \( a \) in terms of \( b \) and \( c \).
First, we convert the logarithms using the change of base formula. For the first equation:
$$
\log_b a = \frac{\log_b a^2}{\log_b c}
$$
Simplifying the right-hand side, we get... | 16,409 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find a in terms of b and c using these two logarithmic equations. Hmm. The equations given are:
First, log base b of a equals log base c of a squared. So, in symbols, that would be:
$$\log_b a = \log_c a^2$$
And the second equation is log base c of a equals log base b of a cubed:
... | [
{
"from": "human",
"value": "If\n$$\\log_ba=\\log_ca^2\\qquad\\mbox{and}\\qquad\\log_ca=\\log_ba^3$$\ndetermine $a$ in terms of $b$ and $c$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find a in terms of b and c using these two logarithmic equations. Hmm. The equations giv... |
The hyperbola given by the equation \[\frac{y^2}{9}-\frac{x^2}{4} = 1\]has asymptotes $y = \pm mx,$ where $m$ is positive. Find $m.$ | The hyperbola given by the equation \[\frac{y^2}{9}-\frac{x^2}{4} = 1\]has asymptotes $y = \pm mx,$ where $m$ is positive. Find $m.$ | Okay, so I need to find the value of m for the asymptotes of the hyperbola given by the equation y²/9 - x²/4 = 1. The asymptotes are given as y = ±mx, and m is positive. Alright, let me think about how to approach this.
First, I remember that hyperbolas have asymptotes which are straight lines that the hyperbola appr... | To find the value of \( m \) for the asymptotes of the hyperbola given by the equation \(\frac{y^2}{9} - \frac{x^2}{4} = 1\), we start by noting that this is a vertically oriented hyperbola. The standard form of a vertical hyperbola is \(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\), where the center is \((h, k)\... | 28,669 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of m for the asymptotes of the hyperbola given by the equation y²/9 - x²/4 = 1. The asymptotes are given as y = ±mx, and m is positive. Alright, let me think about how to approach this.
First, I remember that hyperbolas have asymptotes which are straight lines that the hyperb... | [
{
"from": "human",
"value": "The hyperbola given by the equation \\[\\frac{y^2}{9}-\\frac{x^2}{4} = 1\\]has asymptotes $y = \\pm mx,$ where $m$ is positive. Find $m.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of m for the asymptotes of the hyperbola given by the e... |
Suppose the cost of the registration fee for a certain summer camp is $30 per person for the first 20 people, and $15 per person for any additional people beyond the initial 20. What would be the total registration cost for a group consisting of 35 people? | Suppose the cost of the registration fee for a certain summer camp is $30 per person for the first 20 people, and $15 per person for any additional people beyond the initial 20. What would be the total registration cost for a group consisting of 35 people? | Okay, let's see. I need to figure out the total registration cost for a group of 35 people attending a summer camp. The fee structure is $30 per person for the first 20 people and then $15 per person for any additional people beyond that initial 20. Hmm, so basically, the first 20 are at a higher rate, and then it's di... | The cost of the registration fee for the summer camp is $30 per person for the first 20 people and $15 per person for any additional people beyond the initial 20.
To calculate the total registration cost for a group of 35 people:
1. The first 20 people are charged $30 each:
\[
20 \times 30 = 600
\]
2. The re... | 14,943 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to figure out the total registration cost for a group of 35 people attending a summer camp. The fee structure is $30 per person for the first 20 people and then $15 per person for any additional people beyond that initial 20. Hmm, so basically, the first 20 are at a higher rate, and then... | [
{
"from": "human",
"value": "Suppose the cost of the registration fee for a certain summer camp is $30 per person for the first 20 people, and $15 per person for any additional people beyond the initial 20. What would be the total registration cost for a group consisting of 35 people?"
},
{
"from": ... |
Suppose $279_{11}=4cd_{10}$, where $c$ and $d$ represent base-10 digits. Find $\frac{c \cdot d}{12}$. | Suppose $279_{11}=4cd_{10}$, where $c$ and $d$ represent base-10 digits. Find $\frac{c \cdot d}{12}$. | Alright, so I have this problem here: Suppose 279 in base 11 is equal to 4cd in base 10, where c and d are base-10 digits. I need to find the value of (c·d)/12. Hmm, okay. Let's break this down step by step. First, I need to convert the base 11 number 279₁₁ to base 10 and set it equal to 4cd₁₀. Then, once I have that, ... | To solve the problem, we need to convert the base-11 number \(279_{11}\) to base-10 and set it equal to \(4cd_{10}\), where \(c\) and \(d\) are base-10 digits.
First, we convert \(279_{11}\) to base-10:
\[
279_{11} = 2 \times 11^2 + 7 \times 11^1 + 9 \times 11^0
\]
Calculating each term:
\[
2 \times 121 = 242
\]
\[
7... | 11,305 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I have this problem here: Suppose 279 in base 11 is equal to 4cd in base 10, where c and d are base-10 digits. I need to find the value of (c·d)/12. Hmm, okay. Let's break this down step by step. First, I need to convert the base 11 number 279₁₁ to base 10 and set it equal to 4cd₁₀. Then, once I hav... | [
{
"from": "human",
"value": "Suppose $279_{11}=4cd_{10}$, where $c$ and $d$ represent base-10 digits. Find $\\frac{c \\cdot d}{12}$."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this problem here: Suppose 279 in base 11 is equal to 4cd in base 10, where c and d are base-10 digits. ... |
Let $P(x)$ be a polynomial with real coefficients that has a factor of $x^3 + 2x^2 + x + 1$. Find the polynomial $P(x)$ such that $$P(x) = (x^3 + 2x^2 + x + 1) \cdot Q(x)$$ for some polynomial $Q(x)$ and that $$P(x) \equiv 0 \pmod{x^3 + 2x^2 + x + 1}$$ where $P(x)$ has a degree of at most 5. | Let $P(x)$ be a polynomial with real coefficients that has a factor of $x^3 + 2x^2 + x + 1$. Find the polynomial $P(x)$ such that $$P(x) = (x^3 + 2x^2 + x + 1) \cdot Q(x)$$ for some polynomial $Q(x)$ and that $$P(x) \equiv 0 \pmod{x^3 + 2x^2 + x + 1}$$ where $P(x)$ has a degree of at most 5. | Okay, so I have this problem here where I need to find a polynomial P(x) of degree at most 5 that is divisible by x³ + 2x² + x + 1. That means when I divide P(x) by this cubic polynomial, there's no remainder. The problem also states that P(x) is equal to (x³ + 2x² + x + 1) multiplied by another polynomial Q(x). Since ... | Given the polynomial \( P(x) \) with real coefficients that has a factor of \( x^3 + 2x^2 + x + 1 \), we need to find \( P(x) \) such that \( P(x) = (x^3 + 2x^2 + x + 1) \cdot Q(x) \) for some polynomial \( Q(x) \) and that \( P(x) \equiv 0 \pmod{x^3 + 2x^2 + x + 1} \) with \( P(x) \) having a degree of at most 5.
1. ... | 17,403 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here where I need to find a polynomial P(x) of degree at most 5 that is divisible by x³ + 2x² + x + 1. That means when I divide P(x) by this cubic polynomial, there's no remainder. The problem also states that P(x) is equal to (x³ + 2x² + x + 1) multiplied by another polynomial Q(x)... | [
{
"from": "human",
"value": "Let $P(x)$ be a polynomial with real coefficients that has a factor of $x^3 + 2x^2 + x + 1$. Find the polynomial $P(x)$ such that $$P(x) = (x^3 + 2x^2 + x + 1) \\cdot Q(x)$$ for some polynomial $Q(x)$ and that $$P(x) \\equiv 0 \\pmod{x^3 + 2x^2 + x + 1}$$ where $P(x)$ has a degr... |
A local bakery has a daily special on croissants, with a 60% chance of having chocolate-filled croissants on any given day. If they don't have chocolate-filled croissants, they will have plain croissants instead. A group of friends wants to visit the bakery on a 3-day weekend and have exactly one day with chocolate-fil... | A local bakery has a daily special on croissants, with a 60% chance of having chocolate-filled croissants on any given day. If they don't have chocolate-filled croissants, they will have plain croissants instead. A group of friends wants to visit the bakery on a 3-day weekend and have exactly one day with chocolate-fil... | Okay, so there's this bakery that has a daily special on croissants. Each day, there's a 60% chance they have chocolate-filled ones, and if they don't, they have plain instead. A group of friends wants to visit over a 3-day weekend and have exactly one day with chocolate-filled croissants. I need to find the probabilit... | The bakery has a 60% chance of offering chocolate-filled croissants each day and a 40% chance of offering plain croissants. The friends want exactly one day with chocolate-filled croissants over a 3-day weekend. This can be modeled using a binomial distribution with parameters \( n = 3 \), \( k = 1 \), and \( p = 0.6 \... | 17,193 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this bakery that has a daily special on croissants. Each day, there's a 60% chance they have chocolate-filled ones, and if they don't, they have plain instead. A group of friends wants to visit over a 3-day weekend and have exactly one day with chocolate-filled croissants. I need to find the pr... | [
{
"from": "human",
"value": "A local bakery has a daily special on croissants, with a 60% chance of having chocolate-filled croissants on any given day. If they don't have chocolate-filled croissants, they will have plain croissants instead. A group of friends wants to visit the bakery on a 3-day weekend an... |
Determine the range of the function $f(x) = \frac{x^2 + 2x + 1}{x^2 - 1}$, where $x$ is a real number. | Determine the range of the function $f(x) = \frac{x^2 + 2x + 1}{x^2 - 1}$, where $x$ is a real number. | Okay, so I need to find the range of the function f(x) = (x² + 2x + 1)/(x² - 1). Hmm, let's start by recalling that the range of a function is the set of all possible output values (y-values) it can produce. To find the range, sometimes it's helpful to express x in terms of y and then determine for which y the equation... | To determine the range of the function \( f(x) = \frac{x^2 + 2x + 1}{x^2 - 1} \), we start by simplifying the function. The numerator \( x^2 + 2x + 1 \) factors to \( (x + 1)^2 \), and the denominator \( x^2 - 1 \) factors to \( (x - 1)(x + 1) \). Simplifying, we get:
\[
f(x) = \frac{(x + 1)^2}{(x - 1)(x + 1)} = \frac... | 7,820 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the range of the function f(x) = (x² + 2x + 1)/(x² - 1). Hmm, let's start by recalling that the range of a function is the set of all possible output values (y-values) it can produce. To find the range, sometimes it's helpful to express x in terms of y and then determine for which y the ... | [
{
"from": "human",
"value": "Determine the range of the function $f(x) = \\frac{x^2 + 2x + 1}{x^2 - 1}$, where $x$ is a real number."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the range of the function f(x) = (x² + 2x + 1)/(x² - 1). Hmm, let's start by recalling that the ran... |
Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three dice show the same value)? | Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three dice show the same value)? | Okay, so I have this probability problem here: when rolling five standard six-sided dice, what's the probability that there's at least one pair but no three-of-a-kind? Hmm, let's see. I need to find the probability that among the five dice, there's at least one pair (so two dice show the same number), but there isn't a... | To find the probability that there is at least one pair but no three-of-a-kind when rolling five standard six-sided dice, we need to consider the favorable outcomes and divide by the total number of possible outcomes.
1. **Total number of possible outcomes**: \(6^5 = 7776\).
2. **Favorable outcomes**:
- **One pair... | 17,227 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this probability problem here: when rolling five standard six-sided dice, what's the probability that there's at least one pair but no three-of-a-kind? Hmm, let's see. I need to find the probability that among the five dice, there's at least one pair (so two dice show the same number), but there... | [
{
"from": "human",
"value": "Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three dice show the same value)?"
},
{
"from": "gpt",
"value": "<think... |
A bakery produces 480 identical bread rolls per day. The baker wants to package the rolls in boxes that hold either 12, 15, or 20 rolls each. What is the smallest number of boxes needed to package all the rolls without leaving any rolls unpackaged? | A bakery produces 480 identical bread rolls per day. The baker wants to package the rolls in boxes that hold either 12, 15, or 20 rolls each. What is the smallest number of boxes needed to package all the rolls without leaving any rolls unpackaged? | Okay, so I need to figure out the smallest number of boxes required to package 480 bread rolls. The boxes can hold either 12, 15, or 20 rolls each. Hmm, let's see. The goal is to use as few boxes as possible. That probably means using more of the larger boxes since they can hold more rolls, right? But I have to make su... | To determine the smallest number of boxes needed to package 480 bread rolls using boxes that hold 12, 15, or 20 rolls each, we start by considering the largest boxes first, as they hold the most rolls and thus minimize the number of boxes.
1. **Using only 20-roll boxes**:
- The number of 20-roll boxes needed is \( ... | 17,146 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the smallest number of boxes required to package 480 bread rolls. The boxes can hold either 12, 15, or 20 rolls each. Hmm, let's see. The goal is to use as few boxes as possible. That probably means using more of the larger boxes since they can hold more rolls, right? But I have to... | [
{
"from": "human",
"value": "A bakery produces 480 identical bread rolls per day. The baker wants to package the rolls in boxes that hold either 12, 15, or 20 rolls each. What is the smallest number of boxes needed to package all the rolls without leaving any rolls unpackaged?"
},
{
"from": "gpt",
... |
Solve for $n$, if $8^n\cdot8^n\cdot8^n=64^3$. | Solve for $n$, if $8^n\cdot8^n\cdot8^n=64^3$. | Okay, so I need to solve for n in the equation 8^n multiplied by itself three times equals 64 cubed. Let me write that out first to make sure I understand the problem correctly. The equation is 8^n * 8^n * 8^n = 64^3. Hmm. Alright, let's break this down step by step.
First, let me recall some exponent rules. When you ... | To solve for \( n \) in the equation \( 8^n \cdot 8^n \cdot 8^n = 64^3 \), we start by simplifying both sides using exponent rules and expressing the bases as powers of 2.
First, we simplify the left side:
\[
8^n \cdot 8^n \cdot 8^n = 8^{n + n + n} = 8^{3n}
\]
Next, we express 8 and 64 as powers of 2:
\[
8 = 2^3 \qua... | 10,810 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve for n in the equation 8^n multiplied by itself three times equals 64 cubed. Let me write that out first to make sure I understand the problem correctly. The equation is 8^n * 8^n * 8^n = 64^3. Hmm. Alright, let's break this down step by step.
First, let me recall some exponent rules. W... | [
{
"from": "human",
"value": "Solve for $n$, if $8^n\\cdot8^n\\cdot8^n=64^3$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve for n in the equation 8^n multiplied by itself three times equals 64 cubed. Let me write that out first to make sure I understand the problem correctly. T... |
If $\begin{vmatrix} e & f \\ g & h \end{vmatrix} = -3,$ then find
\[\begin{vmatrix} e - g & f - h \\ g & h \end{vmatrix}.\] | If $\begin{vmatrix} e & f \\ g & h \end{vmatrix} = -3,$ then find
\[\begin{vmatrix} e - g & f - h \\ g & h \end{vmatrix}.\] | Okay, so I need to find the determinant of this matrix:
\[
\begin{vmatrix} e - g & f - h \\ g & h \end{vmatrix}
\]
And I know that the determinant of the original matrix
\[
\begin{vmatrix} e & f \\ g & h \end{vmatrix} = -3.
\]
Hmm, determinants. Let me remember. The determinant of a 2x2 matrix \(\begin{vmatrix} a ... | Given that the determinant of the matrix \(\begin{vmatrix} e & f \\ g & h \end{vmatrix} = -3\), we need to find the determinant of the matrix \(\begin{vmatrix} e - g & f - h \\ g & h \end{vmatrix}\).
First, recall that the determinant of a \(2 \times 2\) matrix \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) is calcu... | 27,255 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the determinant of this matrix:
\[
\begin{vmatrix} e - g & f - h \\ g & h \end{vmatrix}
\]
And I know that the determinant of the original matrix
\[
\begin{vmatrix} e & f \\ g & h \end{vmatrix} = -3.
\]
Hmm, determinants. Let me remember. The determinant of a 2x2 matrix \(\begin{vma... | [
{
"from": "human",
"value": "If $\\begin{vmatrix} e & f \\\\ g & h \\end{vmatrix} = -3,$ then find\n\\[\\begin{vmatrix} e - g & f - h \\\\ g & h \\end{vmatrix}.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the determinant of this matrix:\n\n\\[\n\\begin{vmatrix} e - g & f -... |
In a library, books are organized on shelves based on the first letter of the author's last name. Each shelf can hold exactly 10 books. If there are 20 shelves, and the books are evenly distributed among the shelves, what is the probability that a randomly chosen book will have an author whose last name starts with the... | In a library, books are organized on shelves based on the first letter of the author's last name. Each shelf can hold exactly 10 books. If there are 20 shelves, and the books are evenly distributed among the shelves, what is the probability that a randomly chosen book will have an author whose last name starts with the... | Okay, let me try to figure out this probability problem. So, the question is about a library where books are organized on shelves based on the first letter of the author's last name. Each shelf can hold exactly 10 books, and there are 20 shelves. The books are evenly distributed among the shelves. We need to find the p... | The library has 20 shelves, each holding exactly 10 books, resulting in a total of \(20 \times 10 = 200\) books. The books are organized on shelves based on the first letter of the author's last name, and they are evenly distributed among the shelves.
Given that there are 20 shelves, each shelf is dedicated to a diff... | 12,271 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this probability problem. So, the question is about a library where books are organized on shelves based on the first letter of the author's last name. Each shelf can hold exactly 10 books, and there are 20 shelves. The books are evenly distributed among the shelves. We need to fi... | [
{
"from": "human",
"value": "In a library, books are organized on shelves based on the first letter of the author's last name. Each shelf can hold exactly 10 books. If there are 20 shelves, and the books are evenly distributed among the shelves, what is the probability that a randomly chosen book will have ... |
What is the coefficient of the $x^2$ term in the expansion of $\left(2x + \dfrac{3}{x}\right)^8$? | What is the coefficient of the $x^2$ term in the expansion of $\left(2x + \dfrac{3}{x}\right)^8$? | Okay, let me try to figure out the coefficient of the \( x^2 \) term in the expansion of \( \left(2x + \dfrac{3}{x}\right)^8 \). Hmm, I remember that when expanding expressions like \( (a + b)^n \), we can use the binomial theorem. The binomial theorem states that each term in the expansion is of the form \( \dbinom{n}... | To find the coefficient of the \( x^2 \) term in the expansion of \( \left(2x + \dfrac{3}{x}\right)^8 \), we use the binomial theorem. The general term in the expansion is given by:
\[
\dbinom{8}{k} (2x)^{8 - k} \left(\dfrac{3}{x}\right)^k
\]
We need the exponent of \( x \) to be 2. The exponent of \( x \) in the ter... | 10,770 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out the coefficient of the \( x^2 \) term in the expansion of \( \left(2x + \dfrac{3}{x}\right)^8 \). Hmm, I remember that when expanding expressions like \( (a + b)^n \), we can use the binomial theorem. The binomial theorem states that each term in the expansion is of the form \( \d... | [
{
"from": "human",
"value": "What is the coefficient of the $x^2$ term in the expansion of $\\left(2x + \\dfrac{3}{x}\\right)^8$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out the coefficient of the \\( x^2 \\) term in the expansion of \\( \\left(2x + \\dfrac{3}{x}\\righ... |
Express the value of the following as a common fraction:
$\left(1-\frac12\right)\cdot\left(1-\frac13\right)\cdot\left(1-\frac14\right) \dotsm \left(1-\frac1{n+1}\right) \dotsm \left(1-\frac1{100}\right)$ | Express the value of the following as a common fraction:
$\left(1-\frac12\right)\cdot\left(1-\frac13\right)\cdot\left(1-\frac14\right) \dotsm \left(1-\frac1{n+1}\right) \dotsm \left(1-\frac1{100}\right)$ | Okay, so I need to find the value of this product: starting from (1 - 1/2) times (1 - 1/3) times (1 - 1/4) all the way up to (1 - 1/100). And express it as a common fraction. Hmm, let me think step by step.
First, let me write down the first few terms to see if there's a pattern. The first term is (1 - 1/2) which is 1... | The given product is \(\left(1-\frac{1}{2}\right)\cdot\left(1-\frac{1}{3}\right)\cdot\left(1-\frac{1}{4}\right) \dotsm \left(1-\frac{1}{100}\right)\).
First, we express each term as a fraction:
\[
\left(1 - \frac{1}{2}\right) = \frac{1}{2}, \quad \left(1 - \frac{1}{3}\right) = \frac{2}{3}, \quad \left(1 - \frac{1}{4}\... | 9,016 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of this product: starting from (1 - 1/2) times (1 - 1/3) times (1 - 1/4) all the way up to (1 - 1/100). And express it as a common fraction. Hmm, let me think step by step.
First, let me write down the first few terms to see if there's a pattern. The first term is (1 - 1/2) wh... | [
{
"from": "human",
"value": "Express the value of the following as a common fraction:\n$\\left(1-\\frac12\\right)\\cdot\\left(1-\\frac13\\right)\\cdot\\left(1-\\frac14\\right) \\dotsm \\left(1-\\frac1{n+1}\\right) \\dotsm \\left(1-\\frac1{100}\\right)$"
},
{
"from": "gpt",
"value": "<think>\nOka... |
Three different switches control three light bulbs in a room. Each switch corresponds to one of the light bulbs, but none of them are labelled. You can turn the lights on and off as many times as you want, but you can only enter the room once to observe the bulbs. How many minimum number of times do you need to flip th... | Three different switches control three light bulbs in a room. Each switch corresponds to one of the light bulbs, but none of them are labelled. You can turn the lights on and off as many times as you want, but you can only enter the room once to observe the bulbs. How many minimum number of times do you need to flip th... | Okay, so I have this problem where there are three switches outside a room that control three light bulbs inside the room. Each switch corresponds to one bulb, but they aren't labeled. I need to figure out which switch controls which bulb. The catch is, I can only enter the room once to check the bulbs, but I can flip ... | To determine which switch controls each light bulb with only one entry into the room, we need to use a strategy that encodes information about the switches into the state of the bulbs (on or off). The key steps involve flipping switches in a specific sequence to create a unique pattern of bulb states.
1. **Flip Switch... | 16,815 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem where there are three switches outside a room that control three light bulbs inside the room. Each switch corresponds to one bulb, but they aren't labeled. I need to figure out which switch controls which bulb. The catch is, I can only enter the room once to check the bulbs, but I c... | [
{
"from": "human",
"value": "Three different switches control three light bulbs in a room. Each switch corresponds to one of the light bulbs, but none of them are labelled. You can turn the lights on and off as many times as you want, but you can only enter the room once to observe the bulbs. How many minim... |
Find the smallest prime number $n$ such that the sum of its digits is equal to the number of its digits plus 2. | Find the smallest prime number $n$ such that the sum of its digits is equal to the number of its digits plus 2. | Okay, so I need to find the smallest prime number n where the sum of its digits is equal to the number of digits plus 2. Let me break this down step by step.
First, let me make sure I understand the problem correctly. We have a prime number n. Let S(n) be the sum of its digits, and D(n) be the number of digits in n. T... | To find the smallest prime number \( n \) such that the sum of its digits is equal to the number of its digits plus 2, we start by checking the condition for single-digit primes first.
For a single-digit prime, the number of digits \( D(n) \) is 1. Therefore, the sum of the digits \( S(n) \) should be \( 1 + 2 = 3 \)... | 31,431 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest prime number n where the sum of its digits is equal to the number of digits plus 2. Let me break this down step by step.
First, let me make sure I understand the problem correctly. We have a prime number n. Let S(n) be the sum of its digits, and D(n) be the number of digits... | [
{
"from": "human",
"value": "Find the smallest prime number $n$ such that the sum of its digits is equal to the number of its digits plus 2."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest prime number n where the sum of its digits is equal to the number of digits pl... |
Let $p$ and $q$ be two positive integers such that $p^2 - q^2 = 20$. Find the smallest possible value of $p + q$. | Let $p$ and $q$ be two positive integers such that $p^2 - q^2 = 20$. Find the smallest possible value of $p + q$. | Okay, so I need to find the smallest possible value of p + q where p and q are positive integers satisfying p² - q² = 20. Hmm, let's start by recalling that p² - q² can be factored as (p - q)(p + q). So, that equation becomes (p - q)(p + q) = 20.
Since p and q are positive integers, both (p - q) and (p + q) must also... | Given two positive integers \( p \) and \( q \) such that \( p^2 - q^2 = 20 \), we need to find the smallest possible value of \( p + q \).
First, we factor the equation as a difference of squares:
\[
p^2 - q^2 = (p - q)(p + q) = 20
\]
We need to find pairs of factors \((a, b)\) of 20 such that \( a = p - q \) and \( ... | 14,568 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest possible value of p + q where p and q are positive integers satisfying p² - q² = 20. Hmm, let's start by recalling that p² - q² can be factored as (p - q)(p + q). So, that equation becomes (p - q)(p + q) = 20.
Since p and q are positive integers, both (p - q) and (p + q) m... | [
{
"from": "human",
"value": "Let $p$ and $q$ be two positive integers such that $p^2 - q^2 = 20$. Find the smallest possible value of $p + q$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest possible value of p + q where p and q are positive integers satisfying p² -... |
The sequence $(a_n)$ is defined by $a_1 = 3$ and $a_k = a_{k - 1}^3 - 2a_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{a_1 - 1} + \frac{1}{a_2 - 1} + \frac{1}{a_3 - 1} + \dotsb.\] | The sequence $(a_n)$ is defined by $a_1 = 3$ and $a_k = a_{k - 1}^3 - 2a_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{a_1 - 1} + \frac{1}{a_2 - 1} + \frac{1}{a_3 - 1} + \dotsb.\] | Alright, so I need to compute the sum of the series 1/(a₁ - 1) + 1/(a₂ - 1) + 1/(a₃ - 1) + ... where the sequence (aₙ) is defined by a₁ = 3 and the recursive formula aₖ = aₖ₋₁³ - 2aₖ₋₁ for k ≥ 2. Hmm, okay. Let me think about how to approach this.
First, let me write down the first few terms of the sequence to see if ... | The sequence \((a_n)\) is defined by \(a_1 = 3\) and \(a_k = a_{k-1}^3 - 2a_{k-1}\) for all \(k \ge 2\). We need to compute the sum \(\frac{1}{a_1 - 1} + \frac{1}{a_2 - 1} + \frac{1}{a_3 - 1} + \dotsb\).
First, we compute the first few terms of the sequence:
- \(a_1 = 3\)
- \(a_2 = 3^3 - 2 \cdot 3 = 27 - 6 = 21\)
- \(... | 25,056 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to compute the sum of the series 1/(a₁ - 1) + 1/(a₂ - 1) + 1/(a₃ - 1) + ... where the sequence (aₙ) is defined by a₁ = 3 and the recursive formula aₖ = aₖ₋₁³ - 2aₖ₋₁ for k ≥ 2. Hmm, okay. Let me think about how to approach this.
First, let me write down the first few terms of the sequence to... | [
{
"from": "human",
"value": "The sequence $(a_n)$ is defined by $a_1 = 3$ and $a_k = a_{k - 1}^3 - 2a_{k - 1}$ for all $k \\ge 2.$ Compute\n\\[\\frac{1}{a_1 - 1} + \\frac{1}{a_2 - 1} + \\frac{1}{a_3 - 1} + \\dotsb.\\]"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to compute the sum... |
A right cylinder has a base with a radius of 8 inches and a height of 25 inches. The radius of this cylinder is reduced while the height stays the same. The volume of the smaller cylinder is $300\pi$ cubic inches. What is the ratio of the smaller radius to the original radius? Express your answer as a common fraction. | A right cylinder has a base with a radius of 8 inches and a height of 25 inches. The radius of this cylinder is reduced while the height stays the same. The volume of the smaller cylinder is $300\pi$ cubic inches. What is the ratio of the smaller radius to the original radius? Express your answer as a common fraction. | Okay, so I have this problem here about a right cylinder. The original cylinder has a radius of 8 inches and a height of 25 inches. Then they reduce the radius while keeping the height the same, and the volume of the smaller cylinder is 300π cubic inches. I need to find the ratio of the smaller radius to the original r... | The original cylinder has a radius of 8 inches and a height of 25 inches. The volume of the original cylinder is calculated as follows:
\[
V = \pi r^2 h = \pi (8)^2 (25) = 1600\pi \text{ cubic inches}
\]
For the smaller cylinder with the same height of 25 inches and a volume of \(300\pi\) cubic inches, we denote the s... | 1,178 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here about a right cylinder. The original cylinder has a radius of 8 inches and a height of 25 inches. Then they reduce the radius while keeping the height the same, and the volume of the smaller cylinder is 300π cubic inches. I need to find the ratio of the smaller radius to the or... | [
{
"from": "human",
"value": "A right cylinder has a base with a radius of 8 inches and a height of 25 inches. The radius of this cylinder is reduced while the height stays the same. The volume of the smaller cylinder is $300\\pi$ cubic inches. What is the ratio of the smaller radius to the original radius? ... |
Find the largest positive integer x which is less than 10 and relatively prime to 48.
To solve this problem, you will need to use the Euclidean algorithm or another method to find the GCD of x and 48, and determine which values of x are relatively prime to 48. This problem requires a different approach than the origin... | Find the largest positive integer x which is less than 10 and relatively prime to 48.
To solve this problem, you will need to use the Euclidean algorithm or another method to find the GCD of x and 48, and determine which values of x are relatively prime to 48. This problem requires a different approach than the origin... | Okay, so I need to find the largest positive integer x that's less than 10 and relatively prime to 48. Hmm, let me start by understanding what "relatively prime" means. If two numbers are relatively prime, their greatest common divisor (GCD) is 1. So, I need to check numbers less than 10 and see which ones have a GCD o... | To find the largest positive integer \( x \) which is less than 10 and relatively prime to 48, we need to check the greatest common divisor (GCD) of each number from 9 down to 1 with 48. A number is relatively prime to 48 if its GCD with 48 is 1.
1. **Check GCD(48, 9)**:
- Using the Euclidean algorithm: \( 48 \div ... | 7,470 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the largest positive integer x that's less than 10 and relatively prime to 48. Hmm, let me start by understanding what "relatively prime" means. If two numbers are relatively prime, their greatest common divisor (GCD) is 1. So, I need to check numbers less than 10 and see which ones have... | [
{
"from": "human",
"value": "Find the largest positive integer x which is less than 10 and relatively prime to 48.\n\nTo solve this problem, you will need to use the Euclidean algorithm or another method to find the GCD of x and 48, and determine which values of x are relatively prime to 48. This problem re... |
Consider the rational function \[f(x)=\frac{2x^2+Ax+B}{x^3-3x^2-4x+12}.\]Given that \[f(x)=\frac{2}{x-3}+\frac{2}{x+2}+\frac{1}{x-2},\]find $A+B$. | Consider the rational function \[f(x)=\frac{2x^2+Ax+B}{x^3-3x^2-4x+12}.\]Given that \[f(x)=\frac{2}{x-3}+\frac{2}{x+2}+\frac{1}{x-2},\]find $A+B$. | Okay, so I need to find A and B in the numerator of the rational function f(x) such that when expressed as a single fraction, it equals the given partial fractions. Then, sum A and B to get the answer. Let me start by recalling that partial fractions decomposition is the process of breaking down a complex rational func... | Given the rational function \( f(x) = \frac{2x^2 + Ax + B}{x^3 - 3x^2 - 4x + 12} \) and its partial fraction decomposition \( f(x) = \frac{2}{x-3} + \frac{2}{x+2} + \frac{1}{x-2} \), we need to find \( A + B \).
First, we factor the denominator of the original function:
\[ x^3 - 3x^2 - 4x + 12 = (x - 3)(x - 2)(x + 2) ... | 18,803 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find A and B in the numerator of the rational function f(x) such that when expressed as a single fraction, it equals the given partial fractions. Then, sum A and B to get the answer. Let me start by recalling that partial fractions decomposition is the process of breaking down a complex ratio... | [
{
"from": "human",
"value": "Consider the rational function \\[f(x)=\\frac{2x^2+Ax+B}{x^3-3x^2-4x+12}.\\]Given that \\[f(x)=\\frac{2}{x-3}+\\frac{2}{x+2}+\\frac{1}{x-2},\\]find $A+B$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find A and B in the numerator of the rational functio... |
Triangle $PQR$ is inscribed in a sphere with centre $O$. If $\angle PQO$ is right and $PQ=7$, find the volume of the sphere. | Triangle $PQR$ is inscribed in a sphere with centre $O$. If $\angle PQO$ is right and $PQ=7$, find the volume of the sphere. | Alright, so I need to find the volume of a sphere in which triangle PQR is inscribed. The problem states that angle PQO is a right angle, and PQ is 7 units long. The center of the sphere is O. Hmm, okay, let me think through this step by step.
First, since the sphere's center is O, and triangle PQR is inscribed in the... | Given triangle \(PQR\) inscribed in a sphere with center \(O\), where \(\angle PQO\) is a right angle and \(PQ = 7\), we need to find the volume of the sphere.
1. **Identify the problem constraints**:
- Triangle \(PQR\) is inscribed in the sphere, meaning all vertices \(P\), \(Q\), and \(R\) lie on the sphere's sur... | 21,030 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to find the volume of a sphere in which triangle PQR is inscribed. The problem states that angle PQO is a right angle, and PQ is 7 units long. The center of the sphere is O. Hmm, okay, let me think through this step by step.
First, since the sphere's center is O, and triangle PQR is inscribe... | [
{
"from": "human",
"value": "Triangle $PQR$ is inscribed in a sphere with centre $O$. If $\\angle PQO$ is right and $PQ=7$, find the volume of the sphere."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to find the volume of a sphere in which triangle PQR is inscribed. The problem sta... |
What is the greatest common divisor of $4455$ and $2471$? | What is the greatest common divisor of $4455$ and $2471$? | Okay, so I need to find the greatest common divisor (GCD) of 4455 and 2471. Hmm, GCD, right? That's the largest number that divides both numbers without leaving a remainder. I remember there are a few methods to find the GCD. The most efficient one is the Euclidean algorithm, right? Let me try to recall how that works.... | To find the greatest common divisor (GCD) of 4455 and 2471 using the Euclidean algorithm, we follow these steps:
1. Divide 4455 by 2471, which gives a quotient of 1 and a remainder of 1984.
\[
4455 = 2471 \times 1 + 1984
\]
Now, we need to find GCD(2471, 1984).
2. Divide 2471 by 1984, which gives a quotie... | 17,768 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the greatest common divisor (GCD) of 4455 and 2471. Hmm, GCD, right? That's the largest number that divides both numbers without leaving a remainder. I remember there are a few methods to find the GCD. The most efficient one is the Euclidean algorithm, right? Let me try to recall how tha... | [
{
"from": "human",
"value": "What is the greatest common divisor of $4455$ and $2471$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the greatest common divisor (GCD) of 4455 and 2471. Hmm, GCD, right? That's the largest number that divides both numbers without leaving a remain... |
Let's create a new math problem inspired by the given one.
Problem:
A bakery sells two types of cakes: chocolate and vanilla. The number of chocolate cakes sold is 20% more than the number of vanilla cakes sold. If the total number of cakes sold is 120, how many vanilla cakes were sold?
This problem requires a differ... | Let's create a new math problem inspired by the given one.
Problem:
A bakery sells two types of cakes: chocolate and vanilla. The number of chocolate cakes sold is 20% more than the number of vanilla cakes sold. If the total number of cakes sold is 120, how many vanilla cakes were sold?
This problem requires a differ... | Okay, let's try to figure out this bakery problem. So, they sell chocolate and vanilla cakes. The chocolate cakes sold are 20% more than vanilla. Total sold is 120. Need to find how many vanilla cakes were sold.
Hmm, let's start by understanding what 20% more means. If vanilla cakes are a certain number, then chocolat... | **Problem:**
A sports store sells basketballs and soccer balls. The number of basketballs sold is 50% more than the number of soccer balls sold. If the total number of balls sold is 200, how many soccer balls were sold?
**Solution:**
Let \( S \) represent the number of soccer balls sold. Since basketballs sold are... | 20,580 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to figure out this bakery problem. So, they sell chocolate and vanilla cakes. The chocolate cakes sold are 20% more than vanilla. Total sold is 120. Need to find how many vanilla cakes were sold.
Hmm, let's start by understanding what 20% more means. If vanilla cakes are a certain number, then ... | [
{
"from": "human",
"value": "Let's create a new math problem inspired by the given one.\n\nProblem:\nA bakery sells two types of cakes: chocolate and vanilla. The number of chocolate cakes sold is 20% more than the number of vanilla cakes sold. If the total number of cakes sold is 120, how many vanilla cake... |
A book is on sale for 15% off its original price. A sales tax of 8% will be added to the sale price. If the final price of the book is $135.50, what was the original price of the book? | A book is on sale for 15% off its original price. A sales tax of 8% will be added to the sale price. If the final price of the book is $135.50, what was the original price of the book? | Okay, so I need to figure out the original price of the book. The problem says that the book is on sale for 15% off its original price, and then there's an 8% sales tax added to the sale price. The final price after the discount and tax is $135.50. Hmm, let me think about how to reverse-engineer this.
First, let's rec... | Let \( P \) be the original price of the book. The book is on sale for 15% off, so the sale price is \( 0.85P \). An 8% sales tax is then added to the sale price, resulting in the final price of \( 0.85P \times 1.08 \). Given that the final price is $135.50, we set up the equation:
\[
135.50 = P \times 0.85 \times 1.0... | 6,388 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the original price of the book. The problem says that the book is on sale for 15% off its original price, and then there's an 8% sales tax added to the sale price. The final price after the discount and tax is $135.50. Hmm, let me think about how to reverse-engineer this.
First, l... | [
{
"from": "human",
"value": "A book is on sale for 15% off its original price. A sales tax of 8% will be added to the sale price. If the final price of the book is $135.50, what was the original price of the book?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the original... |
Suppose we roll three fair dice simultaneously: a red die, a green die, and a blue die. What is the probability that the red and green dice both show the same number? | Suppose we roll three fair dice simultaneously: a red die, a green die, and a blue die. What is the probability that the red and green dice both show the same number? | Okay, so I need to find the probability that when I roll three dice—red, green, and blue—the red and green ones show the same number. Hmm, let's think. First, probability is generally favorable outcomes over total possible outcomes. So maybe I should figure out how many total possible outcomes there are when rolling th... | To find the probability that the red and green dice both show the same number when rolling three fair dice (red, green, and blue), we start by determining the total number of possible outcomes. Each die has 6 faces, so the total number of outcomes is:
\[
6 \times 6 \times 6 = 216
\]
Next, we calculate the number of f... | 9,287 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that when I roll three dice—red, green, and blue—the red and green ones show the same number. Hmm, let's think. First, probability is generally favorable outcomes over total possible outcomes. So maybe I should figure out how many total possible outcomes there are when ro... | [
{
"from": "human",
"value": "Suppose we roll three fair dice simultaneously: a red die, a green die, and a blue die. What is the probability that the red and green dice both show the same number?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the probability that when I roll thr... |
Determine the smallest positive integer x such that 7x divided by 1000 leaves a remainder of 1. | Determine the smallest positive integer x such that 7x divided by 1000 leaves a remainder of 1. | Okay, so I need to find the smallest positive integer x such that when 7x is divided by 1000, the remainder is 1. Hmm, let's see. That means 7x ≡ 1 mod 1000. Right? So, I have to solve this congruence equation for x.
I remember that solving an equation like 7x ≡ 1 mod 1000 is about finding the multiplicative inverse ... | To determine the smallest positive integer \( x \) such that \( 7x \) divided by 1000 leaves a remainder of 1, we need to solve the congruence equation \( 7x \equiv 1 \pmod{1000} \). This is equivalent to finding the multiplicative inverse of 7 modulo 1000.
Using the Extended Euclidean Algorithm, we start by dividing ... | 25,790 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest positive integer x such that when 7x is divided by 1000, the remainder is 1. Hmm, let's see. That means 7x ≡ 1 mod 1000. Right? So, I have to solve this congruence equation for x.
I remember that solving an equation like 7x ≡ 1 mod 1000 is about finding the multiplicative ... | [
{
"from": "human",
"value": "Determine the smallest positive integer x such that 7x divided by 1000 leaves a remainder of 1."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest positive integer x such that when 7x is divided by 1000, the remainder is 1. Hmm, let's see. T... |
How many ways can you arrange 4 red marbles, 2 blue marbles, and 2 green marbles into a single row of 8 marbles, if the marbles of the same color are indistinguishable? | How many ways can you arrange 4 red marbles, 2 blue marbles, and 2 green marbles into a single row of 8 marbles, if the marbles of the same color are indistinguishable? | Okay, so I need to figure out how many different ways I can arrange these marbles in a row. There are 4 red marbles, 2 blue marbles, and 2 green marbles. All the marbles of the same color are indistinct, meaning I can't tell them apart. Hmm, this seems like a permutations problem with repeated elements. Let me recall t... | To determine the number of ways to arrange 4 red marbles, 2 blue marbles, and 2 green marbles into a single row of 8 marbles, we use the formula for permutations of multiset:
$$
\frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}
$$
where \( n \) is the total number of items, and \( n_1, n_2, \ldots, n_k \) are the co... | 35,974 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many different ways I can arrange these marbles in a row. There are 4 red marbles, 2 blue marbles, and 2 green marbles. All the marbles of the same color are indistinct, meaning I can't tell them apart. Hmm, this seems like a permutations problem with repeated elements. Let me ... | [
{
"from": "human",
"value": "How many ways can you arrange 4 red marbles, 2 blue marbles, and 2 green marbles into a single row of 8 marbles, if the marbles of the same color are indistinguishable?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many different ways I ca... |
In cylindrical coordinates $(r,\theta,z)$, find the shape described by the equation
$$z=r^2\cos 2\theta.$$
What is the shape described by this equation? | In cylindrical coordinates $(r,\theta,z)$, find the shape described by the equation
$$z=r^2\cos 2\theta.$$
What is the shape described by this equation? | Okay, so I need to figure out what shape is described by the equation z = r² cos 2θ in cylindrical coordinates. Hmm, cylindrical coordinates... right, that's (r, θ, z), where r is the radius in the xy-plane, θ is the angle, and z is the height.
First, let me recall that in cylindrical coordinates, there are conversio... | To determine the shape described by the equation \( z = r^2 \cos 2\theta \) in cylindrical coordinates, we start by converting the equation to Cartesian coordinates.
First, recall the double-angle identity for cosine:
\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]
Substituting \(\cos \theta = \frac{x}{r}\) and \... | 21,951 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out what shape is described by the equation z = r² cos 2θ in cylindrical coordinates. Hmm, cylindrical coordinates... right, that's (r, θ, z), where r is the radius in the xy-plane, θ is the angle, and z is the height.
First, let me recall that in cylindrical coordinates, there are c... | [
{
"from": "human",
"value": "In cylindrical coordinates $(r,\\theta,z)$, find the shape described by the equation\n\n$$z=r^2\\cos 2\\theta.$$\n\nWhat is the shape described by this equation?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out what shape is described by the equa... |
Find the value of x such that the arithmetic mean of x^2, 2x, and 7 is equal to the geometric mean of 4, 6, and 9. | Find the value of x such that the arithmetic mean of x^2, 2x, and 7 is equal to the geometric mean of 4, 6, and 9. | Okay, let me try to solve this problem. The question says: Find the value of x such that the arithmetic mean of x², 2x, and 7 is equal to the geometric mean of 4, 6, and 9.
First, I need to recall what arithmetic mean (AM) and geometric mean (GM) are. The arithmetic mean of a set of numbers is the sum of the numbers d... | To find the value of \( x \) such that the arithmetic mean of \( x^2 \), \( 2x \), and 7 is equal to the geometric mean of 4, 6, and 9, we start by calculating the arithmetic mean (AM) and geometric mean (GM).
The arithmetic mean of \( x^2 \), \( 2x \), and 7 is:
\[
\frac{x^2 + 2x + 7}{3}
\]
The geometric mean of 4, ... | 39,380 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to solve this problem. The question says: Find the value of x such that the arithmetic mean of x², 2x, and 7 is equal to the geometric mean of 4, 6, and 9.
First, I need to recall what arithmetic mean (AM) and geometric mean (GM) are. The arithmetic mean of a set of numbers is the sum of the n... | [
{
"from": "human",
"value": "Find the value of x such that the arithmetic mean of x^2, 2x, and 7 is equal to the geometric mean of 4, 6, and 9."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to solve this problem. The question says: Find the value of x such that the arithmetic mean of ... |
A quadratic equation $2x^2 + 5x + c = 0$ has two real roots. What is the sum of these two roots? | A quadratic equation $2x^2 + 5x + c = 0$ has two real roots. What is the sum of these two roots? | Okay, so I need to find the sum of the two real roots of the quadratic equation \(2x^2 + 5x + c = 0\). Hmm, let me remember how to approach this.
First off, quadratic equations are of the form \(ax^2 + bx + c = 0\), right? And I recall there's something called Vieta's formulas that relates the coefficients of the equ... | To find the sum of the two real roots of the quadratic equation \(2x^2 + 5x + c = 0\), we use Vieta's formulas. For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots is given by \(-b/a\).
In the given equation, the coefficients are \(a = 2\), \(b = 5\), and \(c = c\). According to Vieta's formula, the s... | 17,703 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the sum of the two real roots of the quadratic equation \(2x^2 + 5x + c = 0\). Hmm, let me remember how to approach this.
First off, quadratic equations are of the form \(ax^2 + bx + c = 0\), right? And I recall there's something called Vieta's formulas that relates the coefficients of... | [
{
"from": "human",
"value": "A quadratic equation $2x^2 + 5x + c = 0$ has two real roots. What is the sum of these two roots?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of the two real roots of the quadratic equation \\(2x^2 + 5x + c = 0\\). Hmm, let me remember how ... |
Tom's weight is $25\%$ more than twice the weight of his younger brother Alex. If Tom's weight is 65 kg, what was Alex's weight 2 years ago if we know that Alex's weight increases by 5 kg every year? | Tom's weight is $25\%$ more than twice the weight of his younger brother Alex. If Tom's weight is 65 kg, what was Alex's weight 2 years ago if we know that Alex's weight increases by 5 kg every year? | Okay, let me try to solve this problem step by step. First, the question is about Tom and his younger brother Alex's weights. The problem states that Tom's weight is 25% more than twice Alex's weight. It also tells us that Tom currently weighs 65 kg. We need to find Alex's weight two years ago, given that Alex's weight... | Given that Tom's weight is 25% more than twice the weight of his younger brother Alex, and Tom's weight is 65 kg, we need to find Alex's weight two years ago. Alex's weight increases by 5 kg every year.
1. Let Alex's current weight be \( A \) kg.
2. Tom's weight is 25% more than twice Alex's weight, which can be expre... | 19,431 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to solve this problem step by step. First, the question is about Tom and his younger brother Alex's weights. The problem states that Tom's weight is 25% more than twice Alex's weight. It also tells us that Tom currently weighs 65 kg. We need to find Alex's weight two years ago, given that Alex'... | [
{
"from": "human",
"value": "Tom's weight is $25\\%$ more than twice the weight of his younger brother Alex. If Tom's weight is 65 kg, what was Alex's weight 2 years ago if we know that Alex's weight increases by 5 kg every year?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to solve ... |
A quadratic equation $x^2 + bx + c = 0$ has roots whose product is equal to the sum of their squares. If the sum of the roots is equal to $-3$, find the value of $b + c$. | A quadratic equation $x^2 + bx + c = 0$ has roots whose product is equal to the sum of their squares. If the sum of the roots is equal to $-3$, find the value of $b + c$. | Okay, so I have this quadratic equation: \(x^2 + bx + c = 0\). The problem says that the product of the roots is equal to the sum of their squares. Also, the sum of the roots is \(-3\). I need to find the value of \(b + c\). Hmm, let me think about how to approach this.
First, I remember that for a quadratic equation ... | Given the quadratic equation \(x^2 + bx + c = 0\) with roots whose product is equal to the sum of their squares and the sum of the roots is \(-3\), we need to find the value of \(b + c\).
First, we use the fact that the sum of the roots of the quadratic equation \(x^2 + bx + c = 0\) is given by \(-b\). Since the probl... | 16,349 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this quadratic equation: \(x^2 + bx + c = 0\). The problem says that the product of the roots is equal to the sum of their squares. Also, the sum of the roots is \(-3\). I need to find the value of \(b + c\). Hmm, let me think about how to approach this.
First, I remember that for a quadratic e... | [
{
"from": "human",
"value": "A quadratic equation $x^2 + bx + c = 0$ has roots whose product is equal to the sum of their squares. If the sum of the roots is equal to $-3$, find the value of $b + c$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this quadratic equation: \\(x^2 + bx + c... |
Compute the exact value of the expression $|\sqrt{\pi} - |\sqrt{\pi} - e||$. Write your answer using only integers, $\pi$, and $e$, without any absolute value signs. | Compute the exact value of the expression $|\sqrt{\pi} - |\sqrt{\pi} - e||$. Write your answer using only integers, $\pi$, and $e$, without any absolute value signs. | Okay, so I need to compute the exact value of the expression |√π - |√π - e||. Hmm, let's break this down step by step. First, there are two absolute value signs here, one outside and one inside. The expression inside the outer absolute value is √π minus the absolute value of √π minus e. So, maybe I should start by eval... | To compute the exact value of the expression \( |\sqrt{\pi} - |\sqrt{\pi} - e|| \), we start by evaluating the inner absolute value \( |\sqrt{\pi} - e| \).
Since \( e \approx 2.718 \) is greater than \( \sqrt{\pi} \approx 1.772 \), we have:
\[
|\sqrt{\pi} - e| = e - \sqrt{\pi}
\]
Next, we substitute this result back ... | 39,102 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to compute the exact value of the expression |√π - |√π - e||. Hmm, let's break this down step by step. First, there are two absolute value signs here, one outside and one inside. The expression inside the outer absolute value is √π minus the absolute value of √π minus e. So, maybe I should start... | [
{
"from": "human",
"value": "Compute the exact value of the expression $|\\sqrt{\\pi} - |\\sqrt{\\pi} - e||$. Write your answer using only integers, $\\pi$, and $e$, without any absolute value signs."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to compute the exact value of the expres... |
Convert the point $\left( 4, \frac{\pi}{6}, 2 \right)$ in cylindrical coordinates to rectangular coordinates. | Convert the point $\left( 4, \frac{\pi}{6}, 2 \right)$ in cylindrical coordinates to rectangular coordinates. | Okay, so I need to convert the cylindrical coordinates (4, π/6, 2) to rectangular coordinates. Hmm, let me remember how that conversion works. Cylindrical coordinates are given as (r, θ, z), right? And rectangular coordinates are (x, y, z). So, the z-coordinate stays the same, so z is 2. That part is easy. Now, for the... | To convert the cylindrical coordinates \(\left( 4, \frac{\pi}{6}, 2 \right)\) to rectangular coordinates, we use the conversion formulas \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z\) remains the same.
1. **Calculate \(x\):**
\[
x = 4 \cos \left( \frac{\pi}{6} \right) = 4 \times \frac{\sqrt{3}}{2} = 2\s... | 11,623 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to convert the cylindrical coordinates (4, π/6, 2) to rectangular coordinates. Hmm, let me remember how that conversion works. Cylindrical coordinates are given as (r, θ, z), right? And rectangular coordinates are (x, y, z). So, the z-coordinate stays the same, so z is 2. That part is easy. Now,... | [
{
"from": "human",
"value": "Convert the point $\\left( 4, \\frac{\\pi}{6}, 2 \\right)$ in cylindrical coordinates to rectangular coordinates."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to convert the cylindrical coordinates (4, π/6, 2) to rectangular coordinates. Hmm, let me rememb... |
Evaluate
\[i^{14762} + i^{14763} + i^{14764} + i^{14765}.\] | Evaluate
\[i^{14762} + i^{14763} + i^{14764} + i^{14765}.\] | Okay, so I need to evaluate the sum \(i^{14762} + i^{14763} + i^{14764} + i^{14765}\). Hmm, let me think. I remember that the powers of \(i\) cycle every four exponents. So \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then \(i^5 = i\) again, right? So this cycle repeats every 4. That means if I can find th... | To evaluate the sum \(i^{14762} + i^{14763} + i^{14764} + i^{14765}\), we use the cyclical nature of the powers of \(i\), which repeat every 4 terms: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then the cycle repeats.
First, we find the remainders when each exponent is divided by 4:
- \(14762 \div 4\) le... | 24,227 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to evaluate the sum \(i^{14762} + i^{14763} + i^{14764} + i^{14765}\). Hmm, let me think. I remember that the powers of \(i\) cycle every four exponents. So \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then \(i^5 = i\) again, right? So this cycle repeats every 4. That means if I can... | [
{
"from": "human",
"value": "Evaluate\n\\[i^{14762} + i^{14763} + i^{14764} + i^{14765}.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to evaluate the sum \\(i^{14762} + i^{14763} + i^{14764} + i^{14765}\\). Hmm, let me think. I remember that the powers of \\(i\\) cycle every four e... |
Find all solutions to
\[x^3 + 2x^2 - 5x - 2 \sqrt{x^2 + 1} = 0.\]
Enter all the solutions, separated by commas. | Find all solutions to
\[x^3 + 2x^2 - 5x - 2 \sqrt{x^2 + 1} = 0.\]
Enter all the solutions, separated by commas. | Okay, so I have this equation to solve: \(x^3 + 2x^2 - 5x - 2 \sqrt{x^2 + 1} = 0\). Hmm, let's see. First, I need to find all real solutions. Since there's a square root term, \(\sqrt{x^2 + 1}\), I know that the domain is all real numbers because \(x^2 + 1\) is always non-negative. So, no restrictions there. But dealin... | To solve the equation \(x^3 + 2x^2 - 5x - 2 \sqrt{x^2 + 1} = 0\), we start by isolating the square root term:
\[x^3 + 2x^2 - 5x = 2 \sqrt{x^2 + 1}\]
Next, we square both sides to eliminate the square root:
\[(x^3 + 2x^2 - 5x)^2 = (2 \sqrt{x^2 + 1})^2\]
Expanding both sides, we get:
\[(x^3 + 2x^2 - 5x)^2 = 4(x^2 + ... | 2,340 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this equation to solve: \(x^3 + 2x^2 - 5x - 2 \sqrt{x^2 + 1} = 0\). Hmm, let's see. First, I need to find all real solutions. Since there's a square root term, \(\sqrt{x^2 + 1}\), I know that the domain is all real numbers because \(x^2 + 1\) is always non-negative. So, no restrictions there. Bu... | [
{
"from": "human",
"value": "Find all solutions to\n\\[x^3 + 2x^2 - 5x - 2 \\sqrt{x^2 + 1} = 0.\\]\n\nEnter all the solutions, separated by commas."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this equation to solve: \\(x^3 + 2x^2 - 5x - 2 \\sqrt{x^2 + 1} = 0\\). Hmm, let's see. First... |
A regular icosahedron has 20 triangular faces, and each face is painted with one of three colors: red, blue, or green. Each face is equally likely to be painted with any of the three colors, and the colors of the faces are determined independently. What is the probability that the painted icosahedron has at least one p... | A regular icosahedron has 20 triangular faces, and each face is painted with one of three colors: red, blue, or green. Each face is equally likely to be painted with any of the three colors, and the colors of the faces are determined independently. What is the probability that the painted icosahedron has at least one p... | Okay, so I need to find the probability that a randomly painted regular icosahedron has at least one pair of adjacent faces with the same color. Let me start by understanding the problem better.
First, a regular icosahedron has 20 triangular faces, each of which is an equilateral triangle. Each face is adjacent to thr... | To find the probability that a randomly painted regular icosahedron has at least one pair of adjacent faces with the same color, we start by considering the complementary event where all adjacent faces have different colors.
A regular icosahedron has 20 triangular faces, each adjacent to three others. Each face is pa... | 37,148 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that a randomly painted regular icosahedron has at least one pair of adjacent faces with the same color. Let me start by understanding the problem better.
First, a regular icosahedron has 20 triangular faces, each of which is an equilateral triangle. Each face is adjacen... | [
{
"from": "human",
"value": "A regular icosahedron has 20 triangular faces, and each face is painted with one of three colors: red, blue, or green. Each face is equally likely to be painted with any of the three colors, and the colors of the faces are determined independently. What is the probability that t... |
What is the height of a building, in meters, if it casts a shadow 75 meters long at the same time a 15-meter statue casts a shadow that is 40 meters long? Express your answer to the nearest whole number. | What is the height of a building, in meters, if it casts a shadow 75 meters long at the same time a 15-meter statue casts a shadow that is 40 meters long? Express your answer to the nearest whole number. | Okay, let's see. So, there's a building that casts a shadow 75 meters long. At the same time, a 15-meter statue casts a shadow that's 40 meters long. I need to find the height of the building. Hmm, sounds like similar triangles or something with proportions.
Alright, so when objects cast shadows at the same time of d... | To find the height of the building, we use the concept of similar triangles. The ratio of the height of an object to the length of its shadow is the same for both the statue and the building.
Given:
- The statue is 15 meters tall and casts a shadow of 40 meters.
- The building casts a shadow of 75 meters.
We set up ... | 22,418 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. So, there's a building that casts a shadow 75 meters long. At the same time, a 15-meter statue casts a shadow that's 40 meters long. I need to find the height of the building. Hmm, sounds like similar triangles or something with proportions.
Alright, so when objects cast shadows at the same t... | [
{
"from": "human",
"value": "What is the height of a building, in meters, if it casts a shadow 75 meters long at the same time a 15-meter statue casts a shadow that is 40 meters long? Express your answer to the nearest whole number."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. So, th... |
A florist has 15 identical bouquets to arrange in the display window. Each bouquet is either red, white, or a mix of red and white flowers. In how many ways can the florist arrange the bouquets so that no two bouquets of the same color are next to each other? | A florist has 15 identical bouquets to arrange in the display window. Each bouquet is either red, white, or a mix of red and white flowers. In how many ways can the florist arrange the bouquets so that no two bouquets of the same color are next to each other? | Okay, so I have this problem where a florist needs to arrange 15 identical bouquets in a display window. Each bouquet can be red, white, or a mix of red and white. The key constraint is that no two bouquets of the same color can be next to each other. I need to figure out how many different ways the florist can arrange... | The florist needs to arrange 15 bouquets, each either red (R), white (W), or mixed (M), such that no two bouquets of the same color are adjacent. Mixed bouquets are considered a separate color, so they cannot be adjacent to other mixed bouquets either.
To solve this, we recognize that the problem is equivalent to coun... | 13,831 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem where a florist needs to arrange 15 identical bouquets in a display window. Each bouquet can be red, white, or a mix of red and white. The key constraint is that no two bouquets of the same color can be next to each other. I need to figure out how many different ways the florist can... | [
{
"from": "human",
"value": "A florist has 15 identical bouquets to arrange in the display window. Each bouquet is either red, white, or a mix of red and white flowers. In how many ways can the florist arrange the bouquets so that no two bouquets of the same color are next to each other?"
},
{
"from... |
At a coffee shop, the cost of a coffee is inversely proportional to the number of creamers added to it. If a coffee with 3 creamers costs $2.50, how much will a coffee with 5 creamers cost? | At a coffee shop, the cost of a coffee is inversely proportional to the number of creamers added to it. If a coffee with 3 creamers costs $2.50, how much will a coffee with 5 creamers cost? | Okay, let's see. The problem says that the cost of a coffee is inversely proportional to the number of creamers added. Hmm, inversely proportional. So, I remember that inverse proportionality means that as one quantity increases, the other decreases in such a way that their product is a constant.
Let me write that do... | The cost of a coffee (C) is inversely proportional to the number of creamers (n). This relationship can be written as \( C = \frac{k}{n} \), where \( k \) is the constant of proportionality.
Given that a coffee with 3 creamers costs $2.50, we can find \( k \) by substituting \( n = 3 \) and \( C = 2.50 \) into the equ... | 18,385 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem says that the cost of a coffee is inversely proportional to the number of creamers added. Hmm, inversely proportional. So, I remember that inverse proportionality means that as one quantity increases, the other decreases in such a way that their product is a constant.
Let me write... | [
{
"from": "human",
"value": "At a coffee shop, the cost of a coffee is inversely proportional to the number of creamers added to it. If a coffee with 3 creamers costs $2.50, how much will a coffee with 5 creamers cost?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem says tha... |
The perimeter of a rectangle is 48 units, and its area is 135 square units. The length of one side is 3 times the length of the adjacent side. Find the length of the diagonal of the rectangle. | The perimeter of a rectangle is 48 units, and its area is 135 square units. The length of one side is 3 times the length of the adjacent side. Find the length of the diagonal of the rectangle. | Okay, so I need to find the length of the diagonal of a rectangle. The problem says the perimeter is 48 units, the area is 135 square units, and one side is three times the length of the adjacent side. Hmm, let me break this down step by step.
First, let's recall that in a rectangle, opposite sides are equal. So if on... | Given the problem, we need to find the length of the diagonal of a rectangle with a perimeter of 48 units, an area of 135 square units, and one side being three times the length of the adjacent side.
First, we denote the sides of the rectangle as \( l \) (length) and \( w \) (width). The given conditions are:
1. The p... | 9,311 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the length of the diagonal of a rectangle. The problem says the perimeter is 48 units, the area is 135 square units, and one side is three times the length of the adjacent side. Hmm, let me break this down step by step.
First, let's recall that in a rectangle, opposite sides are equal. ... | [
{
"from": "human",
"value": "The perimeter of a rectangle is 48 units, and its area is 135 square units. The length of one side is 3 times the length of the adjacent side. Find the length of the diagonal of the rectangle."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the length... |
Find all real numbers $x$ such that the product $(x + 1)((x + 2)i + 3)((x + 4)i - 5)$ is a real number. Enter all the solutions, separated by commas. | Find all real numbers $x$ such that the product $(x + 1)((x + 2)i + 3)((x + 4)i - 5)$ is a real number. Enter all the solutions, separated by commas. | Okay, so I need to find all real numbers x such that the product (x + 1)*((x + 2)i + 3)*((x + 4)i - 5) is a real number. Hmm, complex numbers... the product of three terms. For the product to be real, the imaginary part must be zero. So, maybe I can expand the expression and then set the imaginary part to zero? Let's s... | To find all real numbers \( x \) such that the product \((x + 1)((x + 2)i + 3)((x + 4)i - 5)\) is a real number, we need to ensure the imaginary part of the product is zero.
First, we consider the product of the two complex numbers \( A = 3 + (x + 2)i \) and \( B = -5 + (x + 4)i \):
\[
A \cdot B = (3 + (x + 2)i)(-5 ... | 38,561 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find all real numbers x such that the product (x + 1)*((x + 2)i + 3)*((x + 4)i - 5) is a real number. Hmm, complex numbers... the product of three terms. For the product to be real, the imaginary part must be zero. So, maybe I can expand the expression and then set the imaginary part to zero?... | [
{
"from": "human",
"value": "Find all real numbers $x$ such that the product $(x + 1)((x + 2)i + 3)((x + 4)i - 5)$ is a real number. Enter all the solutions, separated by commas."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find all real numbers x such that the product (x + 1)*((x ... |
Find the product of the roots of the equation
$$\frac{x^2}{x+1} - 2x + \sqrt{x} + 1 = 0,$$
given that all of the roots are real and positive. | Find the product of the roots of the equation
$$\frac{x^2}{x+1} - 2x + \sqrt{x} + 1 = 0,$$
given that all of the roots are real and positive. | Okay, so I need to find the product of the roots of this equation: (x²)/(x + 1) - 2x + sqrt(x) + 1 = 0. All the roots are real and positive, which is good to know. Hmm, let me start by trying to simplify the equation.
First, maybe I should get rid of the fractions by multiplying both sides by (x + 1). Let me check if... | To find the product of the roots of the equation
\[
\frac{x^2}{x+1} - 2x + \sqrt{x} + 1 = 0,
\]
given that all roots are real and positive, we start by manipulating the equation.
First, we multiply both sides by \(x + 1\) to eliminate the fraction:
\[
x^2 - 2x(x + 1) + \sqrt{x}(x + 1) + (x + 1) = 0.
\]
Simplifying the... | 20,082 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the product of the roots of this equation: (x²)/(x + 1) - 2x + sqrt(x) + 1 = 0. All the roots are real and positive, which is good to know. Hmm, let me start by trying to simplify the equation.
First, maybe I should get rid of the fractions by multiplying both sides by (x + 1). Let me ... | [
{
"from": "human",
"value": "Find the product of the roots of the equation\n$$\\frac{x^2}{x+1} - 2x + \\sqrt{x} + 1 = 0,$$\ngiven that all of the roots are real and positive."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the product of the roots of this equation: (x²)/(x + 1) -... |
A triangle has a perimeter of 90 centimeters and each side has an integer length. How many non-congruent triangles meet these criteria? | A triangle has a perimeter of 90 centimeters and each side has an integer length. How many non-congruent triangles meet these criteria? | Okay, so I need to figure out how many non-congruent triangles there are with a perimeter of 90 centimeters and each side having an integer length. Hmm, triangles with integer sides and a specific perimeter. Let me recall the triangle inequality theorem here. It states that the sum of the lengths of any two sides must ... | To determine the number of non-congruent triangles with a perimeter of 90 centimeters and integer side lengths, we need to consider triangles with sides \(a \leq b \leq c\) such that \(a + b + c = 90\) and satisfy the triangle inequality \(a + b > c\).
1. **Triangle Inequality and Perimeter Constraint**:
- For side... | 30,595 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many non-congruent triangles there are with a perimeter of 90 centimeters and each side having an integer length. Hmm, triangles with integer sides and a specific perimeter. Let me recall the triangle inequality theorem here. It states that the sum of the lengths of any two sid... | [
{
"from": "human",
"value": "A triangle has a perimeter of 90 centimeters and each side has an integer length. How many non-congruent triangles meet these criteria?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many non-congruent triangles there are with a perimeter o... |
The sequence $b_1,$ $b_2,$ $b_3,$ $\dots$ satisfies $b_1 = 3,$ $b_4 = 15,$ and for all $n \ge 3,$ $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2.$ | The sequence $b_1,$ $b_2,$ $b_3,$ $\dots$ satisfies $b_1 = 3,$ $b_4 = 15,$ and for all $n \ge 3,$ $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2.$ | Okay, let's see. I need to find the value of b₂ in this sequence where b₁ is 3, b₄ is 15, and each term from b₃ onwards is the geometric mean of the previous terms. Hmm. Let me start by recalling what a geometric mean is. The geometric mean of k numbers is the kth root of their product. So, for example, the geometric m... | Given the sequence \( b_1, b_2, b_3, \ldots \) where \( b_1 = 3 \), \( b_4 = 15 \), and for all \( n \ge 3 \), \( b_n \) is the geometric mean of the first \( n - 1 \) terms, we need to find \( b_2 \).
1. Denote \( b_2 \) as \( x \).
2. Calculate \( b_3 \) as the geometric mean of \( b_1 \) and \( b_2 \):
\[
b_3... | 27,884 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the value of b₂ in this sequence where b₁ is 3, b₄ is 15, and each term from b₃ onwards is the geometric mean of the previous terms. Hmm. Let me start by recalling what a geometric mean is. The geometric mean of k numbers is the kth root of their product. So, for example, the geo... | [
{
"from": "human",
"value": "The sequence $b_1,$ $b_2,$ $b_3,$ $\\dots$ satisfies $b_1 = 3,$ $b_4 = 15,$ and for all $n \\ge 3,$ $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the value of b₂ in this se... |
What is the minimum value of the function $f(x) = \frac{x^2-2x+1}{x+1}$, for $x> -1$? | What is the minimum value of the function $f(x) = \frac{x^2-2x+1}{x+1}$, for $x> -1$? | Alright, so I need to find the minimum value of the function f(x) = (x² - 2x + 1)/(x + 1) for x > -1. Hmm, okay. Let me start by recalling how to find minima of functions. Usually, for calculus, we take the derivative, set it equal to zero, and solve for x. Then check if it's a minimum using the second derivative or so... | To find the minimum value of the function \( f(x) = \frac{x^2 - 2x + 1}{x + 1} \) for \( x > -1 \), we start by simplifying the function. The numerator \( x^2 - 2x + 1 \) factors as \( (x - 1)^2 \), so the function can be rewritten as \( f(x) = \frac{(x - 1)^2}{x + 1} \).
Next, we use the quotient rule to find the der... | 12,511 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to find the minimum value of the function f(x) = (x² - 2x + 1)/(x + 1) for x > -1. Hmm, okay. Let me start by recalling how to find minima of functions. Usually, for calculus, we take the derivative, set it equal to zero, and solve for x. Then check if it's a minimum using the second derivati... | [
{
"from": "human",
"value": "What is the minimum value of the function $f(x) = \\frac{x^2-2x+1}{x+1}$, for $x> -1$?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to find the minimum value of the function f(x) = (x² - 2x + 1)/(x + 1) for x > -1. Hmm, okay. Let me start by recalling h... |
Determine the range of $b$ such that the quadratic equation $x^2 + bx + (b-2) = 0$ has at least one real root. Express your answer in interval notation. | Determine the range of $b$ such that the quadratic equation $x^2 + bx + (b-2) = 0$ has at least one real root. Express your answer in interval notation. | Okay, so I need to figure out the range of values for b that makes the quadratic equation x² + bx + (b - 2) = 0 have at least one real root. Hmm, let's see. I remember that for a quadratic equation ax² + bx + c = 0, the discriminant D is b² - 4ac. If the discriminant is non-negative (D ≥ 0), then the equation has at le... | To determine the range of \( b \) such that the quadratic equation \( x^2 + bx + (b - 2) = 0 \) has at least one real root, we need to analyze the discriminant of the quadratic equation. The discriminant \( D \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \). For the equation to have a... | 12,539 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the range of values for b that makes the quadratic equation x² + bx + (b - 2) = 0 have at least one real root. Hmm, let's see. I remember that for a quadratic equation ax² + bx + c = 0, the discriminant D is b² - 4ac. If the discriminant is non-negative (D ≥ 0), then the equation h... | [
{
"from": "human",
"value": "Determine the range of $b$ such that the quadratic equation $x^2 + bx + (b-2) = 0$ has at least one real root. Express your answer in interval notation."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the range of values for b that makes the qua... |
The function $f(n)$ is defined as follows: $f(1) = 3$, and for $n > 1$, $f(n)$ is the sum of all positive integers $k$ such that $f(n-1)$ is divisible by $k$. What is the smallest value of $n$ such that $f(n) > 10$? | The function $f(n)$ is defined as follows: $f(1) = 3$, and for $n > 1$, $f(n)$ is the sum of all positive integers $k$ such that $f(n-1)$ is divisible by $k$. What is the smallest value of $n$ such that $f(n) > 10$? | Okay, let's try to figure out this problem step by step. The function f(n) is defined recursively. We start with f(1) = 3. For each subsequent n > 1, f(n) is the sum of all positive divisors of f(n-1). We need to find the smallest n where f(n) exceeds 10.
First, let me make sure I understand the problem correctly. Eac... | The function \( f(n) \) is defined as follows: \( f(1) = 3 \), and for \( n > 1 \), \( f(n) \) is the sum of all positive divisors of \( f(n-1) \). We need to find the smallest value of \( n \) such that \( f(n) > 10 \).
Starting with \( f(1) = 3 \):
1. To find \( f(2) \), we calculate the sum of the divisors of 3. T... | 11,530 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to figure out this problem step by step. The function f(n) is defined recursively. We start with f(1) = 3. For each subsequent n > 1, f(n) is the sum of all positive divisors of f(n-1). We need to find the smallest n where f(n) exceeds 10.
First, let me make sure I understand the problem correc... | [
{
"from": "human",
"value": "The function $f(n)$ is defined as follows: $f(1) = 3$, and for $n > 1$, $f(n)$ is the sum of all positive integers $k$ such that $f(n-1)$ is divisible by $k$. What is the smallest value of $n$ such that $f(n) > 10$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's... |
Let $z_{1},z_{2},z_{3},\ldots,z_{n}$ be the $n$th roots of unity. That is, they satisfy $z^{n}-1=0.$ For any integer $k$, consider the polynomial $$p_{k}(z)=(z-z_{1})(z-z_{2})\cdots(z-z_{n})+z^{k}.$$ For how many integer values of $k$, $1\leq k\leq 1000$, is $p_{k}(z)$ a perfect square? | Let $z_{1},z_{2},z_{3},\ldots,z_{n}$ be the $n$th roots of unity. That is, they satisfy $z^{n}-1=0.$ For any integer $k$, consider the polynomial $$p_{k}(z)=(z-z_{1})(z-z_{2})\cdots(z-z_{n})+z^{k}.$$ For how many integer values of $k$, $1\leq k\leq 1000$, is $p_{k}(z)$ a perfect square? | Okay, let's see. The problem is about polynomials constructed from the nth roots of unity and adding z^k, and we need to determine for how many integer values of k between 1 and 1000 the polynomial p_k(z) is a perfect square.
First, I need to recall that the nth roots of unity are the solutions to the equation z^n = ... | Given the polynomial \( p_k(z) = (z - z_1)(z - z_2) \cdots (z - z_n) + z^k \) where \( z_1, z_2, \ldots, z_n \) are the \( n \)-th roots of unity, we need to determine how many integer values of \( k \) between 1 and 1000 make \( p_k(z) \) a perfect square.
1. **Roots of Unity and Polynomial Form**:
- The polynomia... | 12,188 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is about polynomials constructed from the nth roots of unity and adding z^k, and we need to determine for how many integer values of k between 1 and 1000 the polynomial p_k(z) is a perfect square.
First, I need to recall that the nth roots of unity are the solutions to the equatio... | [
{
"from": "human",
"value": "Let $z_{1},z_{2},z_{3},\\ldots,z_{n}$ be the $n$th roots of unity. That is, they satisfy $z^{n}-1=0.$ For any integer $k$, consider the polynomial $$p_{k}(z)=(z-z_{1})(z-z_{2})\\cdots(z-z_{n})+z^{k}.$$ For how many integer values of $k$, $1\\leq k\\leq 1000$, is $p_{k}(z)$ a p... |
What is the volume, in terms of π, of a cylindrical barrel that has a diameter of 10 feet and a height of 8 feet? | What is the volume, in terms of π, of a cylindrical barrel that has a diameter of 10 feet and a height of 8 feet? | Okay, let's see. I need to find the volume of a cylindrical barrel. The problem gives me the diameter as 10 feet and the height as 8 feet. Hmm, volume of a cylinder. I remember there's a formula for that. Let me recall... I think it's π times the radius squared times the height. So, Volume = πr²h. Yeah, that sounds rig... | To find the volume of a cylindrical barrel with a diameter of 10 feet and a height of 8 feet, we use the formula for the volume of a cylinder, which is \( V = \pi r^2 h \).
1. **Calculate the radius**: The radius is half of the diameter.
\[
r = \frac{10}{2} = 5 \text{ feet}
\]
2. **Substitute the values into... | 31,970 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the volume of a cylindrical barrel. The problem gives me the diameter as 10 feet and the height as 8 feet. Hmm, volume of a cylinder. I remember there's a formula for that. Let me recall... I think it's π times the radius squared times the height. So, Volume = πr²h. Yeah, that so... | [
{
"from": "human",
"value": "What is the volume, in terms of π, of a cylindrical barrel that has a diameter of 10 feet and a height of 8 feet?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the volume of a cylindrical barrel. The problem gives me the diameter as 10 feet ... |
Find: $\frac{2}{7}+\frac{8}{10}$ | Find: $\frac{2}{7}+\frac{8}{10}$ | Okay, let me figure out how to solve \(\frac{2}{7} + \frac{8}{10}\). Hmm, adding fractions. Right, I remember that to add fractions, they need to have the same denominator. So first, I should find a common denominator for 7 and 10. Let me think... The denominators are 7 and 10. The least common multiple of 7 and 10. Si... | To solve \(\frac{2}{7} + \frac{8}{10}\), we first need to find a common denominator. The least common multiple of 7 and 10 is 70.
Convert each fraction to have 70 as the denominator:
- \(\frac{2}{7}\) is converted to \(\frac{20}{70}\) by multiplying the numerator and denominator by 10.
- \(\frac{8}{10}\) is converted... | 2,754 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me figure out how to solve \(\frac{2}{7} + \frac{8}{10}\). Hmm, adding fractions. Right, I remember that to add fractions, they need to have the same denominator. So first, I should find a common denominator for 7 and 10. Let me think... The denominators are 7 and 10. The least common multiple of 7 an... | [
{
"from": "human",
"value": "Find: $\\frac{2}{7}+\\frac{8}{10}$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me figure out how to solve \\(\\frac{2}{7} + \\frac{8}{10}\\). Hmm, adding fractions. Right, I remember that to add fractions, they need to have the same denominator. So first, I sho... |
Solve for $y$: $$\frac{1}{2} + \frac{1}{3} = \frac{2}{y-1}$$ | Solve for $y$: $$\frac{1}{2} + \frac{1}{3} = \frac{2}{y-1}$$ | Okay, so I need to solve this equation: 1/2 + 1/3 = 2/(y-1). Hmm, let me think. First, I should probably handle the left side of the equation because there are two fractions there that I can combine. If I add 1/2 and 1/3 together, that should simplify things a bit.
To add fractions, they need a common denominator. Th... | To solve the equation \(\frac{1}{2} + \frac{1}{3} = \frac{2}{y-1}\):
1. First, add the fractions on the left side. The least common denominator of 2 and 3 is 6:
\[
\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}
\]
This simplifies the equation to:
\[
\frac{5}{6} = \frac{2}{y-1}
... | 25,796 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this equation: 1/2 + 1/3 = 2/(y-1). Hmm, let me think. First, I should probably handle the left side of the equation because there are two fractions there that I can combine. If I add 1/2 and 1/3 together, that should simplify things a bit.
To add fractions, they need a common denomin... | [
{
"from": "human",
"value": "Solve for $y$: $$\\frac{1}{2} + \\frac{1}{3} = \\frac{2}{y-1}$$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this equation: 1/2 + 1/3 = 2/(y-1). Hmm, let me think. First, I should probably handle the left side of the equation because there are two... |
If $b$ is a real number and $x^2+bx+9$ has two distinct real roots, then what are the possible values of $b$? Express your answer in interval notation. | If $b$ is a real number and $x^2+bx+9$ has two distinct real roots, then what are the possible values of $b$? Express your answer in interval notation. | Okay, so I need to figure out the possible values of \( b \) such that the quadratic equation \( x^2 + bx + 9 \) has two distinct real roots. Hmm, let's start by recalling what I know about quadratic equations and their roots.
First, a quadratic equation in the form \( ax^2 + bx + c = 0 \) will have two distinct real ... | To determine the possible values of \( b \) such that the quadratic equation \( x^2 + bx + 9 \) has two distinct real roots, we need to ensure the discriminant is positive. The discriminant \( D \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \).
For the given quadratic equation \( x^2... | 4,482 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the possible values of \( b \) such that the quadratic equation \( x^2 + bx + 9 \) has two distinct real roots. Hmm, let's start by recalling what I know about quadratic equations and their roots.
First, a quadratic equation in the form \( ax^2 + bx + c = 0 \) will have two distin... | [
{
"from": "human",
"value": "If $b$ is a real number and $x^2+bx+9$ has two distinct real roots, then what are the possible values of $b$? Express your answer in interval notation."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the possible values of \\( b \\) such that t... |
A rectangular sheet of paper is folded in half along the diagonal, and then the resulting triangle is folded in half along its altitude. The area of the final triangle is 25 square centimeters. What is the length of the original sheet of paper's diagonal? | A rectangular sheet of paper is folded in half along the diagonal, and then the resulting triangle is folded in half along its altitude. The area of the final triangle is 25 square centimeters. What is the length of the original sheet of paper's diagonal? | Okay, let's see. We have a rectangular sheet of paper. When it's folded along the diagonal, it becomes a triangle. Then, that triangle is folded along its altitude, resulting in another triangle with area 25 cm². We need to find the length of the original rectangle's diagonal. Hmm.
First, let me visualize this. A rect... | Let the original rectangle have length \( l \) and width \( w \). The area of the rectangle is \( l \times w \). When folded along the diagonal, the resulting right-angled triangle has an area of \( \frac{l \times w}{2} \). Folding this triangle along its altitude results in another triangle with an area of 25 square c... | 15,586 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. We have a rectangular sheet of paper. When it's folded along the diagonal, it becomes a triangle. Then, that triangle is folded along its altitude, resulting in another triangle with area 25 cm². We need to find the length of the original rectangle's diagonal. Hmm.
First, let me visualize this... | [
{
"from": "human",
"value": "A rectangular sheet of paper is folded in half along the diagonal, and then the resulting triangle is folded in half along its altitude. The area of the final triangle is 25 square centimeters. What is the length of the original sheet of paper's diagonal?"
},
{
"from": "... |
Count the number of ways to arrange the letters in the word "PERMUTATION" such that all the vowels are in alphabetical order and all the consonants are in reverse alphabetical order. | Count the number of ways to arrange the letters in the word "PERMUTATION" such that all the vowels are in alphabetical order and all the consonants are in reverse alphabetical order. | Okay, let's see. I need to figure out how many ways to arrange the letters in "PERMUTATION" such that all vowels are in alphabetical order and all consonants are in reverse alphabetical order. Hmm. First, let me recall what the word "PERMUTATION" consists of. Let me write down the letters to make sure.
P, E, R, M, U, ... | To solve the problem of arranging the letters in the word "PERMUTATION" such that all vowels are in alphabetical order and all consonants are in reverse alphabetical order, we follow these steps:
1. **Identify the vowels and consonants**:
- Vowels: E, U, A, I, O (5 vowels)
- Consonants: P, R, M, T, T, N (6 conso... | 24,375 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to figure out how many ways to arrange the letters in "PERMUTATION" such that all vowels are in alphabetical order and all consonants are in reverse alphabetical order. Hmm. First, let me recall what the word "PERMUTATION" consists of. Let me write down the letters to make sure.
P, E, R... | [
{
"from": "human",
"value": "Count the number of ways to arrange the letters in the word \"PERMUTATION\" such that all the vowels are in alphabetical order and all the consonants are in reverse alphabetical order."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to figure out how ... |
Tom has been saving money for a new bike and has $\$120$ to spend. There are three bike models available, priced at $\$20$, $\$40$, and $\$60$. However, there's a catch: for every two cheaper bikes he buys, he must also buy one of the next most expensive model. What is the largest number of bikes Tom can purchase while... | Tom has been saving money for a new bike and has $\$120$ to spend. There are three bike models available, priced at $\$20$, $\$40$, and $\$60$. However, there's a catch: for every two cheaper bikes he buys, he must also buy one of the next most expensive model. What is the largest number of bikes Tom can purchase while... | Okay, so Tom wants to buy as many bikes as possible with his $120. The bikes are priced at $20, $40, and $60. But there's a rule: for every two cheaper bikes he buys, he must also buy one of the next most expensive model. Hmm, that sounds a bit complicated. Let me try to break it down.
First, let me list the bike pric... | Tom wants to buy as many bikes as possible with his $120, considering the prices of $20, $40, and $60, and the rule that for every two cheaper bikes he buys, he must also buy one of the next most expensive model.
Key steps:
1. **Understanding the rule**: For every two cheaper bikes, Tom must buy one of the next most ... | 16,497 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Tom wants to buy as many bikes as possible with his $120. The bikes are priced at $20, $40, and $60. But there's a rule: for every two cheaper bikes he buys, he must also buy one of the next most expensive model. Hmm, that sounds a bit complicated. Let me try to break it down.
First, let me list the b... | [
{
"from": "human",
"value": "Tom has been saving money for a new bike and has $\\$120$ to spend. There are three bike models available, priced at $\\$20$, $\\$40$, and $\\$60$. However, there's a catch: for every two cheaper bikes he buys, he must also buy one of the next most expensive model. What is the l... |
What is the value of $x$ if $\frac{x}{x + 3} = \frac{7}{5}$? | What is the value of $x$ if $\frac{x}{x + 3} = \frac{7}{5}$? | Okay, so I need to solve for x in the equation (x)/(x + 3) = 7/5. Hmm, let's see. I remember that when you have a proportion like this, you can cross-multiply to get rid of the fractions. So, if I cross-multiply, that would mean multiplying the numerator of one side by the denominator of the other side. Let me write th... | To solve for \( x \) in the equation \(\frac{x}{x + 3} = \frac{7}{5}\), we start by cross-multiplying to eliminate the fractions:
\[
5x = 7(x + 3)
\]
Expanding the right side gives:
\[
5x = 7x + 21
\]
Next, we isolate \( x \) by subtracting \( 7x \) from both sides:
\[
5x - 7x = 21
\]
Simplifying the left side:
... | 25,266 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve for x in the equation (x)/(x + 3) = 7/5. Hmm, let's see. I remember that when you have a proportion like this, you can cross-multiply to get rid of the fractions. So, if I cross-multiply, that would mean multiplying the numerator of one side by the denominator of the other side. Let me ... | [
{
"from": "human",
"value": "What is the value of $x$ if $\\frac{x}{x + 3} = \\frac{7}{5}$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve for x in the equation (x)/(x + 3) = 7/5. Hmm, let's see. I remember that when you have a proportion like this, you can cross-multiply to ge... |
The decimal representation of a fraction ends in $0.\overline{3}$. When the decimal is changed to a common fraction and reduced to lowest terms, what is the denominator of the fraction? | The decimal representation of a fraction ends in $0.\overline{3}$. When the decimal is changed to a common fraction and reduced to lowest terms, what is the denominator of the fraction? | Okay, so I need to figure out the denominator of the fraction that represents the repeating decimal 0.\overline{3} when it's simplified to its lowest terms. Hmm, let me start by recalling what 0.\overline{3} means. The bar over the 3 indicates that the 3 repeats infinitely, right? So that's 0.3333... and continues fore... | Let \( x = 0.\overline{3} \). This means \( x = 0.333333\ldots \).
To convert this repeating decimal into a fraction, multiply both sides by 10:
\[ 10x = 3.333333\ldots \]
Subtract the original equation from this new equation:
\[ 10x - x = 3.333333\ldots - 0.333333\ldots \]
\[ 9x = 3 \]
Solving for \( x \) gives:
\[... | 28,192 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the denominator of the fraction that represents the repeating decimal 0.\overline{3} when it's simplified to its lowest terms. Hmm, let me start by recalling what 0.\overline{3} means. The bar over the 3 indicates that the 3 repeats infinitely, right? So that's 0.3333... and contin... | [
{
"from": "human",
"value": "The decimal representation of a fraction ends in $0.\\overline{3}$. When the decimal is changed to a common fraction and reduced to lowest terms, what is the denominator of the fraction?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the denomi... |
What is the value of $x$ in the expression $\left(\frac{2}{3}\right)^x \cdot \left(\frac{2}{3}\right)^{-x}$ if the result is equal to $\frac{1}{4}$? | What is the value of $x$ in the expression $\left(\frac{2}{3}\right)^x \cdot \left(\frac{2}{3}\right)^{-x}$ if the result is equal to $\frac{1}{4}$? | Okay, let's see. I have this equation here: (2/3)^x multiplied by (2/3)^{-x} equals 1/4. Hmm, I need to find the value of x. Alright, let's start by recalling some exponent rules. When you multiply two expressions with the same base, you add their exponents. So, (a^b) * (a^c) = a^{b+c}.
In this case, the base is (2/3... | Given the expression \(\left(\frac{2}{3}\right)^x \cdot \left(\frac{2}{3}\right)^{-x}\) and the result is equal to \(\frac{1}{4}\), we need to find the value of \(x\).
First, we use the properties of exponents to simplify the expression. When multiplying two expressions with the same base, we add their exponents:
\[
... | 32,580 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I have this equation here: (2/3)^x multiplied by (2/3)^{-x} equals 1/4. Hmm, I need to find the value of x. Alright, let's start by recalling some exponent rules. When you multiply two expressions with the same base, you add their exponents. So, (a^b) * (a^c) = a^{b+c}.
In this case, the base... | [
{
"from": "human",
"value": "What is the value of $x$ in the expression $\\left(\\frac{2}{3}\\right)^x \\cdot \\left(\\frac{2}{3}\\right)^{-x}$ if the result is equal to $\\frac{1}{4}$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I have this equation here: (2/3)^x multiplied by (2/3... |
Let $a_{n+2}=a_{n+1}^2-a_n$ for all $n\geq 1$, where $a_1=1$ and $a_2=1$. Show that there is no integer $n\geq 3$ such that $a_n$ is negative. | Let $a_{n+2}=a_{n+1}^2-a_n$ for all $n\geq 1$, where $a_1=1$ and $a_2=1$. Show that there is no integer $n\geq 3$ such that $a_n$ is negative. | Okay, so I need to show that for the sequence defined by the recurrence relation \( a_{n+2} = a_{n+1}^2 - a_n \) with initial terms \( a_1 = 1 \) and \( a_2 = 1 \), there is no integer \( n \geq 3 \) such that \( a_n \) is negative. Hmm, interesting. Let me try to work through this step by step.
First, let's write dow... | Given the recurrence relation \( a_{n+2} = a_{n+1}^2 - a_n \) with initial conditions \( a_1 = 1 \) and \( a_2 = 1 \), we need to show that there is no integer \( n \geq 3 \) such that \( a_n \) is negative.
First, we compute the initial terms of the sequence:
- \( a_1 = 1 \)
- \( a_2 = 1 \)
- \( a_3 = a_2^2 - a_1 = 1... | 26,621 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to show that for the sequence defined by the recurrence relation \( a_{n+2} = a_{n+1}^2 - a_n \) with initial terms \( a_1 = 1 \) and \( a_2 = 1 \), there is no integer \( n \geq 3 \) such that \( a_n \) is negative. Hmm, interesting. Let me try to work through this step by step.
First, let's w... | [
{
"from": "human",
"value": "Let $a_{n+2}=a_{n+1}^2-a_n$ for all $n\\geq 1$, where $a_1=1$ and $a_2=1$. Show that there is no integer $n\\geq 3$ such that $a_n$ is negative."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to show that for the sequence defined by the recurrence relation \... |
A bookshelf has 5 shelves, and each shelf can hold a maximum of 3 books. In how many ways can you arrange 3 books on the bookshelf such that no two books are on the same shelf? | A bookshelf has 5 shelves, and each shelf can hold a maximum of 3 books. In how many ways can you arrange 3 books on the bookshelf such that no two books are on the same shelf? | Okay, so I have this problem here: there's a bookshelf with 5 shelves, each shelf can hold up to 3 books. I need to figure out how many ways I can arrange 3 books on the bookshelf such that no two books are on the same shelf. Hmm, let me break this down step by step.
First, let's parse the question. We have 5 shelves,... | To determine the number of ways to arrange 3 books on a bookshelf with 5 shelves such that no two books are on the same shelf, we need to consider the permutations of the books on the shelves.
1. **Key Points**:
- The books are distinct, so the order matters.
- Each shelf can hold a maximum of 3 books, but we a... | 8,255 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: there's a bookshelf with 5 shelves, each shelf can hold up to 3 books. I need to figure out how many ways I can arrange 3 books on the bookshelf such that no two books are on the same shelf. Hmm, let me break this down step by step.
First, let's parse the question. We have 5 ... | [
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a maximum of 3 books. In how many ways can you arrange 3 books on the bookshelf such that no two books are on the same shelf?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: there's a books... |
Let $m$ and $n$ be positive integers. The sum of the reciprocals of the positive divisors of $m^n$ is equal to $\frac{13}{6}$. Determine $m+n$. | Let $m$ and $n$ be positive integers. The sum of the reciprocals of the positive divisors of $m^n$ is equal to $\frac{13}{6}$. Determine $m+n$. | Okay, let's try to tackle this problem. So, we have positive integers m and n. The sum of the reciprocals of the positive divisors of m^n is 13/6. We need to find m + n. Hmm, alright. Let me start by recalling some number theory concepts here.
First off, the sum of reciprocals of the divisors of a number. I remember t... | Given positive integers \( m \) and \( n \), the sum of the reciprocals of the positive divisors of \( m^n \) is \(\frac{13}{6}\). We need to determine \( m + n \).
First, recall that the sum of the reciprocals of the divisors of a number \( k \) is given by \(\frac{\sigma(k)}{k}\), where \(\sigma(k)\) is the sum of t... | 26,061 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to tackle this problem. So, we have positive integers m and n. The sum of the reciprocals of the positive divisors of m^n is 13/6. We need to find m + n. Hmm, alright. Let me start by recalling some number theory concepts here.
First off, the sum of reciprocals of the divisors of a number. I re... | [
{
"from": "human",
"value": "Let $m$ and $n$ be positive integers. The sum of the reciprocals of the positive divisors of $m^n$ is equal to $\\frac{13}{6}$. Determine $m+n$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to tackle this problem. So, we have positive integers m and n. The... |
What is the perimeter of a regular hexagon with side length $4$ meters? Express your answer in terms of $\sqrt{3}$. | What is the perimeter of a regular hexagon with side length $4$ meters? Express your answer in terms of $\sqrt{3}$. | Okay, so I need to find the perimeter of a regular hexagon where each side is 4 meters long. Hmm, let me think. A regular hexagon has all sides equal and all angles equal, right? So, perimeter is just the total distance around the hexagon. Since all six sides are the same length, I guess I just multiply the length of o... | To find the perimeter of a regular hexagon with a side length of 4 meters, we use the fact that a regular hexagon has six equal sides. The perimeter is calculated by multiplying the length of one side by the number of sides:
\[
\text{Perimeter} = 6 \times \text{side length}
\]
Given the side length is 4 meters, we su... | 15,088 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the perimeter of a regular hexagon where each side is 4 meters long. Hmm, let me think. A regular hexagon has all sides equal and all angles equal, right? So, perimeter is just the total distance around the hexagon. Since all six sides are the same length, I guess I just multiply the len... | [
{
"from": "human",
"value": "What is the perimeter of a regular hexagon with side length $4$ meters? Express your answer in terms of $\\sqrt{3}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the perimeter of a regular hexagon where each side is 4 meters long. Hmm, let me think... |
Solve the inequality $\frac{x^2+1}{(x+2)^2} \le 0.$ (Give your answer in interval notation.) | Solve the inequality $\frac{x^2+1}{(x+2)^2} \le 0.$ (Give your answer in interval notation.) | Alright, so I need to solve the inequality \(\frac{x^2 + 1}{(x + 2)^2} \le 0\). Hmm, let me think about how to approach this.
First, I remember that when dealing with rational inequalities, the key is to determine where the expression is negative or zero. Since this is a fraction, the sign of the numerator and the de... | To solve the inequality \(\frac{x^2 + 1}{(x + 2)^2} \le 0\), we analyze the numerator and the denominator separately.
1. **Numerator Analysis**: The numerator is \(x^2 + 1\). Since \(x^2\) is always non-negative and adding 1 makes it strictly positive for all real \(x\). Therefore, the numerator is always positive and... | 8,380 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to solve the inequality \(\frac{x^2 + 1}{(x + 2)^2} \le 0\). Hmm, let me think about how to approach this.
First, I remember that when dealing with rational inequalities, the key is to determine where the expression is negative or zero. Since this is a fraction, the sign of the numerator an... | [
{
"from": "human",
"value": "Solve the inequality $\\frac{x^2+1}{(x+2)^2} \\le 0.$ (Give your answer in interval notation.)"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to solve the inequality \\(\\frac{x^2 + 1}{(x + 2)^2} \\le 0\\). Hmm, let me think about how to approach this. \n... |
Find the greatest common divisor (GCD) of two single-digit integers $X$ and $Y$ such that the GCD is the remainder when $XY$ is divided by $15$.
$X$ and $Y$ are distinct single-digit integers with $X < Y$.
$XY$ is a two-digit number when written in base $10$.
Express your answer as the remainder when the GCD is divi... | Find the greatest common divisor (GCD) of two single-digit integers $X$ and $Y$ such that the GCD is the remainder when $XY$ is divided by $15$.
$X$ and $Y$ are distinct single-digit integers with $X < Y$.
$XY$ is a two-digit number when written in base $10$.
Express your answer as the remainder when the GCD is divi... | Okay, let's see. The problem is asking for the greatest common divisor (GCD) of two distinct single-digit integers X and Y, where X is less than Y. The condition given is that this GCD should be equal to the remainder when the product XY (which is a two-digit number in base 10) is divided by 15. Then, we need to expres... | Given two distinct single-digit integers \(X\) and \(Y\) such that \(X < Y\) and their product \(XY\) (interpreted as a two-digit number in base 10) leaves a remainder equal to their greatest common divisor (GCD) when divided by 15, we need to find this GCD and express it as the remainder when divided by 15.
1. **Unde... | 26,759 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is asking for the greatest common divisor (GCD) of two distinct single-digit integers X and Y, where X is less than Y. The condition given is that this GCD should be equal to the remainder when the product XY (which is a two-digit number in base 10) is divided by 15. Then, we need t... | [
{
"from": "human",
"value": "Find the greatest common divisor (GCD) of two single-digit integers $X$ and $Y$ such that the GCD is the remainder when $XY$ is divided by $15$.\n\n$X$ and $Y$ are distinct single-digit integers with $X < Y$.\n\n$XY$ is a two-digit number when written in base $10$.\n\nExpress yo... |
Problem:
A group of friends went to a movie theater and bought a combination of large and small popcorns. The total cost was $120, and they consumed a total of 30 bags of popcorn. If each large popcorn costs $6 and each small popcorn costs $4, and the number of small popcorns is twice the number of large popcorns, how ... | Problem:
A group of friends went to a movie theater and bought a combination of large and small popcorns. The total cost was $120, and they consumed a total of 30 bags of popcorn. If each large popcorn costs $6 and each small popcorn costs $4, and the number of small popcorns is twice the number of large popcorns, how ... | Okay, let's see. So, the problem is about a group of friends who bought large and small popcorns. The total cost was $120, and they had 30 bags in total. Each large popcorn is $6, each small is $4. Also, the number of small popcorns is twice the number of large ones. We need to find how many large popcorns they bought.... | Given the problem, we need to determine the number of large popcorns bought by the group of friends. We denote the number of large popcorns as \( L \) and the number of small popcorns as \( S \). The problem provides the following information:
1. The total number of popcorn bags is 30.
2. The total cost is $120.
3. Ea... | 6,643 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. So, the problem is about a group of friends who bought large and small popcorns. The total cost was $120, and they had 30 bags in total. Each large popcorn is $6, each small is $4. Also, the number of small popcorns is twice the number of large ones. We need to find how many large popcorns they... | [
{
"from": "human",
"value": "Problem:\nA group of friends went to a movie theater and bought a combination of large and small popcorns. The total cost was $120, and they consumed a total of 30 bags of popcorn. If each large popcorn costs $6 and each small popcorn costs $4, and the number of small popcorns i... |
A convex hexagon has three diagonals that connect opposite vertices. A convex nonagon has 27 diagonals. How many diagonals does a convex dodecagon have? | A convex hexagon has three diagonals that connect opposite vertices. A convex nonagon has 27 diagonals. How many diagonals does a convex dodecagon have? | Alright, so I need to find the number of diagonals in a convex dodecagon. Hmm, let me start by recalling what a dodecagon is. A dodecagon is a 12-sided polygon, right? And since it's convex, all its interior angles are less than 180 degrees, and all the vertices point outward.
The problem mentions that a convex hexag... | To find the number of diagonals in a convex dodecagon (12-sided polygon), we use the formula for the number of diagonals in a convex polygon, which is \(\frac{n(n - 3)}{2}\), where \(n\) is the number of sides.
1. For a dodecagon, \(n = 12\).
2. Applying the formula:
\[
\frac{12(12 - 3)}{2} = \frac{12 \times 9}{... | 12,028 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to find the number of diagonals in a convex dodecagon. Hmm, let me start by recalling what a dodecagon is. A dodecagon is a 12-sided polygon, right? And since it's convex, all its interior angles are less than 180 degrees, and all the vertices point outward.
The problem mentions that a conv... | [
{
"from": "human",
"value": "A convex hexagon has three diagonals that connect opposite vertices. A convex nonagon has 27 diagonals. How many diagonals does a convex dodecagon have?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to find the number of diagonals in a convex dodecagon. ... |
Let $O$ be the origin. There exists a scalar $k$ so that for any points $A,$ $B,$ $C,$ $D,$ and $E$ such that
\[2 \overrightarrow{OA} + 3 \overrightarrow{OB} - \overrightarrow{OC} + 4 \overrightarrow{OD} + k \overrightarrow{OE} = \mathbf{0},\]the five points $A,$ $B,$ $C,$ $D,$ and $E$ are coplanar. Find $k$. | Let $O$ be the origin. There exists a scalar $k$ so that for any points $A,$ $B,$ $C,$ $D,$ and $E$ such that
\[2 \overrightarrow{OA} + 3 \overrightarrow{OB} - \overrightarrow{OC} + 4 \overrightarrow{OD} + k \overrightarrow{OE} = \mathbf{0},\]the five points $A,$ $B,$ $C,$ $D,$ and $E$ are coplanar. Find $k$. | Okay, so I need to find the scalar \( k \) such that if the equation \( 2 \overrightarrow{OA} + 3 \overrightarrow{OB} - \overrightarrow{OC} + 4 \overrightarrow{OD} + k \overrightarrow{OE} = \mathbf{0} \) holds, then the points \( A, B, C, D, E \) are coplanar. Hmm, coplanar points... that means all the points lie on th... | To find the scalar \( k \) such that the points \( A, B, C, D, \) and \( E \) are coplanar given the equation \( 2 \overrightarrow{OA} + 3 \overrightarrow{OB} - \overrightarrow{OC} + 4 \overrightarrow{OD} + k \overrightarrow{OE} = \mathbf{0} \), we need to ensure that the linear combination of vectors enforces coplanar... | 26,360 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the scalar \( k \) such that if the equation \( 2 \overrightarrow{OA} + 3 \overrightarrow{OB} - \overrightarrow{OC} + 4 \overrightarrow{OD} + k \overrightarrow{OE} = \mathbf{0} \) holds, then the points \( A, B, C, D, E \) are coplanar. Hmm, coplanar points... that means all the points l... | [
{
"from": "human",
"value": "Let $O$ be the origin. There exists a scalar $k$ so that for any points $A,$ $B,$ $C,$ $D,$ and $E$ such that\n\\[2 \\overrightarrow{OA} + 3 \\overrightarrow{OB} - \\overrightarrow{OC} + 4 \\overrightarrow{OD} + k \\overrightarrow{OE} = \\mathbf{0},\\]the five points $A,$ $B,$ $... |
In the Cartesian plane, a point $(x,y)$ is said to be equidistant from a pair of points $(a,b)$ and $(c,d)$ if the distance from $(x,y)$ to $(a,b)$ is equal to the distance from $(x,y)$ to $(c,d)$. For example, the point $(2,2)$ is equidistant from the points $(1,1)$ and $(3,3)$ because all three points lie on the same... | In the Cartesian plane, a point $(x,y)$ is said to be equidistant from a pair of points $(a,b)$ and $(c,d)$ if the distance from $(x,y)$ to $(a,b)$ is equal to the distance from $(x,y)$ to $(c,d)$. For example, the point $(2,2)$ is equidistant from the points $(1,1)$ and $(3,3)$ because all three points lie on the same... | Okay, so I need to find the number of points (x, y) that are equidistant from both (0,1) and (1,0), and also equidistant from (1,0) and (0,-1). Hmm, let me think about how to approach this.
First, I remember that the set of all points equidistant from two points is the perpendicular bisector of the segment connecting... | To find the number of points \((x, y)\) that are equidistant from the points \((0, 1)\) and \((1, 0)\), and also from the points \((1, 0)\) and \((0, -1)\), we need to determine the perpendicular bisectors for each pair of points and find their intersection.
1. **Equidistant from \((0, 1)\) and \((1, 0)\)**:
- The ... | 275 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the number of points (x, y) that are equidistant from both (0,1) and (1,0), and also equidistant from (1,0) and (0,-1). Hmm, let me think about how to approach this.
First, I remember that the set of all points equidistant from two points is the perpendicular bisector of the segment co... | [
{
"from": "human",
"value": "In the Cartesian plane, a point $(x,y)$ is said to be equidistant from a pair of points $(a,b)$ and $(c,d)$ if the distance from $(x,y)$ to $(a,b)$ is equal to the distance from $(x,y)$ to $(c,d)$. For example, the point $(2,2)$ is equidistant from the points $(1,1)$ and $(3,3)$... |
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