instruction_seed stringlengths 17 6.02k | response_seed stringlengths 17 6.02k | reasoning stringlengths 1.28k 104k | deepseek_solution stringlengths 7 3.7k | __original_row_idx int64 0 40k | source stringclasses 1
value | final_reasoning_trace stringlengths 1.61k 106k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
Find the range of the function
\[g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}}.\] | Find the range of the function
\[g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}}.\] | Alright, let's try to figure out the range of this function \( g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}} \). Hmm, okay, so the range is all the possible y-values that this function can take. To find the range, I need to determine all real numbers y such that there exists some x where \( y = \frac{2x + 1}{\sqrt{x^2 + 4x... | To find the range of the function \( g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}} \), we start by determining the domain. The expression inside the square root, \( x^2 + 4x + 1 \), must be positive. Solving the quadratic inequality, we find the domain is \( (-\infty, -2 - \sqrt{3}) \cup (-2 + \sqrt{3}, \infty) \).
Next, ... | 10,835 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's try to figure out the range of this function \( g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}} \). Hmm, okay, so the range is all the possible y-values that this function can take. To find the range, I need to determine all real numbers y such that there exists some x where \( y = \frac{2x + 1}{\sqrt{... | [
{
"from": "human",
"value": "Find the range of the function\n\\[g(x) = \\frac{2x + 1}{\\sqrt{x^2 + 4x + 1}}.\\]"
},
{
"from": "gpt",
"value": "<think>\nAlright, let's try to figure out the range of this function \\( g(x) = \\frac{2x + 1}{\\sqrt{x^2 + 4x + 1}} \\). Hmm, okay, so the range is all ... |
Find the range of the function $f(x) = \arctan\left(\frac{x^3-x}{x^2+1}\right)$, where $x$ is any real number. | Find the range of the function $f(x) = \arctan\left(\frac{x^3-x}{x^2+1}\right)$, where $x$ is any real number. | Okay, so I need to find the range of the function \( f(x) = \arctan\left(\frac{x^3 - x}{x^2 + 1}\right) \). Hmm, the range of an arctan function is usually between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), right? But since the argument here is another function \( \frac{x^3 - x}{x^2 + 1} \), I need to figure out wha... | To find the range of the function \( f(x) = \arctan\left(\frac{x^3 - x}{x^2 + 1}\right) \), we need to analyze the inner function \( g(x) = \frac{x^3 - x}{x^2 + 1} \).
First, we simplify \( g(x) \):
\[
g(x) = \frac{x^3 - x}{x^2 + 1} = x - \frac{2x}{x^2 + 1}
\]
Next, we analyze the behavior of \( g(x) \):
- As \( x \t... | 2,042 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the range of the function \( f(x) = \arctan\left(\frac{x^3 - x}{x^2 + 1}\right) \). Hmm, the range of an arctan function is usually between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), right? But since the argument here is another function \( \frac{x^3 - x}{x^2 + 1} \), I need to figure... | [
{
"from": "human",
"value": "Find the range of the function $f(x) = \\arctan\\left(\\frac{x^3-x}{x^2+1}\\right)$, where $x$ is any real number."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the range of the function \\( f(x) = \\arctan\\left(\\frac{x^3 - x}{x^2 + 1}\\right) \\)... |
Find the volume of the largest cone that can be inscribed in a right square pyramid with base edges of length $8\sqrt{2}$ units each and slant edges of length 10 units each, so that the base of the cone lies on the base of the pyramid and the vertex of the cone coincides with the vertex of the pyramid. | Find the volume of the largest cone that can be inscribed in a right square pyramid with base edges of length $8\sqrt{2}$ units each and slant edges of length 10 units each, so that the base of the cone lies on the base of the pyramid and the vertex of the cone coincides with the vertex of the pyramid. | Okay, so I need to find the volume of the largest cone that can be inscribed in a right square pyramid. The pyramid has a base with edges of length \(8\sqrt{2}\) units and slant edges (the edges from the base to the apex) of 10 units each. The cone's base has to lie on the pyramid's base, and the cone's tip has to be a... | To find the volume of the largest cone that can be inscribed in a right square pyramid with base edges of length \(8\sqrt{2}\) units and slant edges of length 10 units, we start by determining the height of the pyramid. The slant edge of the pyramid (10 units) forms a right triangle with half the diagonal of the base a... | 14,593 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the volume of the largest cone that can be inscribed in a right square pyramid. The pyramid has a base with edges of length \(8\sqrt{2}\) units and slant edges (the edges from the base to the apex) of 10 units each. The cone's base has to lie on the pyramid's base, and the cone's tip has... | [
{
"from": "human",
"value": "Find the volume of the largest cone that can be inscribed in a right square pyramid with base edges of length $8\\sqrt{2}$ units each and slant edges of length 10 units each, so that the base of the cone lies on the base of the pyramid and the vertex of the cone coincides with t... |
Problem:
A water tank can hold 1000 liters of water. Water flows into the tank at a rate of 2 liters per minute and flows out of the tank at a rate of 1.5 liters per minute. How many minutes will it take to fill the tank?
Solution:
Let's analyze the situation:
* Water flows into the tank at a rate of 2 liters per min... | Problem:
A water tank can hold 1000 liters of water. Water flows into the tank at a rate of 2 liters per minute and flows out of the tank at a rate of 1.5 liters per minute. How many minutes will it take to fill the tank?
Solution:
Let's analyze the situation:
* Water flows into the tank at a rate of 2 liters per min... | Okay, so there's this problem about a water tank, right? Let me try to figure it out step by step. The tank can hold 1000 liters. Water is flowing in at 2 liters per minute and flowing out at 1.5 liters per minute. The question is how long it will take to fill the tank.
Hmm, first, I need to understand the rates here.... | To determine how long it takes to fill the 1000-liter tank, we first calculate the net inflow rate by subtracting the outflow from the inflow:
**Net flow rate** = Inflow rate - Outflow rate
= 2 L/min - 1.5 L/min
= **0.5 L/min**
Next, divide the tank's capacity by the net flow rate to find the time required to fil... | 19,453 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this problem about a water tank, right? Let me try to figure it out step by step. The tank can hold 1000 liters. Water is flowing in at 2 liters per minute and flowing out at 1.5 liters per minute. The question is how long it will take to fill the tank.
Hmm, first, I need to understand the rat... | [
{
"from": "human",
"value": "Problem:\nA water tank can hold 1000 liters of water. Water flows into the tank at a rate of 2 liters per minute and flows out of the tank at a rate of 1.5 liters per minute. How many minutes will it take to fill the tank?\n\nSolution:\nLet's analyze the situation:\n\n* Water fl... |
A right circular cone has a slant height of $15$ and a base radius of $6$. Determine the total surface area of the cone, including the base. | A right circular cone has a slant height of $15$ and a base radius of $6$. Determine the total surface area of the cone, including the base. | Okay, so I need to find the total surface area of a right circular cone. The problem gives me the slant height, which is 15, and the base radius, which is 6. Hmm, total surface area includes both the lateral (side) surface area and the area of the base, right? Let me remember the formulas for these.
First, the lateral... | To determine the total surface area of a right circular cone with a slant height of 15 and a base radius of 6, we use the formulas for the lateral surface area and the base area.
1. **Lateral Surface Area (LSA)**: The formula is \( \pi r l \), where \( r \) is the radius and \( l \) is the slant height.
\[
\text... | 19,439 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the total surface area of a right circular cone. The problem gives me the slant height, which is 15, and the base radius, which is 6. Hmm, total surface area includes both the lateral (side) surface area and the area of the base, right? Let me remember the formulas for these.
First, the... | [
{
"from": "human",
"value": "A right circular cone has a slant height of $15$ and a base radius of $6$. Determine the total surface area of the cone, including the base."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the total surface area of a right circular cone. The problem g... |
Let a sequence of consecutive natural numbers be given. Determine $n$ if the sum of all the terms is to be a divisor of the sum of all the even terms. | Let a sequence of consecutive natural numbers be given. Determine $n$ if the sum of all the terms is to be a divisor of the sum of all the even terms. | Okay, so I have this problem here: Given a sequence of consecutive natural numbers, determine n if the sum of all the terms is to be a divisor of the sum of all the even terms. Hmm, let me try to wrap my head around this.
First, let me restate the problem in my own words. We have a sequence of consecutive natural num... | Given a sequence of consecutive natural numbers, we need to determine \( n \) such that the sum of all the terms is a divisor of the sum of all the even terms.
1. **Understanding the Problem**:
- We need to find \( n \) such that for any sequence of \( n \) consecutive natural numbers, the sum of all terms (total s... | 16,236 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: Given a sequence of consecutive natural numbers, determine n if the sum of all the terms is to be a divisor of the sum of all the even terms. Hmm, let me try to wrap my head around this.
First, let me restate the problem in my own words. We have a sequence of consecutive nat... | [
{
"from": "human",
"value": "Let a sequence of consecutive natural numbers be given. Determine $n$ if the sum of all the terms is to be a divisor of the sum of all the even terms."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: Given a sequence of consecutive natural n... |
Let $p(x)$ be a monic cubic polynomial such that $p(1) = 3$, $p(2) = 4$, $p(3) = 5$, and $p(4) = 6$. Find the remainder when $p(1)+p(2)+p(3)+\cdots+p(100)$ is divided by $101$. | Let $p(x)$ be a monic cubic polynomial such that $p(1) = 3$, $p(2) = 4$, $p(3) = 5$, and $p(4) = 6$. Find the remainder when $p(1)+p(2)+p(3)+\cdots+p(100)$ is divided by $101$. | Okay, so I need to find the remainder when the sum of p(1) to p(100) is divided by 101, where p(x) is a monic cubic polynomial that satisfies p(1)=3, p(2)=4, p(3)=5, and p(4)=6. Hmm, monic cubic means the leading coefficient is 1, so the polynomial looks like p(x) = x³ + ax² + bx + c.
First, maybe I can find the coef... | Given a monic cubic polynomial \( p(x) \) such that \( p(1) = 3 \), \( p(2) = 4 \), \( p(3) = 5 \), and \( p(4) = 6 \), we need to find the remainder when \( p(1) + p(2) + \cdots + p(100) \) is divided by 101.
1. **Formulating the Polynomial**:
- We start by noting that \( p(x) - (x + 2) \) must be a cubic polynomi... | 31,416 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the remainder when the sum of p(1) to p(100) is divided by 101, where p(x) is a monic cubic polynomial that satisfies p(1)=3, p(2)=4, p(3)=5, and p(4)=6. Hmm, monic cubic means the leading coefficient is 1, so the polynomial looks like p(x) = x³ + ax² + bx + c.
First, maybe I can find ... | [
{
"from": "human",
"value": "Let $p(x)$ be a monic cubic polynomial such that $p(1) = 3$, $p(2) = 4$, $p(3) = 5$, and $p(4) = 6$. Find the remainder when $p(1)+p(2)+p(3)+\\cdots+p(100)$ is divided by $101$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the remainder when the su... |
A regular polyhedron has faces that are regular polygons, and each face is surrounded by exactly 5 edges. What is the minimum number of edges that the polyhedron can have? | A regular polyhedron has faces that are regular polygons, and each face is surrounded by exactly 5 edges. What is the minimum number of edges that the polyhedron can have? | Okay, so I need to figure out the minimum number of edges a regular polyhedron can have if each face is surrounded by exactly 5 edges. Hmm, let's start by recalling what a regular polyhedron is. A regular polyhedron, also known as a Platonic solid, is a convex polyhedron where all faces are congruent regular polygons, ... | To determine the minimum number of edges a regular polyhedron can have if each face is surrounded by exactly 5 edges, we start by noting that each face must be a regular pentagon. A regular polyhedron with all faces as regular pentagons is a regular dodecahedron, which is a Platonic solid.
For a regular dodecahedron:
... | 3,985 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the minimum number of edges a regular polyhedron can have if each face is surrounded by exactly 5 edges. Hmm, let's start by recalling what a regular polyhedron is. A regular polyhedron, also known as a Platonic solid, is a convex polyhedron where all faces are congruent regular po... | [
{
"from": "human",
"value": "A regular polyhedron has faces that are regular polygons, and each face is surrounded by exactly 5 edges. What is the minimum number of edges that the polyhedron can have?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the minimum number of edg... |
What is the product of the natural-number factors of 8 that are also perfect squares? | What is the product of the natural-number factors of 8 that are also perfect squares? | Okay, let's see. The question is asking for the product of the natural-number factors of 8 that are also perfect squares. Hmm, natural-number factors... So first, I need to find all the factors of 8, and then among those, pick out the ones that are perfect squares. After that, multiply them together to get the product.... | To find the product of the natural-number factors of 8 that are also perfect squares, we start by identifying all factors of 8. The factors of 8 are 1, 2, 4, and 8.
Next, we determine which of these factors are perfect squares. A perfect square is a number that is the square of an integer. Checking each factor:
- 1 i... | 15,645 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The question is asking for the product of the natural-number factors of 8 that are also perfect squares. Hmm, natural-number factors... So first, I need to find all the factors of 8, and then among those, pick out the ones that are perfect squares. After that, multiply them together to get the ... | [
{
"from": "human",
"value": "What is the product of the natural-number factors of 8 that are also perfect squares?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The question is asking for the product of the natural-number factors of 8 that are also perfect squares. Hmm, natural-number... |
On the Cartesian plane, the equation of a conic section is given by
\[\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1.\]A reflecting wall is located at $x = 2$ which reflects any light coming from the conic section in a way that the incidence angle equals the reflection angle. The conic section and the wall intersect at a single... | On the Cartesian plane, the equation of a conic section is given by
\[\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1.\]A reflecting wall is located at $x = 2$ which reflects any light coming from the conic section in a way that the incidence angle equals the reflection angle. The conic section and the wall intersect at a single... | Okay, let me try to figure out this problem step by step. So, we have a conic section given by the equation \(\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1\). That looks like an ellipse to me because both terms are positive and added together. The standard form of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where ... | The given equation of the conic section is \(\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1\), which represents an ellipse centered at the origin with a semi-major axis of 4 and a semi-minor axis of 3. The reflecting wall is located at \(x = 2\).
To find the intersection points of the ellipse and the vertical line \(x = 2\), w... | 12,737 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem step by step. So, we have a conic section given by the equation \(\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1\). That looks like an ellipse to me because both terms are positive and added together. The standard form of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)... | [
{
"from": "human",
"value": "On the Cartesian plane, the equation of a conic section is given by\n\\[\\frac{x^2}{3^2} + \\frac{y^2}{4^2} = 1.\\]A reflecting wall is located at $x = 2$ which reflects any light coming from the conic section in a way that the incidence angle equals the reflection angle. The co... |
In a triangle $ABC$ with $\angle B = 30^\circ$, the length of the median to side $AC$ is $8$ and the length of the altitude to side $AC$ is $5$. Find $\tan A$. | In a triangle $ABC$ with $\angle B = 30^\circ$, the length of the median to side $AC$ is $8$ and the length of the altitude to side $AC$ is $5$. Find $\tan A$. | Okay, so I have this geometry problem here. Let me try to parse it first. It's about triangle ABC where angle B is 30 degrees. They mention the median to side AC is 8 units long, and the altitude to side AC is 5 units. I need to find tan A. Hmm. Let me sketch this in my mind. Triangle ABC, angle at B is 30 degrees. The... | Given triangle \(ABC\) with \(\angle B = 30^\circ\), the median to side \(AC\) is 8, and the altitude to side \(AC\) is 5. We need to find \(\tan A\).
1. **Identify key elements and equations:**
- The median \(BM = 8\) from \(B\) to \(AC\).
- The altitude \(BH = 5\) from \(B\) to \(AC\).
- Use the median form... | 5,597 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this geometry problem here. Let me try to parse it first. It's about triangle ABC where angle B is 30 degrees. They mention the median to side AC is 8 units long, and the altitude to side AC is 5 units. I need to find tan A. Hmm. Let me sketch this in my mind. Triangle ABC, angle at B is 30 degr... | [
{
"from": "human",
"value": "In a triangle $ABC$ with $\\angle B = 30^\\circ$, the length of the median to side $AC$ is $8$ and the length of the altitude to side $AC$ is $5$. Find $\\tan A$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this geometry problem here. Let me try to parse ... |
A circular garden is surrounded by a path that is 2 meters wide. The total area of the garden and the path together is 64π square meters. What is the radius of the garden? | A circular garden is surrounded by a path that is 2 meters wide. The total area of the garden and the path together is 64π square meters. What is the radius of the garden? | Okay, let me try to figure this out. So, there's a circular garden with a path around it that's 2 meters wide. The total area of both the garden and the path is 64π square meters. I need to find the radius of just the garden itself.
First, let me visualize this. Imagine a circle (the garden) and then a bigger circle ... | Let the radius of the circular garden be \( r \) meters. The path surrounding the garden is 2 meters wide, so the radius of the entire area (garden plus path) is \( r + 2 \) meters.
The total area of the garden and the path together is given as \( 64\pi \) square meters. Using the formula for the area of a circle, we ... | 14,525 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure this out. So, there's a circular garden with a path around it that's 2 meters wide. The total area of both the garden and the path is 64π square meters. I need to find the radius of just the garden itself.
First, let me visualize this. Imagine a circle (the garden) and then a bigger... | [
{
"from": "human",
"value": "A circular garden is surrounded by a path that is 2 meters wide. The total area of the garden and the path together is 64π square meters. What is the radius of the garden?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure this out. So, there's a circ... |
A certain country has a currency that comes in denominations of $2, $5, and $10 bills. A teller at a bank needs to dispense a customer's withdrawal of $247 using the fewest possible number of bills. How many bills of each denomination should the teller give to the customer? | A certain country has a currency that comes in denominations of $2, $5, and $10 bills. A teller at a bank needs to dispense a customer's withdrawal of $247 using the fewest possible number of bills. How many bills of each denomination should the teller give to the customer? | Okay, so I need to figure out how to make $247 using the least number of bills possible with denominations of $2, $5, and $10. Let me think. The goal is to minimize the total number of bills, so I should use as many of the highest denomination as possible, which is $10, then $5, and then $2. But I need to make sure tha... | To dispense $247 using the fewest possible number of bills with denominations of $2, $5, and $10, we start by maximizing the number of $10 bills, then $5 bills, and finally $2 bills.
1. **Maximizing $10 bills**:
- Divide 247 by 10: \(247 \div 10 = 24.7\). The maximum number of $10 bills is 24.
- Calculate the r... | 7,200 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how to make $247 using the least number of bills possible with denominations of $2, $5, and $10. Let me think. The goal is to minimize the total number of bills, so I should use as many of the highest denomination as possible, which is $10, then $5, and then $2. But I need to make ... | [
{
"from": "human",
"value": "A certain country has a currency that comes in denominations of $2, $5, and $10 bills. A teller at a bank needs to dispense a customer's withdrawal of $247 using the fewest possible number of bills. How many bills of each denomination should the teller give to the customer?"
}... |
Solve the inequality $|2x-5| + x^2 < 11$ for $x$. | Solve the inequality $|2x-5| + x^2 < 11$ for $x$. | Okay, so I need to solve the inequality |2x - 5| + x² < 11. Hmm, absolute value inequalities can sometimes be tricky because you have to consider different cases. Let me start by recalling that |2x - 5| means the expression inside can be either positive or negative, so I might need to split this into two separate inequ... | To solve the inequality \( |2x - 5| + x^2 < 11 \), we consider two cases based on the absolute value expression.
**Case 1: \( 2x - 5 \geq 0 \) (i.e., \( x \geq \frac{5}{2} \))**
In this case, \( |2x - 5| = 2x - 5 \). Substituting this into the inequality gives:
\[
(2x - 5) + x^2 < 11 \implies x^2 + 2x - 16 < 0
\]
Sol... | 9,237 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve the inequality |2x - 5| + x² < 11. Hmm, absolute value inequalities can sometimes be tricky because you have to consider different cases. Let me start by recalling that |2x - 5| means the expression inside can be either positive or negative, so I might need to split this into two separa... | [
{
"from": "human",
"value": "Solve the inequality $|2x-5| + x^2 < 11$ for $x$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve the inequality |2x - 5| + x² < 11. Hmm, absolute value inequalities can sometimes be tricky because you have to consider different cases. Let me start b... |
A shirt is on sale for 15% off its original price. If the sale price is $34, what was the original price of the shirt? | A shirt is on sale for 15% off its original price. If the sale price is $34, what was the original price of the shirt? | Okay, so there's this shirt that's on sale for 15% off its original price, and after the discount, it's priced at $34. I need to find out what the original price was before the discount. Hmm, let's think.
First, I know that when something is on sale for a certain percentage off, the sale price is the original price m... | Let the original price of the shirt be \( P \). The shirt is on sale for 15% off, so the sale price is 85% of the original price. This can be expressed as:
\[ 0.85P = 34 \]
To find \( P \), we divide both sides of the equation by 0.85:
\[ P = \frac{34}{0.85} \]
Calculating the division:
\[ P = \frac{34}{0.85} = 40... | 7,884 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this shirt that's on sale for 15% off its original price, and after the discount, it's priced at $34. I need to find out what the original price was before the discount. Hmm, let's think.
First, I know that when something is on sale for a certain percentage off, the sale price is the original... | [
{
"from": "human",
"value": "A shirt is on sale for 15% off its original price. If the sale price is $34, what was the original price of the shirt?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so there's this shirt that's on sale for 15% off its original price, and after the discount, it's pric... |
Consider a convex quadrilateral $ABCD$ with all sides positive integers, where $AB < BC < CD < AD$. The diagonals $AC$ and $BD$ are also positive integers, such that $AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + AD^2$. How many possible quadrilaterals satisfying these conditions exist? | Consider a convex quadrilateral $ABCD$ with all sides positive integers, where $AB < BC < CD < AD$. The diagonals $AC$ and $BD$ are also positive integers, such that $AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + AD^2$. How many possible quadrilaterals satisfying these conditions exist? | Okay, let's see. The problem is about a convex quadrilateral ABCD with all sides positive integers, where the sides are ordered AB < BC < CD < AD. The diagonals AC and BD are also positive integers, and they satisfy the equation AC² + BD² = AB² + BC² + CD² + AD². We need to find how many such quadrilaterals exist.
Fir... | Given a convex quadrilateral \(ABCD\) with all sides as positive integers, where \(AB < BC < CD < AD\), and the diagonals \(AC\) and \(BD\) are also positive integers such that \(AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + AD^2\), we need to determine how many such quadrilaterals exist.
First, we recall the British flag theore... | 15,841 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is about a convex quadrilateral ABCD with all sides positive integers, where the sides are ordered AB < BC < CD < AD. The diagonals AC and BD are also positive integers, and they satisfy the equation AC² + BD² = AB² + BC² + CD² + AD². We need to find how many such quadrilaterals exi... | [
{
"from": "human",
"value": "Consider a convex quadrilateral $ABCD$ with all sides positive integers, where $AB < BC < CD < AD$. The diagonals $AC$ and $BD$ are also positive integers, such that $AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + AD^2$. How many possible quadrilaterals satisfying these conditions exist?"
... |
The sum of the squares of two consecutive integers is 365. What is the product of the two integers? | The sum of the squares of two consecutive integers is 365. What is the product of the two integers? | Okay, so I need to find two consecutive integers whose squares add up to 365, and then multiply them together to get the product. Hmm, let's start by breaking down the problem.
First, let's define the integers. Since they are consecutive, if one integer is, say, \( n \), then the next one should be \( n + 1 \). That m... | Let the two consecutive integers be \( n \) and \( n + 1 \). The sum of their squares is given as 365:
\[
n^2 + (n + 1)^2 = 365
\]
Expanding and simplifying the equation:
\[
n^2 + n^2 + 2n + 1 = 365 \\
2n^2 + 2n + 1 = 365 \\
2n^2 + 2n - 364 = 0 \\
n^2 + n - 182 = 0
\]
Factoring the quadratic equation:
\[
(n + 14)(... | 33,650 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find two consecutive integers whose squares add up to 365, and then multiply them together to get the product. Hmm, let's start by breaking down the problem.
First, let's define the integers. Since they are consecutive, if one integer is, say, \( n \), then the next one should be \( n + 1 \)... | [
{
"from": "human",
"value": "The sum of the squares of two consecutive integers is 365. What is the product of the two integers?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find two consecutive integers whose squares add up to 365, and then multiply them together to get the produc... |
What is the ones digit of $1^{2009} + 2^{2009} + 3^{2009} + \cdots + 2009^{2009}?$ | What is the ones digit of $1^{2009} + 2^{2009} + 3^{2009} + \cdots + 2009^{2009}?$ | Okay, so I need to find the ones digit of the sum \(1^{2009} + 2^{2009} + \cdots + 2009^{2009}\). Hmm, that's a pretty big exponent and a lot of terms. But since we're only interested in the ones digit of the total sum, maybe I can find a pattern or use modular arithmetic with modulo 10. Let me recall that the ones dig... | To find the ones digit of \(1^{2009} + 2^{2009} + 3^{2009} + \cdots + 2009^{2009}\), we need to consider each term modulo 10.
First, we observe that for any digit \(d\) (0-9), \(d^{2009} \mod 10\) results in \(d\) itself. This is because the cycles of exponents modulo 10 for each digit \(d\) repeat such that \(d^{200... | 9,323 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the ones digit of the sum \(1^{2009} + 2^{2009} + \cdots + 2009^{2009}\). Hmm, that's a pretty big exponent and a lot of terms. But since we're only interested in the ones digit of the total sum, maybe I can find a pattern or use modular arithmetic with modulo 10. Let me recall that the ... | [
{
"from": "human",
"value": "What is the ones digit of $1^{2009} + 2^{2009} + 3^{2009} + \\cdots + 2009^{2009}?$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the ones digit of the sum \\(1^{2009} + 2^{2009} + \\cdots + 2009^{2009}\\). Hmm, that's a pretty big exponent and a lo... |
Let $n$ be a positive number such that when $179$ is divided by $n$, the remainder is $7.$ Compute the sum of all possible two-digit values of $n$. | Let $n$ be a positive number such that when $179$ is divided by $n$, the remainder is $7.$ Compute the sum of all possible two-digit values of $n$. | Alright, let's see. I have this problem here: Find all two-digit numbers n such that when 179 is divided by n, the remainder is 7. Then sum all those n's. Hmm, okay. Let's break this down step by step.
First, I remember that when a number is divided by another number and there's a remainder, there's a relationship bet... | Given that when 179 is divided by \( n \), the remainder is 7, we can express this relationship as:
\[
179 = n \times q + 7
\]
where \( q \) is the quotient. Rearranging this equation, we get:
\[
n \times q = 179 - 7 = 172
\]
This implies that \( n \) must be a divisor of 172. The prime factorization of 172 is:
\[
172 ... | 4,109 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's see. I have this problem here: Find all two-digit numbers n such that when 179 is divided by n, the remainder is 7. Then sum all those n's. Hmm, okay. Let's break this down step by step.
First, I remember that when a number is divided by another number and there's a remainder, there's a relation... | [
{
"from": "human",
"value": "Let $n$ be a positive number such that when $179$ is divided by $n$, the remainder is $7.$ Compute the sum of all possible two-digit values of $n$."
},
{
"from": "gpt",
"value": "<think>\nAlright, let's see. I have this problem here: Find all two-digit numbers n such... |
At a party, there are 9 guests who wish to participate in a game that involves standing in a line. Due to the rules of the game, no guest can stand next to someone wearing the same color shirt. How many ways can the guests arrange themselves in a line if 3 of them are wearing red shirts, 3 are wearing blue shirts, and ... | At a party, there are 9 guests who wish to participate in a game that involves standing in a line. Due to the rules of the game, no guest can stand next to someone wearing the same color shirt. How many ways can the guests arrange themselves in a line if 3 of them are wearing red shirts, 3 are wearing blue shirts, and ... | Okay, so there's this problem about arranging 9 guests in a line for a game, and they can't stand next to someone with the same color shirt. The shirts are 3 red, 3 blue, and 3 green. Hmm, interesting. Let me try to figure out how to approach this.
First, I know that without any restrictions, the number of ways to arr... | To determine the number of ways to arrange 9 guests wearing red, blue, and green shirts (3 each) such that no two guests with the same color shirt stand next to each other, we need to consider the following steps:
1. **Color Permutations**: First, we need to find the number of valid color permutations where no two adj... | 15,585 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this problem about arranging 9 guests in a line for a game, and they can't stand next to someone with the same color shirt. The shirts are 3 red, 3 blue, and 3 green. Hmm, interesting. Let me try to figure out how to approach this.
First, I know that without any restrictions, the number of way... | [
{
"from": "human",
"value": "At a party, there are 9 guests who wish to participate in a game that involves standing in a line. Due to the rules of the game, no guest can stand next to someone wearing the same color shirt. How many ways can the guests arrange themselves in a line if 3 of them are wearing re... |
Compute the largest integer k such that 2019^k divides the sum of all integers from 1 to 2019^3. | Compute the largest integer k such that 2019^k divides the sum of all integers from 1 to 2019^3. | Alright, let's tackle this problem: Compute the largest integer k such that 2019^k divides the sum of all integers from 1 to 2019³. Hmm, okay. So, first, I need to understand what exactly is being asked here. The question is about finding the highest power of 2019 that divides the sum S = 1 + 2 + 3 + ... + 2019³. Then,... | To compute the largest integer \( k \) such that \( 2019^k \) divides the sum of all integers from 1 to \( 2019^3 \), we start by noting that the sum of the first \( n \) integers is given by \( S = \frac{n(n+1)}{2} \). Here, \( n = 2019^3 \), so the sum is:
\[
S = \frac{2019^3 \cdot (2019^3 + 1)}{2}
\]
We need to de... | 3,358 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's tackle this problem: Compute the largest integer k such that 2019^k divides the sum of all integers from 1 to 2019³. Hmm, okay. So, first, I need to understand what exactly is being asked here. The question is about finding the highest power of 2019 that divides the sum S = 1 + 2 + 3 + ... + 2019... | [
{
"from": "human",
"value": "Compute the largest integer k such that 2019^k divides the sum of all integers from 1 to 2019^3."
},
{
"from": "gpt",
"value": "<think>\nAlright, let's tackle this problem: Compute the largest integer k such that 2019^k divides the sum of all integers from 1 to 2019³... |
Let $m,$ $n,$ $p,$ $q$ be nonzero integers such that
\[\begin{pmatrix} m & n \\ p & q \end{pmatrix}^3 = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}.\]Find the smallest possible value of $|m| + |n| + |p| + |q|.$ | Let $m,$ $n,$ $p,$ $q$ be nonzero integers such that
\[\begin{pmatrix} m & n \\ p & q \end{pmatrix}^3 = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}.\]Find the smallest possible value of $|m| + |n| + |p| + |q|.$ | Okay, so I need to find nonzero integers m, n, p, q such that when you cube the matrix \(\begin{pmatrix} m & n \\ p & q \end{pmatrix}\), you get \(\begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}\). Then, among all such possible matrices, I need to find the one where the sum of the absolute values of its entries, |m| + |n|... | Given the problem of finding nonzero integers \(m, n, p, q\) such that the matrix \(\begin{pmatrix} m & n \\ p & q \end{pmatrix}^3 = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}\), we need to find the smallest possible value of \(|m| + |n| + |p| + |q|\).
Key steps and reasoning:
1. **Determinant Analysis**: The determ... | 17,348 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find nonzero integers m, n, p, q such that when you cube the matrix \(\begin{pmatrix} m & n \\ p & q \end{pmatrix}\), you get \(\begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}\). Then, among all such possible matrices, I need to find the one where the sum of the absolute values of its entries, |... | [
{
"from": "human",
"value": "Let $m,$ $n,$ $p,$ $q$ be nonzero integers such that\n\\[\\begin{pmatrix} m & n \\\\ p & q \\end{pmatrix}^3 = \\begin{pmatrix} 5 & 0 \\\\ 0 & 5 \\end{pmatrix}.\\]Find the smallest possible value of $|m| + |n| + |p| + |q|.$"
},
{
"from": "gpt",
"value": "<think>\nOkay... |
In a rectangular coordinate system, the equations of two circles are given by
(x - 2)^2 + (y - 3)^2 = r^2
and
(x + 2)^2 + (y + 3)^2 = r^2.
If the circles intersect at exactly one point, determine the radius, r, of the circles. | In a rectangular coordinate system, the equations of two circles are given by
(x - 2)^2 + (y - 3)^2 = r^2
and
(x + 2)^2 + (y + 3)^2 = r^2.
If the circles intersect at exactly one point, determine the radius, r, of the circles. | Okay, so I need to figure out the radius r such that these two circles intersect at exactly one point. Hmm, let's see. Both circles have the same radius r, right? Their centers are at (2, 3) and (-2, -3). First, maybe I should visualize this. The first circle is in the first quadrant, and the second one is in the third... | To determine the radius \( r \) such that the circles \((x - 2)^2 + (y - 3)^2 = r^2\) and \((x + 2)^2 + (y + 3)^2 = r^2\) intersect at exactly one point, we start by calculating the distance between their centers. The centers are \((2, 3)\) and \((-2, -3)\). Using the distance formula:
\[
\sqrt[(-2 - 2)^2 + (-3 - 3)^2... | 24,038 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the radius r such that these two circles intersect at exactly one point. Hmm, let's see. Both circles have the same radius r, right? Their centers are at (2, 3) and (-2, -3). First, maybe I should visualize this. The first circle is in the first quadrant, and the second one is in t... | [
{
"from": "human",
"value": "In a rectangular coordinate system, the equations of two circles are given by\n\n(x - 2)^2 + (y - 3)^2 = r^2\n\nand\n\n(x + 2)^2 + (y + 3)^2 = r^2.\n\nIf the circles intersect at exactly one point, determine the radius, r, of the circles."
},
{
"from": "gpt",
"value"... |
A certain library has a collection of books in a specific ratio of fiction to non-fiction books. If the number of fiction books is increased by 20%, and the number of non-fiction books is decreased by 15%, the new ratio of fiction to non-fiction books becomes 2:1. What was the original ratio of fiction to non-fiction b... | A certain library has a collection of books in a specific ratio of fiction to non-fiction books. If the number of fiction books is increased by 20%, and the number of non-fiction books is decreased by 15%, the new ratio of fiction to non-fiction books becomes 2:1. What was the original ratio of fiction to non-fiction b... | Okay, so the problem is about a library's collection of books with a certain ratio of fiction to non-fiction. Then, when the number of fiction books is increased by 20% and non-fiction decreased by 15%, the new ratio becomes 2:1. We need to find the original ratio.
Let me start by setting up variables. Let's say the o... | Let \( F \) be the original number of fiction books and \( N \) be the original number of non-fiction books. The original ratio of fiction to non-fiction books is \( \frac{F}{N} \).
After a 20% increase in fiction books, the number of fiction books becomes \( 1.2F \). After a 15% decrease in non-fiction books, the num... | 1,273 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so the problem is about a library's collection of books with a certain ratio of fiction to non-fiction. Then, when the number of fiction books is increased by 20% and non-fiction decreased by 15%, the new ratio becomes 2:1. We need to find the original ratio.
Let me start by setting up variables. Let's s... | [
{
"from": "human",
"value": "A certain library has a collection of books in a specific ratio of fiction to non-fiction books. If the number of fiction books is increased by 20%, and the number of non-fiction books is decreased by 15%, the new ratio of fiction to non-fiction books becomes 2:1. What was the o... |
In a certain kingdom, each of the 64 cities has its own population, with the population of each city being twice that of the previous one. If the total population of the kingdom is 18446744073709551615, how many people live in the capital city, which is the 20th city in the list of cities by population? | In a certain kingdom, each of the 64 cities has its own population, with the population of each city being twice that of the previous one. If the total population of the kingdom is 18446744073709551615, how many people live in the capital city, which is the 20th city in the list of cities by population? | Okay, let me try to figure out this problem step by step. So, there are 64 cities in the kingdom, each with a population that's twice the previous one. The total population is given as 18,446,744,073,709,551,615. We need to find the population of the 20th city, which is the capital. Hmm, okay.
First, let's make sure I... | Given that each of the 64 cities in the kingdom has a population twice that of the previous one, and the total population is 18,446,744,073,709,551,615, we need to find the population of the 20th city.
1. Recognize that the populations form a geometric series where each term is twice the previous one. The total popula... | 7,329 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem step by step. So, there are 64 cities in the kingdom, each with a population that's twice the previous one. The total population is given as 18,446,744,073,709,551,615. We need to find the population of the 20th city, which is the capital. Hmm, okay.
First, let's mak... | [
{
"from": "human",
"value": "In a certain kingdom, each of the 64 cities has its own population, with the population of each city being twice that of the previous one. If the total population of the kingdom is 18446744073709551615, how many people live in the capital city, which is the 20th city in the list... |
A regular polygon has vertices at $(0,0)$, $(1,0)$, and $(x,y)$. If the area of the triangle formed by these vertices is 4 square units, find the length of the shortest diagonal of the polygon.
(No further text is needed here.) | A regular polygon has vertices at $(0,0)$, $(1,0)$, and $(x,y)$. If the area of the triangle formed by these vertices is 4 square units, find the length of the shortest diagonal of the polygon.
(No further text is needed here.) | Okay, let's tackle this problem step by step. First, let's make sure I understand what's being asked. We have a regular polygon with three vertices given: (0,0), (1,0), and (x,y). The area of the triangle formed by these three points is 4 square units. We need to find the length of the shortest diagonal of this polygon... | Given a regular polygon with vertices at \((0,0)\), \((1,0)\), and \((x,y)\), and the area of the triangle formed by these vertices is 4 square units, we need to find the length of the shortest diagonal of the polygon.
1. **Area Calculation**:
The area of the triangle formed by the vertices \((0,0)\), \((1,0)\), an... | 36,941 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's tackle this problem step by step. First, let's make sure I understand what's being asked. We have a regular polygon with three vertices given: (0,0), (1,0), and (x,y). The area of the triangle formed by these three points is 4 square units. We need to find the length of the shortest diagonal of this... | [
{
"from": "human",
"value": "A regular polygon has vertices at $(0,0)$, $(1,0)$, and $(x,y)$. If the area of the triangle formed by these vertices is 4 square units, find the length of the shortest diagonal of the polygon.\n\n(No further text is needed here.)"
},
{
"from": "gpt",
"value": "<thin... |
A five-digit integer $A8C35$ is divisible by 6. What is the smallest possible value of $A+C$? | A five-digit integer $A8C35$ is divisible by 6. What is the smallest possible value of $A+C$? | Okay, so I have this problem here: A five-digit integer A8C35 is divisible by 6. I need to find the smallest possible value of A + C. Alright, let me think step by step.
First, let me recall that a number divisible by 6 must satisfy two conditions: it has to be divisible by both 2 and 3. So, I need to check what that... | To determine the smallest possible value of \(A + C\) for the five-digit integer \(A8C35\) to be divisible by 6, we need to check the conditions for divisibility by both 2 and 3.
1. **Divisibility by 2**: The last digit must be even. However, the last digit is 5, which is odd. This means the number cannot be divisible... | 21,387 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: A five-digit integer A8C35 is divisible by 6. I need to find the smallest possible value of A + C. Alright, let me think step by step.
First, let me recall that a number divisible by 6 must satisfy two conditions: it has to be divisible by both 2 and 3. So, I need to check w... | [
{
"from": "human",
"value": "A five-digit integer $A8C35$ is divisible by 6. What is the smallest possible value of $A+C$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: A five-digit integer A8C35 is divisible by 6. I need to find the smallest possible value of A + C.... |
Consider a triangular array of numbers where each number is the sum of the two numbers directly above it, but with the added condition that every third row is reversed. Determine the fifth element in Row 21 of this modified triangle. | Consider a triangular array of numbers where each number is the sum of the two numbers directly above it, but with the added condition that every third row is reversed. Determine the fifth element in Row 21 of this modified triangle. | Okay, so I need to figure out the fifth element in Row 21 of this modified triangular array. The triangle is similar to Pascal's Triangle, where each number is the sum of the two above it, but with a twist: every third row is reversed. Hmm, interesting. Let me break this down step by step.
First, let me recall how Pas... | To determine the fifth element in Row 21 of the modified triangular array where every third row is reversed, we start by noting that each number in the triangle is the sum of the two numbers directly above it, similar to Pascal's Triangle. However, every third row is reversed.
Key observations:
1. In Pascal's Triangl... | 29,429 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the fifth element in Row 21 of this modified triangular array. The triangle is similar to Pascal's Triangle, where each number is the sum of the two above it, but with a twist: every third row is reversed. Hmm, interesting. Let me break this down step by step.
First, let me recall... | [
{
"from": "human",
"value": "Consider a triangular array of numbers where each number is the sum of the two numbers directly above it, but with the added condition that every third row is reversed. Determine the fifth element in Row 21 of this modified triangle."
},
{
"from": "gpt",
"value": "<t... |
What is the coefficient of $x^3$ in the expansion of the expression $$2(x^2+3x-4)(x+5)-(3x^2-x+6)(x^2+2)?$$ | What is the coefficient of $x^3$ in the expansion of the expression $$2(x^2+3x-4)(x+5)-(3x^2-x+6)(x^2+2)?$$ | Okay, so I need to find the coefficient of \( x^3 \) in the expansion of the expression \( 2(x^2 + 3x - 4)(x + 5) - (3x^2 - x + 6)(x^2 + 2) \). Hmm, let's break this down step by step.
First, I know that to find the coefficient of a specific term, I don't have to expand the entire expression. Instead, I can focus on ... | To find the coefficient of \(x^3\) in the expansion of the expression \(2(x^2 + 3x - 4)(x + 5) - (3x^2 - x + 6)(x^2 + 2)\), we focus on the terms that contribute to \(x^3\).
First, consider the expansion of \(2(x^2 + 3x - 4)(x + 5)\):
1. The term contributing to \(x^3\) is \(x^2 \cdot x = x^3\).
2. After expanding an... | 27,796 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the coefficient of \( x^3 \) in the expansion of the expression \( 2(x^2 + 3x - 4)(x + 5) - (3x^2 - x + 6)(x^2 + 2) \). Hmm, let's break this down step by step.
First, I know that to find the coefficient of a specific term, I don't have to expand the entire expression. Instead, I can f... | [
{
"from": "human",
"value": "What is the coefficient of $x^3$ in the expansion of the expression $$2(x^2+3x-4)(x+5)-(3x^2-x+6)(x^2+2)?$$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the coefficient of \\( x^3 \\) in the expansion of the expression \\( 2(x^2 + 3x - 4)(x + 5) - ... |
If $m=2^2 \cdot 3^3 \cdot 7$, how many odd positive factors does $m$ have? | If $m=2^2 \cdot 3^3 \cdot 7$, how many odd positive factors does $m$ have? | Okay, so I need to find how many odd positive factors the number m has, where m is given as 2 squared times 3 cubed times 7. Let me start by understanding the problem step by step.
First, let me write down the prime factorization of m again to make sure I have it right. The problem says m = 2² · 3³ · 7. So that breaks... | To determine the number of odd positive factors of \( m = 2^2 \cdot 3^3 \cdot 7 \), we need to exclude the prime factor 2, as any factor including 2 would be even.
The remaining prime factors are \( 3^3 \) and \( 7^1 \). The number of factors of a number is found by adding 1 to each of the exponents in its prime fact... | 36,961 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find how many odd positive factors the number m has, where m is given as 2 squared times 3 cubed times 7. Let me start by understanding the problem step by step.
First, let me write down the prime factorization of m again to make sure I have it right. The problem says m = 2² · 3³ · 7. So tha... | [
{
"from": "human",
"value": "If $m=2^2 \\cdot 3^3 \\cdot 7$, how many odd positive factors does $m$ have?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find how many odd positive factors the number m has, where m is given as 2 squared times 3 cubed times 7. Let me start by understan... |
Eugene, Brianna, and Katie are going on a run. Eugene runs at a rate of 4 miles per hour. If Brianna runs $\frac{2}{3}$ as fast as Eugene, and Katie runs $\frac{7}{5}$ as fast as Brianna, how fast does Katie run? | Eugene, Brianna, and Katie are going on a run. Eugene runs at a rate of 4 miles per hour. If Brianna runs $\frac{2}{3}$ as fast as Eugene, and Katie runs $\frac{7}{5}$ as fast as Brianna, how fast does Katie run? | Okay, let's see. The problem is about Eugene, Brianna, and Katie going on a run. We need to find out how fast Katie runs. The given information is that Eugene runs at 4 miles per hour. Brianna runs two-thirds as fast as Eugene, and Katie runs seven-fifths as fast as Brianna. Hmm, so it's like a chain of rates here. Eug... | Eugene runs at a rate of 4 miles per hour. Brianna runs \(\frac{2}{3}\) as fast as Eugene, so we calculate Brianna's speed as follows:
\[
\text{Brianna's speed} = \frac{2}{3} \times 4 = \frac{8}{3} \text{ miles per hour}
\]
Katie runs \(\frac{7}{5}\) as fast as Brianna. Using Brianna's speed, we calculate Katie's spe... | 32,783 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is about Eugene, Brianna, and Katie going on a run. We need to find out how fast Katie runs. The given information is that Eugene runs at 4 miles per hour. Brianna runs two-thirds as fast as Eugene, and Katie runs seven-fifths as fast as Brianna. Hmm, so it's like a chain of rates h... | [
{
"from": "human",
"value": "Eugene, Brianna, and Katie are going on a run. Eugene runs at a rate of 4 miles per hour. If Brianna runs $\\frac{2}{3}$ as fast as Eugene, and Katie runs $\\frac{7}{5}$ as fast as Brianna, how fast does Katie run?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's ... |
Find the greatest possible value of $n$ such that $8^{n}$ divides $100!$ ($100!=100\times99\times98\times\cdots\times 3\times 2\times 1$). | Find the greatest possible value of $n$ such that $8^{n}$ divides $100!$ ($100!=100\times99\times98\times\cdots\times 3\times 2\times 1$). | Okay, so I need to find the greatest possible value of n such that 8^n divides 100!. Hmm, 8 is 2 cubed, so maybe I need to figure out how many times 2 appears as a factor in 100! and then divide that by 3? Because 8 is 2^3, right? So each 8 requires three 2s. That makes sense.
First, I remember that to find the expone... | To find the greatest possible value of \( n \) such that \( 8^n \) divides \( 100! \), we need to determine the exponent of 2 in the prime factorization of \( 100! \) using Legendre's formula. This is because \( 8 = 2^3 \), and the exponent of 8 in \( 100! \) will be the integer division of the exponent of 2 by 3.
Usi... | 15,845 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the greatest possible value of n such that 8^n divides 100!. Hmm, 8 is 2 cubed, so maybe I need to figure out how many times 2 appears as a factor in 100! and then divide that by 3? Because 8 is 2^3, right? So each 8 requires three 2s. That makes sense.
First, I remember that to find th... | [
{
"from": "human",
"value": "Find the greatest possible value of $n$ such that $8^{n}$ divides $100!$ ($100!=100\\times99\\times98\\times\\cdots\\times 3\\times 2\\times 1$)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the greatest possible value of n such that 8^n divides 10... |
If $y$ is a number between 2 and 3, which of the following represents the largest value?
A). $y$
B). $y^2$
C). $2y$
D). $\sqrt{y}$
E). $\frac{1}{y}$
Express your answer as A, B, C, D, or E. | If $y$ is a number between 2 and 3, which of the following represents the largest value?
A). $y$
B). $y^2$
C). $2y$
D). $\sqrt{y}$
E). $\frac{1}{y}$
Express your answer as A, B, C, D, or E. | Okay, so I need to figure out which of the options A through E is the largest when y is a number between 2 and 3. Let's take it step by step.
First, let me restate the problem to make sure I understand. We have a variable y that's greater than 2 but less than 3. Then there are five expressions: A is just y, B is y sq... | To determine which of the given options represents the largest value when \( y \) is a number between 2 and 3, we evaluate each option:
- **Option A (\( y \))**: Ranges from just above 2 to just below 3.
- **Option B (\( y^2 \))**: Ranges from just above 4 to just below 9.
- **Option C (\( 2y \))**: Ranges from just a... | 12,327 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out which of the options A through E is the largest when y is a number between 2 and 3. Let's take it step by step.
First, let me restate the problem to make sure I understand. We have a variable y that's greater than 2 but less than 3. Then there are five expressions: A is just y, B... | [
{
"from": "human",
"value": "If $y$ is a number between 2 and 3, which of the following represents the largest value?\n\nA). $y$\nB). $y^2$\nC). $2y$\nD). $\\sqrt{y}$\nE). $\\frac{1}{y}$\n\nExpress your answer as A, B, C, D, or E."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure... |
Find the sum of all even numbers from -20 to 40, inclusive. | Find the sum of all even numbers from -20 to 40, inclusive. | Okay, so I need to find the sum of all even numbers from -20 to 40, inclusive. Hmm, let's see. First, I should make sure I understand the problem correctly. They want all even numbers in that range, including both -20 and 40 if they are even. Let me confirm: -20 is even, right? Yes, because it's divisible by 2. And 40 ... | To find the sum of all even numbers from -20 to 40, inclusive, we recognize that this is an arithmetic sequence with the first term \(a_1 = -20\), the last term \(a_n = 40\), and a common difference \(d = 2\).
First, we determine the number of terms \(n\) using the formula for the number of terms in an arithmetic sequ... | 4,573 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the sum of all even numbers from -20 to 40, inclusive. Hmm, let's see. First, I should make sure I understand the problem correctly. They want all even numbers in that range, including both -20 and 40 if they are even. Let me confirm: -20 is even, right? Yes, because it's divisible by 2.... | [
{
"from": "human",
"value": "Find the sum of all even numbers from -20 to 40, inclusive."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of all even numbers from -20 to 40, inclusive. Hmm, let's see. First, I should make sure I understand the problem correctly. They want ... |
Let $S$ be a set of $k$ elements, and let $A_1,\dots, A_k$ be a family of $k$ subsets of $S$, each containing exactly $r$ elements. Moreover, every element of $S$ belongs to exactly $r$ subsets $A_1,\dots, A_k$. We will say that an element $x$ is *'colored"* if $x$ is selected from $S$ and a subset containing $x$ is al... | Let $S$ be a set of $k$ elements, and let $A_1,\dots, A_k$ be a family of $k$ subsets of $S$, each containing exactly $r$ elements. Moreover, every element of $S$ belongs to exactly $r$ subsets $A_1,\dots, A_k$. We will say that an element $x$ is *'colored"* if $x$ is selected from $S$ and a subset containing $x$ is al... | Okay, let me try to figure out this probability problem. So, we have a set S with k elements. Then there are k subsets A₁, A₂, ..., A_k, each of size r. Also, every element in S is in exactly r subsets. That sounds like a kind of regular structure, maybe a block design? Not sure if that's important, but maybe.
So, the... | Given a set \( S \) with \( k \) elements and a family of \( k \) subsets \( A_1, \ldots, A_k \), each containing exactly \( r \) elements, where every element of \( S \) belongs to exactly \( r \) subsets, we need to find the probability that an element \( y \) is the element \( x \) that has been colored, given that ... | 6,677 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this probability problem. So, we have a set S with k elements. Then there are k subsets A₁, A₂, ..., A_k, each of size r. Also, every element in S is in exactly r subsets. That sounds like a kind of regular structure, maybe a block design? Not sure if that's important, but maybe.
... | [
{
"from": "human",
"value": "Let $S$ be a set of $k$ elements, and let $A_1,\\dots, A_k$ be a family of $k$ subsets of $S$, each containing exactly $r$ elements. Moreover, every element of $S$ belongs to exactly $r$ subsets $A_1,\\dots, A_k$. We will say that an element $x$ is *'colored\"* if $x$ is selecte... |
What is the maximum possible area of a rectangle inscribed in a semicircle with radius 4, where one side of the rectangle lies on the diameter of the semicircle? | What is the maximum possible area of a rectangle inscribed in a semicircle with radius 4, where one side of the rectangle lies on the diameter of the semicircle? | Okay, so I need to find the maximum area of a rectangle inscribed in a semicircle with radius 4, and one side of the rectangle lies on the diameter. Hmm, let me visualize this first. A semicircle is half of a circle, right? So if the radius is 4, the diameter would be 8. The rectangle is sitting inside this semicircle ... | To find the maximum possible area of a rectangle inscribed in a semicircle with radius 4, where one side of the rectangle lies on the diameter, we start by noting the equation of the semicircle: \(x^2 + y^2 = 16\) with \(y \geq 0\). The rectangle has a width of \(2x\) and a height of \(y\), so the area \(A\) is given b... | 31,984 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the maximum area of a rectangle inscribed in a semicircle with radius 4, and one side of the rectangle lies on the diameter. Hmm, let me visualize this first. A semicircle is half of a circle, right? So if the radius is 4, the diameter would be 8. The rectangle is sitting inside this sem... | [
{
"from": "human",
"value": "What is the maximum possible area of a rectangle inscribed in a semicircle with radius 4, where one side of the rectangle lies on the diameter of the semicircle?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the maximum area of a rectangle inscribed... |
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... |
In $\triangle ABC,$ the difference between $\angle A$ and $\angle B$ is $30^\circ.$ What is the measure of $\angle C?$ | In $\triangle ABC,$ the difference between $\angle A$ and $\angle B$ is $30^\circ.$ What is the measure of $\angle C?$ | Okay, so I need to find the measure of angle C in triangle ABC, given that the difference between angle A and angle B is 30 degrees. Hmm, let me think. Alright, triangles have some basic properties. The sum of all angles in a triangle is always 180 degrees. So, if I can express angles A, B, and C in terms of each other... | Given $\triangle ABC$ where the difference between $\angle A$ and $\angle B$ is $30^\circ$, we need to determine the measure of $\angle C$.
1. **Understanding the Problem:**
- The sum of the angles in a triangle is $180^\circ$.
- The difference between $\angle A$ and $\angle B$ is $30^\circ$, which can be expres... | 27,079 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the measure of angle C in triangle ABC, given that the difference between angle A and angle B is 30 degrees. Hmm, let me think. Alright, triangles have some basic properties. The sum of all angles in a triangle is always 180 degrees. So, if I can express angles A, B, and C in terms of ea... | [
{
"from": "human",
"value": "In $\\triangle ABC,$ the difference between $\\angle A$ and $\\angle B$ is $30^\\circ.$ What is the measure of $\\angle C?$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the measure of angle C in triangle ABC, given that the difference between angle... |
Let p(x) be a polynomial such that all its roots are positive integers. If it has two real and two imaginary roots, what is the minimum possible number of real roots that any of its divisors could have? | Let p(x) be a polynomial such that all its roots are positive integers. If it has two real and two imaginary roots, what is the minimum possible number of real roots that any of its divisors could have? | Okay, so I have this problem here: Let p(x) be a polynomial where all its roots are positive integers. It's given that p(x) has two real roots and two imaginary roots. The question is asking for the minimum possible number of real roots that any of its divisors (which I assume means factors, like other polynomials that... | Given a polynomial \( p(x) \) such that all its roots are positive integers, and it has two real roots and two imaginary roots, we need to determine the minimum possible number of real roots that any of its divisors could have.
1. **Interpretation of the Problem**:
- The polynomial \( p(x) \) has two real roots (wh... | 21,545 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: Let p(x) be a polynomial where all its roots are positive integers. It's given that p(x) has two real roots and two imaginary roots. The question is asking for the minimum possible number of real roots that any of its divisors (which I assume means factors, like other polynomi... | [
{
"from": "human",
"value": "Let p(x) be a polynomial such that all its roots are positive integers. If it has two real and two imaginary roots, what is the minimum possible number of real roots that any of its divisors could have?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this pr... |
Prove that for all positive integers $n$, the inequality
$$\lfloor \sqrt{n^2 + n} \rfloor = n$$ holds. | Prove that for all positive integers $n$, the inequality
$$\lfloor \sqrt{n^2 + n} \rfloor = n$$ holds. | Okay, so I need to prove that for all positive integers n, the floor of the square root of (n squared plus n) is equal to n. Let me write that down again to make sure I understand: ⌊√(n² + n)⌋ = n. Hmm, okay.
First, maybe I should try plugging in some small values of n to see if the statement holds. Let's start with ... | To prove that for all positive integers \( n \), the inequality \(\lfloor \sqrt{n^2 + n} \rfloor = n\) holds, we can follow these steps:
1. **Establish Inequalities**:
- For any positive integer \( n \), we have \( n^2 < n^2 + n < (n + 1)^2 \).
- The left inequality \( n^2 < n^2 + n \) is true because \( n \) is... | 32,631 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to prove that for all positive integers n, the floor of the square root of (n squared plus n) is equal to n. Let me write that down again to make sure I understand: ⌊√(n² + n)⌋ = n. Hmm, okay.
First, maybe I should try plugging in some small values of n to see if the statement holds. Let's sta... | [
{
"from": "human",
"value": "Prove that for all positive integers $n$, the inequality\n$$\\lfloor \\sqrt{n^2 + n} \\rfloor = n$$ holds."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to prove that for all positive integers n, the floor of the square root of (n squared plus n) is equal t... |
Determine the number of trailing zeros in the expansion of 150! when written as an integer. | Determine the number of trailing zeros in the expansion of 150! when written as an integer. | Okay, so I need to find out how many trailing zeros there are in 150 factorial. Hmm, I remember that trailing zeros in a factorial are determined by the number of times the number 10 is a factor in that factorial. Since 10 is 2 multiplied by 5, I probably need to figure out how many pairs of 2s and 5s there are in the ... | To determine the number of trailing zeros in 150!, we need to count the number of times 10 is a factor in the prime factorization of 150!. Since 10 is the product of 2 and 5, and there are typically more factors of 2 than 5 in a factorial, the number of trailing zeros is determined by the number of factors of 5.
We us... | 3,704 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find out how many trailing zeros there are in 150 factorial. Hmm, I remember that trailing zeros in a factorial are determined by the number of times the number 10 is a factor in that factorial. Since 10 is 2 multiplied by 5, I probably need to figure out how many pairs of 2s and 5s there are... | [
{
"from": "human",
"value": "Determine the number of trailing zeros in the expansion of 150! when written as an integer."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find out how many trailing zeros there are in 150 factorial. Hmm, I remember that trailing zeros in a factorial are ... |
A curve $C_1$ is the graph of the function $f(x) = \frac{2x^2+1}{x^2+1}$ and a curve $C_2$ is the graph of the function $g(x) = \frac{x^2-1}{2x^2+1}$. For any curve $C$, the transformation $T(C)$ produces another curve as follows: a point $(x, y)$ is on $T(C)$ if and only if the point $(y, x)$ is on $C$. Let $T^{(1)}(C... | A curve $C_1$ is the graph of the function $f(x) = \frac{2x^2+1}{x^2+1}$ and a curve $C_2$ is the graph of the function $g(x) = \frac{x^2-1}{2x^2+1}$. For any curve $C$, the transformation $T(C)$ produces another curve as follows: a point $(x, y)$ is on $T(C)$ if and only if the point $(y, x)$ is on $C$. Let $T^{(1)}(C... | Okay, so I need to find the smallest positive integer m such that applying the transformation T m times to the curve C gives back C itself. The curve C is given by y = (x + 1)/(2x - 1). The transformation T takes a curve and swaps the x and y coordinates. That is, if a point (a, b) is on T(C), then the point (b, a) mus... | To find the smallest positive integer \( m \) for which \( T^{(m)}(C) = C \) where \( C \) is the curve \( y = \frac{x + 1}{2x - 1} \), we need to analyze the transformation \( T \). The transformation \( T \) reflects a curve over the line \( y = x \), meaning if a point \( (x, y) \) is on \( T(C) \), then the point \... | 27,539 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest positive integer m such that applying the transformation T m times to the curve C gives back C itself. The curve C is given by y = (x + 1)/(2x - 1). The transformation T takes a curve and swaps the x and y coordinates. That is, if a point (a, b) is on T(C), then the point (b... | [
{
"from": "human",
"value": "A curve $C_1$ is the graph of the function $f(x) = \\frac{2x^2+1}{x^2+1}$ and a curve $C_2$ is the graph of the function $g(x) = \\frac{x^2-1}{2x^2+1}$. For any curve $C$, the transformation $T(C)$ produces another curve as follows: a point $(x, y)$ is on $T(C)$ if and only if t... |
Count the number of five-digit natural numbers whose digits form a non-strictly increasing sequence. | Count the number of five-digit natural numbers whose digits form a non-strictly increasing sequence. | Okay, so I need to figure out how many five-digit numbers have digits that form a non-strictly increasing sequence. Let me start by understanding what that means. A non-strictly increasing sequence would allow the digits to stay the same or increase as we go from left to right. For example, 11234 is okay because each d... | To count the number of five-digit natural numbers whose digits form a non-strictly increasing sequence, we need to consider sequences of digits where each digit is greater than or equal to the previous one. Since the number must be a five-digit natural number, the first digit cannot be zero, so all digits must be from ... | 27,692 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many five-digit numbers have digits that form a non-strictly increasing sequence. Let me start by understanding what that means. A non-strictly increasing sequence would allow the digits to stay the same or increase as we go from left to right. For example, 11234 is okay becaus... | [
{
"from": "human",
"value": "Count the number of five-digit natural numbers whose digits form a non-strictly increasing sequence."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many five-digit numbers have digits that form a non-strictly increasing sequence. Let me sta... |
If $f(x) = \cos x + i \sin x$, find $f(20^\circ) + f(40^\circ) + f(80^\circ)$. | If $f(x) = \cos x + i \sin x$, find $f(20^\circ) + f(40^\circ) + f(80^\circ)$. | Okay, so I need to find the sum of f(20°), f(40°), and f(80°), where f(x) is defined as cos x plus i times sin x. Hmm, let me start by writing down what each term is.
First, f(20°) would be cos 20° + i sin 20°, right? Similarly, f(40°) is cos 40° + i sin 40°, and f(80°) is cos 80° + i sin 80°. So the sum would be [cos... | To find \( f(20^\circ) + f(40^\circ) + f(80^\circ) \) where \( f(x) = \cos x + i \sin x \), we start by expressing each term using Euler's formula, which states \( f(x) = e^{ix} \). Therefore, we need to compute the sum \( e^{i20^\circ} + e^{i40^\circ} + e^{i80^\circ} \).
First, we convert the sum into trigonometric f... | 6,460 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the sum of f(20°), f(40°), and f(80°), where f(x) is defined as cos x plus i times sin x. Hmm, let me start by writing down what each term is.
First, f(20°) would be cos 20° + i sin 20°, right? Similarly, f(40°) is cos 40° + i sin 40°, and f(80°) is cos 80° + i sin 80°. So the sum would... | [
{
"from": "human",
"value": "If $f(x) = \\cos x + i \\sin x$, find $f(20^\\circ) + f(40^\\circ) + f(80^\\circ)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of f(20°), f(40°), and f(80°), where f(x) is defined as cos x plus i times sin x. Hmm, let me start by writing ... |
Compute the value of $252^2 - 248^2$. | Compute the value of $252^2 - 248^2$. | Okay, so I need to compute the value of 252 squared minus 248 squared. Hmm, let's see. At first glance, this looks like a straightforward subtraction problem, but squaring such large numbers might be time-consuming. There must be a smarter way to do this without having to calculate each square individually.
Wait, I re... | To compute the value of \(252^2 - 248^2\), we can use the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\).
1. Identify \(a = 252\) and \(b = 248\).
2. Calculate \(a - b\):
\[
252 - 248 = 4
\]
3. Calculate \(a + b\):
\[
252 + 248 = 500
\]
4. Multiply the results from steps 2 and 3:
\[
... | 20,145 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to compute the value of 252 squared minus 248 squared. Hmm, let's see. At first glance, this looks like a straightforward subtraction problem, but squaring such large numbers might be time-consuming. There must be a smarter way to do this without having to calculate each square individually.
Wa... | [
{
"from": "human",
"value": "Compute the value of $252^2 - 248^2$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to compute the value of 252 squared minus 248 squared. Hmm, let's see. At first glance, this looks like a straightforward subtraction problem, but squaring such large number... |
Simplify $\frac{\sqrt{20x^3} + \sqrt{45x^5}}{\sqrt{5x}}$. Express your answer in simplest radical form in terms of $x$. | Simplify $\frac{\sqrt{20x^3} + \sqrt{45x^5}}{\sqrt{5x}}$. Express your answer in simplest radical form in terms of $x$. | Okay, so I need to simplify this expression: (√(20x³) + √(45x⁵)) divided by √(5x). Hmm, let me take it step by step. First, maybe I can simplify each of the square roots in the numerator before dealing with the denominator. Let's look at the first term in the numerator: √(20x³).
Breaking down 20 into prime factors, 2... | To simplify the expression \(\frac{\sqrt{20x^3} + \sqrt{45x^5}}{\sqrt{5x}}\), we start by simplifying each term in the numerator.
First, we simplify \(\sqrt{20x^3}\):
\[
\sqrt{20x^3} = \sqrt{4 \cdot 5 \cdot x^2 \cdot x} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{5x} = 2x\sqrt{5x}
\]
Next, we simplify \(\sqrt{45x^5}\):
\... | 1,428 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to simplify this expression: (√(20x³) + √(45x⁵)) divided by √(5x). Hmm, let me take it step by step. First, maybe I can simplify each of the square roots in the numerator before dealing with the denominator. Let's look at the first term in the numerator: √(20x³).
Breaking down 20 into prime fa... | [
{
"from": "human",
"value": "Simplify $\\frac{\\sqrt{20x^3} + \\sqrt{45x^5}}{\\sqrt{5x}}$. Express your answer in simplest radical form in terms of $x$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to simplify this expression: (√(20x³) + √(45x⁵)) divided by √(5x). Hmm, let me take it ... |
A bookstore has three shelves, each with a different number of books. If the books are rearranged so that each shelf has a different number of books, in how many different ways can the books be distributed among the shelves? | A bookstore has three shelves, each with a different number of books. If the books are rearranged so that each shelf has a different number of books, in how many different ways can the books be distributed among the shelves? | Okay, let's see. The problem is about a bookstore with three shelves, each having a different number of books. Then, if the books are rearranged so that each shelf still has a different number of books, how many different ways can the books be distributed among the shelves?
First, I need to understand the problem corr... | The problem involves a bookstore with three shelves, each initially having a different number of books. The task is to determine the number of different ways to rearrange the books such that each shelf still has a different number of books.
1. **Understanding the Problem**: The total number of books remains the same, ... | 16,544 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is about a bookstore with three shelves, each having a different number of books. Then, if the books are rearranged so that each shelf still has a different number of books, how many different ways can the books be distributed among the shelves?
First, I need to understand the prob... | [
{
"from": "human",
"value": "A bookstore has three shelves, each with a different number of books. If the books are rearranged so that each shelf has a different number of books, in how many different ways can the books be distributed among the shelves?"
},
{
"from": "gpt",
"value": "<think>\nOk... |
Let $g(x) = x^2 + 2x\lfloor x \rfloor - \cos(x).$ Compute the range of $g(x).$ | Let $g(x) = x^2 + 2x\lfloor x \rfloor - \cos(x).$ Compute the range of $g(x).$ | Okay, so I have this function here: g(x) = x² + 2x⌊x⌋ - cos(x). I need to find its range. Hmm, let's start by understanding each part of the function and then maybe break it down.
First, let's recall that ⌊x⌋ is the floor function, which gives the greatest integer less than or equal to x. So, for example, if x is 2.3... | To find the range of the function \( g(x) = x^2 + 2x\lfloor x \rfloor - \cos(x) \), we analyze the function piecewise over intervals where the floor function \( \lfloor x \rfloor \) is constant. For each integer \( n \), on the interval \([n, n+1)\), the function simplifies to \( g_n(x) = x^2 + 2nx - \cos(x) \).
### K... | 15,377 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this function here: g(x) = x² + 2x⌊x⌋ - cos(x). I need to find its range. Hmm, let's start by understanding each part of the function and then maybe break it down.
First, let's recall that ⌊x⌋ is the floor function, which gives the greatest integer less than or equal to x. So, for example, if ... | [
{
"from": "human",
"value": "Let $g(x) = x^2 + 2x\\lfloor x \\rfloor - \\cos(x).$ Compute the range of $g(x).$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this function here: g(x) = x² + 2x⌊x⌋ - cos(x). I need to find its range. Hmm, let's start by understanding each part of the fun... |
Find the smallest positive integer whose digits form an arithmetic sequence and the sum of whose digits is equal to 12. | Find the smallest positive integer whose digits form an arithmetic sequence and the sum of whose digits is equal to 12. | Okay, so I need to find the smallest positive integer whose digits form an arithmetic sequence and the sum of the digits is 12. Hmm, let me start by understanding the problem better.
First, an arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. For example, 1, 3, 5,... | To find the smallest positive integer whose digits form an arithmetic sequence and sum to 12, we start by considering the properties of an arithmetic sequence. The digits must have a common difference \(d\) and the sum of the digits must be 12. We need to check sequences with different numbers of digits (starting from ... | 28,498 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest positive integer whose digits form an arithmetic sequence and the sum of the digits is 12. Hmm, let me start by understanding the problem better.
First, an arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. For example, ... | [
{
"from": "human",
"value": "Find the smallest positive integer whose digits form an arithmetic sequence and the sum of whose digits is equal to 12."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest positive integer whose digits form an arithmetic sequence and the sum ... |
Alex has forgotten the combination to his safe. He knows that it consists of 5 digits, with the last digit being either 0, 2, or 5, and the first four digits being the prime numbers less than 10 in some order. If Alex tries a combination at random, what is the probability that he opens the safe on the first try? Expres... | Alex has forgotten the combination to his safe. He knows that it consists of 5 digits, with the last digit being either 0, 2, or 5, and the first four digits being the prime numbers less than 10 in some order. If Alex tries a combination at random, what is the probability that he opens the safe on the first try? Expres... | Okay, let me try to figure out this probability problem. So, Alex has a safe with a 5-digit combination. The last digit is either 0, 2, or 5. The first four digits are prime numbers less than 10, arranged in some order. We need to find the probability that Alex opens the safe on his first try. Hmm, probability is gener... | Alex's safe combination consists of 5 digits. The first four digits are the prime numbers less than 10 (2, 3, 5, 7) in some order, and the last digit is either 0, 2, or 5.
To find the total number of possible combinations:
1. The number of permutations of the four prime numbers (2, 3, 5, 7) is \(4!\) (since each prim... | 395 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this probability problem. So, Alex has a safe with a 5-digit combination. The last digit is either 0, 2, or 5. The first four digits are prime numbers less than 10, arranged in some order. We need to find the probability that Alex opens the safe on his first try. Hmm, probability ... | [
{
"from": "human",
"value": "Alex has forgotten the combination to his safe. He knows that it consists of 5 digits, with the last digit being either 0, 2, or 5, and the first four digits being the prime numbers less than 10 in some order. If Alex tries a combination at random, what is the probability that h... |
I have a bag with only red, blue, and green marbles. The ratio of red marbles to blue marbles to green marbles is $1:5:3$. There are 27 green marbles in the bag. How many marbles are there in the bag? | I have a bag with only red, blue, and green marbles. The ratio of red marbles to blue marbles to green marbles is $1:5:3$. There are 27 green marbles in the bag. How many marbles are there in the bag? | Okay, let's see here. I have a problem about marbles in a bag. The ratio of red to blue to green marbles is 1:5:3, and there are 27 green marbles. I need to find the total number of marbles in the bag. Hmm, ratios can sometimes be tricky, but let me try to work through it step by step.
First, ratios compare the quanti... | Given the ratio of red to blue to green marbles is \(1:5:3\), we denote the number of red, blue, and green marbles as \(1x\), \(5x\), and \(3x\) respectively. We know there are 27 green marbles, so we set up the equation:
\[
3x = 27
\]
Solving for \(x\):
\[
x = \frac{27}{3} = 9
\]
Using this value of \(x\), we find... | 1,263 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see here. I have a problem about marbles in a bag. The ratio of red to blue to green marbles is 1:5:3, and there are 27 green marbles. I need to find the total number of marbles in the bag. Hmm, ratios can sometimes be tricky, but let me try to work through it step by step.
First, ratios compare th... | [
{
"from": "human",
"value": "I have a bag with only red, blue, and green marbles. The ratio of red marbles to blue marbles to green marbles is $1:5:3$. There are 27 green marbles in the bag. How many marbles are there in the bag?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see here. ... |
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If a marble is drawn at random and then replaced, and this process is repeated 3 times, what is the probability that exactly 2 of the 3 marbles drawn are blue? | A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If a marble is drawn at random and then replaced, and this process is repeated 3 times, what is the probability that exactly 2 of the 3 marbles drawn are blue? | Okay, so I have this probability problem here: there's a bag with 5 red marbles, 3 blue marbles, and 2 green marbles. We're drawing a marble at random, replacing it each time, and repeating this process three times. We need to find the probability that exactly 2 out of the 3 marbles drawn are blue. Hmm, let me think ab... | The problem involves finding the probability of drawing exactly 2 blue marbles out of 3 draws with replacement from a bag containing 5 red, 3 blue, and 2 green marbles.
1. **Determine the probability of success (drawing a blue marble):**
- Total marbles = 10
- Probability of drawing a blue marble, \( p = \frac{... | 34,689 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this probability problem here: there's a bag with 5 red marbles, 3 blue marbles, and 2 green marbles. We're drawing a marble at random, replacing it each time, and repeating this process three times. We need to find the probability that exactly 2 out of the 3 marbles drawn are blue. Hmm, let me ... | [
{
"from": "human",
"value": "A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If a marble is drawn at random and then replaced, and this process is repeated 3 times, what is the probability that exactly 2 of the 3 marbles drawn are blue?"
},
{
"from": "gpt",
"value": "<think>\n... |
Consider a polynomial with integer coefficients of the form
\[x^3 + b_2 x^2 + b_1 x + 24.\]
You are told that the integer $s$ is a triple root of this polynomial. (In other words, the polynomial is divisible by $(x - s)^3.$) Find all the possible values of $s.$ | Consider a polynomial with integer coefficients of the form
\[x^3 + b_2 x^2 + b_1 x + 24.\]
You are told that the integer $s$ is a triple root of this polynomial. (In other words, the polynomial is divisible by $(x - s)^3.$) Find all the possible values of $s.$ | Okay, let's see. I need to find all possible integer values of s that are triple roots of the polynomial x³ + b₂x² + b₁x +24, where b₂ and b₁ are integers. Since it's a cubic polynomial with a triple root s, that means the polynomial can be written as (x - s)³. Let me start by expanding (x - s)³ and comparing coefficie... | Given a polynomial with integer coefficients of the form \(x^3 + b_2 x^2 + b_1 x + 24\) and the information that the integer \(s\) is a triple root, we need to find all possible values of \(s\).
A polynomial with a triple root \(s\) can be written as \((x - s)^3\). Expanding this polynomial, we get:
\[
(x - s)^3 = x^3... | 8,805 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find all possible integer values of s that are triple roots of the polynomial x³ + b₂x² + b₁x +24, where b₂ and b₁ are integers. Since it's a cubic polynomial with a triple root s, that means the polynomial can be written as (x - s)³. Let me start by expanding (x - s)³ and comparing c... | [
{
"from": "human",
"value": "Consider a polynomial with integer coefficients of the form\n\\[x^3 + b_2 x^2 + b_1 x + 24.\\]\nYou are told that the integer $s$ is a triple root of this polynomial. (In other words, the polynomial is divisible by $(x - s)^3.$) Find all the possible values of $s.$"
},
{
... |
Suppose that $p$ is prime and $1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p$. How many possible values of $p$ are there? | Suppose that $p$ is prime and $1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p$. How many possible values of $p$ are there? | Okay, so I need to solve this problem where I have an equation involving numbers in base p, and p is a prime number. The equation is:
1007_p + 306_p + 113_p + 125_p + 6_p = 142_p + 271_p + 360_p
And I need to find how many possible prime values of p satisfy this equation. Alright, let's break this down step by step.
... | To solve the given equation involving numbers in base \( p \), we first convert each number from base \( p \) to base 10. The equation is:
\[ 1007_p + 306_p + 113_p + 125_p + 6_p = 142_p + 271_p + 360_p \]
Converting each term on the left side:
- \( 1007_p = p^3 + 7 \)
- \( 306_p = 3p^2 + 6 \)
- \( 113_p = p^2 + p + ... | 21,396 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this problem where I have an equation involving numbers in base p, and p is a prime number. The equation is:
1007_p + 306_p + 113_p + 125_p + 6_p = 142_p + 271_p + 360_p
And I need to find how many possible prime values of p satisfy this equation. Alright, let's break this down step b... | [
{
"from": "human",
"value": "Suppose that $p$ is prime and $1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p$. How many possible values of $p$ are there?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this problem where I have an equation involving numbers in base p, and p is a p... |
What is the greatest common divisor of $5!$ and $\frac{8!}{3!}?$ Express your answer as an integer. (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.) | What is the greatest common divisor of $5!$ and $\frac{8!}{3!}?$ Express your answer as an integer. (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.) | Okay, let me try to figure out the greatest common divisor (GCD) of 5! and 8!/3!. Hmm, first, maybe I should write out what those factorials actually are so I can see the numbers we're dealing with.
Starting with 5!: That's 5 factorial, which is 5 × 4 × 3 × 2 × 1. Let me calculate that: 5×4 is 20, 20×3 is 60, 60×2 is... | To find the greatest common divisor (GCD) of \(5!\) and \(\frac{8!}{3!}\), we start by calculating the values of these factorials.
First, \(5!\) is calculated as:
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]
Next, \(\frac{8!}{3!}\) is calculated as:
\[
\frac{8!}{3!} = \frac{8 \times 7 \times 6 \times 5 \tim... | 151 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out the greatest common divisor (GCD) of 5! and 8!/3!. Hmm, first, maybe I should write out what those factorials actually are so I can see the numbers we're dealing with.
Starting with 5!: That's 5 factorial, which is 5 × 4 × 3 × 2 × 1. Let me calculate that: 5×4 is 20, 20×3 is 60,... | [
{
"from": "human",
"value": "What is the greatest common divisor of $5!$ and $\\frac{8!}{3!}?$ Express your answer as an integer. (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)"
},
{
"from": "gpt",
"value": "<th... |
What is the largest number of distinct positive integers $a$ such that $a^3$ is strictly between 25 and 50? | What is the largest number of distinct positive integers $a$ such that $a^3$ is strictly between 25 and 50? | Okay, let's see. The problem is asking for the largest number of distinct positive integers \( a \) such that \( a^3 \) is strictly between 25 and 50. Hmm. So I need to find all positive integers \( a \) where \( 25 < a^3 < 50 \). Then count how many there are. Alright.
First, let me make sure I understand the questio... | To determine the largest number of distinct positive integers \( a \) such that \( a^3 \) is strictly between 25 and 50, we need to find the integers \( a \) for which \( 25 < a^3 < 50 \).
First, we calculate the cube roots of 25 and 50 to find the range of \( a \):
- The cube root of 25 is approximately 2.924.
- The ... | 14,487 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is asking for the largest number of distinct positive integers \( a \) such that \( a^3 \) is strictly between 25 and 50. Hmm. So I need to find all positive integers \( a \) where \( 25 < a^3 < 50 \). Then count how many there are. Alright.
First, let me make sure I understand the... | [
{
"from": "human",
"value": "What is the largest number of distinct positive integers $a$ such that $a^3$ is strictly between 25 and 50?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem is asking for the largest number of distinct positive integers \\( a \\) such that \\( a^3... |
What is the largest possible mode for the seven number set $\{x, x+2, x+4, 3, 2, 5, 7\}$ if $x$ can be any integer? | What is the largest possible mode for the seven number set $\{x, x+2, x+4, 3, 2, 5, 7\}$ if $x$ can be any integer? | Okay, let me try to figure out this problem. The question is asking for the largest possible mode for the seven number set {x, x+2, x+4, 3, 2, 5, 7} where x can be any integer. Hmm, mode is the number that appears most frequently in a set, right? So I need to determine what integer x can be such that one of the numbers... | To determine the largest possible mode for the set \(\{x, x+2, x+4, 3, 2, 5, 7\}\) where \(x\) is an integer, we need to consider how the values of \(x\), \(x+2\), and \(x+4\) can overlap with the fixed numbers \(2, 3, 5, 7\). The mode is the number that appears most frequently in the set.
1. **Case Analysis for \(x\)... | 24,673 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. The question is asking for the largest possible mode for the seven number set {x, x+2, x+4, 3, 2, 5, 7} where x can be any integer. Hmm, mode is the number that appears most frequently in a set, right? So I need to determine what integer x can be such that one of the... | [
{
"from": "human",
"value": "What is the largest possible mode for the seven number set $\\{x, x+2, x+4, 3, 2, 5, 7\\}$ if $x$ can be any integer?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. The question is asking for the largest possible mode for the sev... |
A deck of 52 playing cards is shuffled and one card is drawn at random. What is the probability that the card drawn is a face card (king, queen, or jack) or a card with a prime number as its rank (2, 3, 5, or 7)? Express your answer as a common fraction. | A deck of 52 playing cards is shuffled and one card is drawn at random. What is the probability that the card drawn is a face card (king, queen, or jack) or a card with a prime number as its rank (2, 3, 5, or 7)? Express your answer as a common fraction. | Alright, let me try to figure out this probability problem. So, the question is: we have a standard deck of 52 playing cards, shuffled, and we draw one card at random. We need to find the probability that the card is either a face card (king, queen, or jack) or a card with a prime number as its rank (2, 3, 5, or 7). Hm... | To determine the probability that a card drawn from a standard deck of 52 playing cards is either a face card (king, queen, or jack) or a card with a prime number (2, 3, 5, or 7), we use the principle of inclusion and exclusion.
1. **Number of face cards**: There are 3 face cards (jack, queen, king) per suit, and 4 su... | 4,214 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let me try to figure out this probability problem. So, the question is: we have a standard deck of 52 playing cards, shuffled, and we draw one card at random. We need to find the probability that the card is either a face card (king, queen, or jack) or a card with a prime number as its rank (2, 3, 5, o... | [
{
"from": "human",
"value": "A deck of 52 playing cards is shuffled and one card is drawn at random. What is the probability that the card drawn is a face card (king, queen, or jack) or a card with a prime number as its rank (2, 3, 5, or 7)? Express your answer as a common fraction."
},
{
"from": "g... |
The lines $y=4x-5$, $y=-x+1$ and $y=2x+m$ intersect at the same point. What is the value of $m$? | The lines $y=4x-5$, $y=-x+1$ and $y=2x+m$ intersect at the same point. What is the value of $m$? | Okay, so I need to find the value of m such that the three lines y=4x-5, y=-x+1, and y=2x+m all intersect at the same point. Hmm, let's see. If all three lines intersect at the same point, that means there's a single coordinate (x, y) that satisfies all three equations. So, the plan is probably to first find the inters... | To find the value of \( m \) such that the lines \( y = 4x - 5 \), \( y = -x + 1 \), and \( y = 2x + m \) intersect at the same point, we first determine the intersection point of the first two lines.
1. Set the equations \( y = 4x - 5 \) and \( y = -x + 1 \) equal to each other:
\[
4x - 5 = -x + 1
\]
Solv... | 5,278 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of m such that the three lines y=4x-5, y=-x+1, and y=2x+m all intersect at the same point. Hmm, let's see. If all three lines intersect at the same point, that means there's a single coordinate (x, y) that satisfies all three equations. So, the plan is probably to first find th... | [
{
"from": "human",
"value": "The lines $y=4x-5$, $y=-x+1$ and $y=2x+m$ intersect at the same point. What is the value of $m$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of m such that the three lines y=4x-5, y=-x+1, and y=2x+m all intersect at the same point. Hmm, ... |
The average height of a group of students in a school is 160 cm. If the average height of the boys is 165 cm and the average height of the girls is 155 cm, what is the ratio of the number of boys to the number of girls in the school? | The average height of a group of students in a school is 160 cm. If the average height of the boys is 165 cm and the average height of the girls is 155 cm, what is the ratio of the number of boys to the number of girls in the school? | Okay, so I need to find the ratio of the number of boys to the number of girls in the school, given the average heights. Let's see. The overall average height is 160 cm. The boys have an average of 165 cm, and the girls 155 cm. Hmm. This seems like a problem involving weighted averages. Right, because the total average... | Given the average height of a group of students is 160 cm, with boys averaging 165 cm and girls averaging 155 cm, we need to find the ratio of the number of boys to the number of girls.
Let \( B \) be the number of boys and \( G \) be the number of girls. The total height of all students is \( 165B + 155G \), and the ... | 33,301 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the ratio of the number of boys to the number of girls in the school, given the average heights. Let's see. The overall average height is 160 cm. The boys have an average of 165 cm, and the girls 155 cm. Hmm. This seems like a problem involving weighted averages. Right, because the total... | [
{
"from": "human",
"value": "The average height of a group of students in a school is 160 cm. If the average height of the boys is 165 cm and the average height of the girls is 155 cm, what is the ratio of the number of boys to the number of girls in the school?"
},
{
"from": "gpt",
"value": "<t... |
Calculate \[\frac{(2n+1)!(n+1)}{(2n+2)!}.\] | Calculate \[\frac{(2n+1)!(n+1)}{(2n+2)!}.\] | Okay, so I have this fraction to calculate: \(\frac{(2n+1)!(n+1)}{(2n+2)!}\). Hmm, factorials can sometimes be tricky, but maybe if I expand them out, I can simplify the expression. Let me start by recalling what factorial means. For any positive integer \(k\), \(k! = k \times (k-1) \times \dots \times 2 \times 1\). So... | To simplify the expression \(\frac{(2n+1)!(n+1)}{(2n+2)!}\), we start by recognizing that the denominator \((2n + 2)!\) can be written as \((2n + 2) \times (2n + 1)!\). This allows us to rewrite the original expression as:
\[
\frac{(2n + 1)! \times (n + 1)}{(2n + 2) \times (2n + 1)!}
\]
Here, the \((2n + 1)!\) terms ... | 24,600 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this fraction to calculate: \(\frac{(2n+1)!(n+1)}{(2n+2)!}\). Hmm, factorials can sometimes be tricky, but maybe if I expand them out, I can simplify the expression. Let me start by recalling what factorial means. For any positive integer \(k\), \(k! = k \times (k-1) \times \dots \times 2 \times... | [
{
"from": "human",
"value": "Calculate \\[\\frac{(2n+1)!(n+1)}{(2n+2)!}.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this fraction to calculate: \\(\\frac{(2n+1)!(n+1)}{(2n+2)!}\\). Hmm, factorials can sometimes be tricky, but maybe if I expand them out, I can simplify the express... |
How many distinct solutions are there to the equation $|x-3| = |x+5|$? | How many distinct solutions are there to the equation $|x-3| = |x+5|$? | Alright, let me try to figure out how many distinct solutions there are to the equation |x - 3| = |x + 5|. Hmm, absolute value equations can sometimes be tricky because they involve distances on the number line. So, the equation is saying that the distance between x and 3 is equal to the distance between x and -5. I re... | To determine the number of distinct solutions to the equation \(|x - 3| = |x + 5|\), we start by recognizing that this equation states that the distance from \(x\) to 3 is equal to the distance from \(x\) to -5. This implies \(x\) must be the midpoint between 3 and -5.
The midpoint between 3 and -5 is calculated as:
... | 830 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let me try to figure out how many distinct solutions there are to the equation |x - 3| = |x + 5|. Hmm, absolute value equations can sometimes be tricky because they involve distances on the number line. So, the equation is saying that the distance between x and 3 is equal to the distance between x and ... | [
{
"from": "human",
"value": "How many distinct solutions are there to the equation $|x-3| = |x+5|$?"
},
{
"from": "gpt",
"value": "<think>\nAlright, let me try to figure out how many distinct solutions there are to the equation |x - 3| = |x + 5|. Hmm, absolute value equations can sometimes be tr... |
Factor completely: $x^6 - 3x^4 + 3x^2 - 1$. | Factor completely: $x^6 - 3x^4 + 3x^2 - 1$. | Okay, so I need to factor the polynomial \( x^6 - 3x^4 + 3x^2 - 1 \) completely. Hmm, let me start by looking at it and trying to see if there's a pattern or something familiar here. The polynomial is of degree 6, which is even, and the coefficients are decreasing from 1, -3, 3, -1. Wait a minute, those coefficients re... | To factor the polynomial \( x^6 - 3x^4 + 3x^2 - 1 \) completely, we start by recognizing the pattern in the coefficients, which resembles the expansion of a binomial cube.
First, we substitute \( y = x^2 \), transforming the polynomial into \( y^3 - 3y^2 + 3y - 1 \). This cubic polynomial can be factored as \( (y - 1... | 19,451 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to factor the polynomial \( x^6 - 3x^4 + 3x^2 - 1 \) completely. Hmm, let me start by looking at it and trying to see if there's a pattern or something familiar here. The polynomial is of degree 6, which is even, and the coefficients are decreasing from 1, -3, 3, -1. Wait a minute, those coeffic... | [
{
"from": "human",
"value": "Factor completely: $x^6 - 3x^4 + 3x^2 - 1$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to factor the polynomial \\( x^6 - 3x^4 + 3x^2 - 1 \\) completely. Hmm, let me start by looking at it and trying to see if there's a pattern or something familiar here... |
The quantity $\tan 7.5^\circ$ can be expressed in the form
\[\tan 7.5^\circ = \sqrt{a} - \sqrt{b} + \sqrt{c} - d,\]where $a \ge b \ge c \ge d$ are positive integers. Find $a + b + c + d.$ | The quantity $\tan 7.5^\circ$ can be expressed in the form
\[\tan 7.5^\circ = \sqrt{a} - \sqrt{b} + \sqrt{c} - d,\]where $a \ge b \ge c \ge d$ are positive integers. Find $a + b + c + d.$ | Okay, let's see. I need to find the value of tan 7.5 degrees expressed in the form sqrt(a) - sqrt(b) + sqrt(c) - d, where a, b, c, d are positive integers with a ≥ b ≥ c ≥ d. Then find their sum. Hmm. Alright, starting off, 7.5 degrees is half of 15 degrees, which is a common angle in trigonometry. Maybe I can use a ha... | To find the value of \(\tan 7.5^\circ\) in the form \(\sqrt{a} - \sqrt{b} + \sqrt{c} - d\) where \(a \ge b \ge c \ge d\) are positive integers, we use the half-angle formula for tangent.
First, we use the half-angle formula:
\[
\tan 7.5^\circ = \tan \left( \frac{15^\circ}{2} \right) = \frac{1 - \cos 15^\circ}{\sin 15... | 8,150 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the value of tan 7.5 degrees expressed in the form sqrt(a) - sqrt(b) + sqrt(c) - d, where a, b, c, d are positive integers with a ≥ b ≥ c ≥ d. Then find their sum. Hmm. Alright, starting off, 7.5 degrees is half of 15 degrees, which is a common angle in trigonometry. Maybe I can ... | [
{
"from": "human",
"value": "The quantity $\\tan 7.5^\\circ$ can be expressed in the form\n\\[\\tan 7.5^\\circ = \\sqrt{a} - \\sqrt{b} + \\sqrt{c} - d,\\]where $a \\ge b \\ge c \\ge d$ are positive integers. Find $a + b + c + d.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need t... |
A bag contains 5 red balls and 7 blue balls. Two balls are drawn at random from the bag, one after the other, without replacement. What is the probability that the first ball drawn is red and the second ball drawn is blue, given that the first ball drawn is not returned to the bag before the second ball is drawn? | A bag contains 5 red balls and 7 blue balls. Two balls are drawn at random from the bag, one after the other, without replacement. What is the probability that the first ball drawn is red and the second ball drawn is blue, given that the first ball drawn is not returned to the bag before the second ball is drawn? | Okay, so there's this problem here about a bag with 5 red balls and 7 blue balls. We need to find the probability that the first ball drawn is red and the second one is blue, right? And they mention that we don't replace the first ball before drawing the second one. Hmm, let me think.
Alright, probability problems can... | The bag contains 5 red balls and 7 blue balls, making a total of 12 balls. We need to find the probability that the first ball drawn is red and the second ball drawn is blue, without replacement.
1. **Probability of drawing a red ball first (Event A):**
\[
P(A) = \frac{5}{12}
\]
There are 5 red balls out o... | 14,081 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this problem here about a bag with 5 red balls and 7 blue balls. We need to find the probability that the first ball drawn is red and the second one is blue, right? And they mention that we don't replace the first ball before drawing the second one. Hmm, let me think.
Alright, probability prob... | [
{
"from": "human",
"value": "A bag contains 5 red balls and 7 blue balls. Two balls are drawn at random from the bag, one after the other, without replacement. What is the probability that the first ball drawn is red and the second ball drawn is blue, given that the first ball drawn is not returned to the b... |
A bakery sells three types of bread: whole wheat, white bread, and rye bread. A batch of whole wheat bread requires 28 cups of flour, a batch of white bread requires 32 cups of flour, and a batch of rye bread requires 42 cups of flour. What is the largest number of cups of flour that can be divided evenly among these t... | A bakery sells three types of bread: whole wheat, white bread, and rye bread. A batch of whole wheat bread requires 28 cups of flour, a batch of white bread requires 32 cups of flour, and a batch of rye bread requires 42 cups of flour. What is the largest number of cups of flour that can be divided evenly among these t... | Okay, so the problem is about a bakery that sells three types of bread: whole wheat, white, and rye. Each batch of these breads requires different amounts of flour. Whole wheat needs 28 cups, white needs 32 cups, and rye needs 42 cups. The question is asking for the largest number of cups of flour that can be divided e... | To determine the largest number of cups of flour that can be divided evenly among the three batches (whole wheat, white bread, and rye bread), we need to find the greatest common divisor (GCD) of the required amounts of flour for each batch: 28, 32, and 42 cups.
1. **List the divisors of each number:**
- Divisors o... | 21,583 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so the problem is about a bakery that sells three types of bread: whole wheat, white, and rye. Each batch of these breads requires different amounts of flour. Whole wheat needs 28 cups, white needs 32 cups, and rye needs 42 cups. The question is asking for the largest number of cups of flour that can be d... | [
{
"from": "human",
"value": "A bakery sells three types of bread: whole wheat, white bread, and rye bread. A batch of whole wheat bread requires 28 cups of flour, a batch of white bread requires 32 cups of flour, and a batch of rye bread requires 42 cups of flour. What is the largest number of cups of flour... |
A bridge is being designed with a length of 2400 feet. A scale model of the bridge has a length of 15 feet. If the designer wants to add a 2-foot wide sidewalk to the bridge, how many inches should be added to the model to represent the sidewalk accurately? | A bridge is being designed with a length of 2400 feet. A scale model of the bridge has a length of 15 feet. If the designer wants to add a 2-foot wide sidewalk to the bridge, how many inches should be added to the model to represent the sidewalk accurately? | Okay, let me try to figure out this problem. So, there's a bridge that's being designed, and the actual length is 2400 feet. They made a scale model of it that's 15 feet long. Now, they want to add a 2-foot wide sidewalk to the real bridge, and we need to figure out how many inches that would be on the model. Hmm, alri... | The actual bridge length is 2400 feet, and the scale model is 15 feet long. To determine the scale factor, we calculate the ratio of the model length to the actual length:
\[
\text{Scale factor} = \frac{\text{Model length}}{\text{Actual length}} = \frac{15 \text{ feet}}{2400 \text{ feet}} = \frac{1}{160}
\]
This scal... | 5,752 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, there's a bridge that's being designed, and the actual length is 2400 feet. They made a scale model of it that's 15 feet long. Now, they want to add a 2-foot wide sidewalk to the real bridge, and we need to figure out how many inches that would be on the model. H... | [
{
"from": "human",
"value": "A bridge is being designed with a length of 2400 feet. A scale model of the bridge has a length of 15 feet. If the designer wants to add a 2-foot wide sidewalk to the bridge, how many inches should be added to the model to represent the sidewalk accurately?"
},
{
"from":... |
Let \[
f(x) =
\begin{cases}
2x & \text{if } x > 5,\\
3-x & \text{if } x \leq 5.
\end{cases}
\]
Compute $\left[f(f(f(f(f(7)))))\right]^2$. | Let \[
f(x) =
\begin{cases}
2x & \text{if } x > 5,\\
3-x & \text{if } x \leq 5.
\end{cases}
\]
Compute $\left[f(f(f(f(f(7)))))\right]^2$. | Okay, let me try to solve this problem. So, we have this function f(x) defined in two parts: if x is greater than 5, then f(x) is 2x, and if x is less than or equal to 5, then f(x) is 3 minus x. The question is asking for [f(f(f(f(f(7)))))] squared. Hmm, that's five applications of the function f starting with 7, and t... | To solve the problem, we need to compute \( \left[f(f(f(f(f(7)))))\right]^2 \) where the function \( f(x) \) is defined as:
\[
f(x) =
\begin{cases}
2x & \text{if } x > 5, \\
3 - x & \text{if } x \leq 5.
\end{cases}
\]
1. Compute \( f(7) \):
- Since \( 7 > 5 \), we use \( f(x) = 2x \).
- \( f(7) = 2 \times 7 = 14... | 31,123 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to solve this problem. So, we have this function f(x) defined in two parts: if x is greater than 5, then f(x) is 2x, and if x is less than or equal to 5, then f(x) is 3 minus x. The question is asking for [f(f(f(f(f(7)))))] squared. Hmm, that's five applications of the function f starting with ... | [
{
"from": "human",
"value": "Let \\[\nf(x) =\n\\begin{cases}\n2x & \\text{if } x > 5,\\\\\n3-x & \\text{if } x \\leq 5.\n\\end{cases}\n\\]\nCompute $\\left[f(f(f(f(f(7)))))\\right]^2$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to solve this problem. So, we have this function f(x) ... |
A bookshelf can be filled with boxes of books in such a way that the number of boxes is evenly divisible by 3, 4, 5, and 7. What is the smallest number of boxes that can fill the bookshelf? | A bookshelf can be filled with boxes of books in such a way that the number of boxes is evenly divisible by 3, 4, 5, and 7. What is the smallest number of boxes that can fill the bookshelf? | Okay, so I have this problem here: A bookshelf can be filled with boxes of books in such a way that the number of boxes is evenly divisible by 3, 4, 5, and 7. What is the smallest number of boxes that can fill the bookshelf?
Hmm, let me think. The question is asking for the smallest number of boxes that can be divided... | To determine the smallest number of boxes that can fill the bookshelf such that the number of boxes is evenly divisible by 3, 4, 5, and 7, we need to find the least common multiple (LCM) of these numbers.
First, we factor each number into its prime factors:
- \(3\) is a prime number: \(3\)
- \(4\) factors into \(2^2\... | 1,977 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: A bookshelf can be filled with boxes of books in such a way that the number of boxes is evenly divisible by 3, 4, 5, and 7. What is the smallest number of boxes that can fill the bookshelf?
Hmm, let me think. The question is asking for the smallest number of boxes that can be... | [
{
"from": "human",
"value": "A bookshelf can be filled with boxes of books in such a way that the number of boxes is evenly divisible by 3, 4, 5, and 7. What is the smallest number of boxes that can fill the bookshelf?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: A ... |
A recipe for making candles calls for a ratio of 3 parts wax to 2 parts fragrance. If you want to make a candle that is 10% larger than a standard candle, and the standard candle requires 120 grams of wax, how many grams of fragrance do you need? | A recipe for making candles calls for a ratio of 3 parts wax to 2 parts fragrance. If you want to make a candle that is 10% larger than a standard candle, and the standard candle requires 120 grams of wax, how many grams of fragrance do you need? | Okay, let's tackle this candle-making problem. Hmm, the question is about adjusting the recipe when making a candle 10% larger than the standard one. The standard requires 120 grams of wax, and we need to find out how much fragrance is needed for the larger candle. Alright, let's break this down step by step.
First, t... | The recipe for making candles requires a ratio of 3 parts wax to 2 parts fragrance. A standard candle uses 120 grams of wax. To make a candle that is 10% larger, we need to determine the amount of fragrance required.
1. **Calculate the increased amount of wax:**
- The standard candle uses 120 grams of wax.
- Inc... | 10,544 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's tackle this candle-making problem. Hmm, the question is about adjusting the recipe when making a candle 10% larger than the standard one. The standard requires 120 grams of wax, and we need to find out how much fragrance is needed for the larger candle. Alright, let's break this down step by step.
... | [
{
"from": "human",
"value": "A recipe for making candles calls for a ratio of 3 parts wax to 2 parts fragrance. If you want to make a candle that is 10% larger than a standard candle, and the standard candle requires 120 grams of wax, how many grams of fragrance do you need?"
},
{
"from": "gpt",
... |
The shaded region in the figure consists of a large rectangle with a smaller rectangle cut out. The area of the shaded region is 120 square units. The dimensions of the large rectangle are $x$ units by $2x$ units. What is the area of the smaller rectangle? [asy]draw((0,0)--(15,0)--(15,20)--(30,20)--(30,0)--(42,0)--(42,... | The shaded region in the figure consists of a large rectangle with a smaller rectangle cut out. The area of the shaded region is 120 square units. The dimensions of the large rectangle are $x$ units by $2x$ units. What is the area of the smaller rectangle? [asy]draw((0,0)--(15,0)--(15,20)--(30,20)--(30,0)--(42,0)--(42,... | Okay, let me try to figure out this problem. So, there's a shaded region which is a large rectangle with a smaller rectangle cut out. The area of this shaded region is 120 square units. The large rectangle has dimensions x by 2x. They want the area of the smaller rectangle.
First, let me visualize this. The Asymptote ... | The shaded region in the figure consists of a large rectangle with a smaller rectangle cut out. The area of the shaded region is 120 square units. The dimensions of the large rectangle are \( x \) units by \( 2x \) units. We need to find the area of the smaller rectangle.
1. **Area of the large rectangle**:
\[
\... | 22,764 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, there's a shaded region which is a large rectangle with a smaller rectangle cut out. The area of this shaded region is 120 square units. The large rectangle has dimensions x by 2x. They want the area of the smaller rectangle.
First, let me visualize this. The As... | [
{
"from": "human",
"value": "The shaded region in the figure consists of a large rectangle with a smaller rectangle cut out. The area of the shaded region is 120 square units. The dimensions of the large rectangle are $x$ units by $2x$ units. What is the area of the smaller rectangle? [asy]draw((0,0)--(15,0... |
Given that $\ds\sum_{k=0}^{\infty} a_kx^k=\ds\f{1}{1-3x}$ for $|x|<\ds\f{1}{3}$
and $a_0=1$, $a_1=3$, find $a_{15}$. | Given that $\ds\sum_{k=0}^{\infty} a_kx^k=\ds\f{1}{1-3x}$ for $|x|<\ds\f{1}{3}$
and $a_0=1$, $a_1=3$, find $a_{15}$. | Okay, so I have this problem here where I need to find the coefficient \( a_{15} \) in the power series expansion of \( \frac{1}{1 - 3x} \). They mentioned that the series is valid for \( |x| < \frac{1}{3} \), and they also gave \( a_0 = 1 \) and \( a_1 = 3 \). Hmm, let me think about how to approach this.
First, I re... | To find the coefficient \( a_{15} \) in the power series expansion of \( \frac{1}{1 - 3x} \), we start by recognizing that this is a geometric series. The given series is valid for \( |x| < \frac{1}{3} \).
The power series expansion for \( \frac{1}{1 - 3x} \) is:
\[
\sum_{k=0}^{\infty} (3x)^k = \sum_{k=0}^{\infty} 3^k... | 25,079 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here where I need to find the coefficient \( a_{15} \) in the power series expansion of \( \frac{1}{1 - 3x} \). They mentioned that the series is valid for \( |x| < \frac{1}{3} \), and they also gave \( a_0 = 1 \) and \( a_1 = 3 \). Hmm, let me think about how to approach this.
Fir... | [
{
"from": "human",
"value": "Given that $\\ds\\sum_{k=0}^{\\infty} a_kx^k=\\ds\\f{1}{1-3x}$ for $|x|<\\ds\\f{1}{3}$\nand $a_0=1$, $a_1=3$, find $a_{15}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here where I need to find the coefficient \\( a_{15} \\) in the power ser... |
Find the largest prime factor of $6783$. | Find the largest prime factor of $6783$. | Okay, so I need to find the largest prime factor of 6783. Hmm, prime factors. Let me remember how to do that. Prime factors are the prime numbers that multiply together to give the original number, right? So I need to break down 6783 into its prime components and then see which one is the biggest.
First off, maybe I ... | To find the largest prime factor of \(6783\), we start by checking divisibility by smaller primes.
First, we check divisibility by 3. The sum of the digits of \(6783\) is \(6 + 7 + 8 + 3 = 24\), which is divisible by 3. Dividing \(6783\) by 3 gives:
\[
6783 \div 3 = 2261
\]
Next, we factor \(2261\). Checking divisibi... | 39 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the largest prime factor of 6783. Hmm, prime factors. Let me remember how to do that. Prime factors are the prime numbers that multiply together to give the original number, right? So I need to break down 6783 into its prime components and then see which one is the biggest.
First off, ... | [
{
"from": "human",
"value": "Find the largest prime factor of $6783$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest prime factor of 6783. Hmm, prime factors. Let me remember how to do that. Prime factors are the prime numbers that multiply together to give the origi... |
What is the value of $y$ in the equation $8^8 + 8^8 + 8^8 + 8^8 + 8^8 = 2^y$? | What is the value of $y$ in the equation $8^8 + 8^8 + 8^8 + 8^8 + 8^8 = 2^y$? | Okay, so I need to find the value of y in the equation 8^8 + 8^8 + 8^8 + 8^8 + 8^8 = 2^y. Hmm, let's see. First, I remember that when you have the same term added multiple times, you can multiply that term by the number of times it's added. In this case, there are five 8^8 terms. So maybe I can rewrite the left side as... | The given equation is \(8^8 + 8^8 + 8^8 + 8^8 + 8^8 = 2^y\).
First, we recognize that adding the same term five times can be rewritten as multiplication:
\[
8^8 + 8^8 + 8^8 + 8^8 + 8^8 = 5 \cdot 8^8
\]
Next, we express \(8\) as a power of \(2\):
\[
8 = 2^3
\]
Thus, \(8^8\) can be rewritten as:
\[
8^8 = (2^3)^8 = 2^... | 16,421 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of y in the equation 8^8 + 8^8 + 8^8 + 8^8 + 8^8 = 2^y. Hmm, let's see. First, I remember that when you have the same term added multiple times, you can multiply that term by the number of times it's added. In this case, there are five 8^8 terms. So maybe I can rewrite the left... | [
{
"from": "human",
"value": "What is the value of $y$ in the equation $8^8 + 8^8 + 8^8 + 8^8 + 8^8 = 2^y$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of y in the equation 8^8 + 8^8 + 8^8 + 8^8 + 8^8 = 2^y. Hmm, let's see. First, I remember that when you have the sa... |
For polynomial $P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}$, define
\[Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\sum_{i=0}^{50} a_ix^{i}.\]Find $\sum_{i=0}^{50} |a_i|.$ | For polynomial $P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}$, define
\[Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\sum_{i=0}^{50} a_ix^{i}.\]Find $\sum_{i=0}^{50} |a_i|.$ | Okay, so I have this problem here where I need to find the sum of the absolute values of the coefficients of the polynomial Q(x), which is defined as the product of several P(x^{k}) terms. Let me break this down step by step.
First, the polynomial P(x) is given as 1 - (1/3)x + (1/6)x². Then Q(x) is the product of P(x)... | To find the sum of the absolute values of the coefficients of the polynomial \( Q(x) = P(x)P(x^3)P(x^5)P(x^7)P(x^9) \), where \( P(x) = 1 - \frac{1}{3}x + \frac{1}{6}x^2 \), we need to consider the absolute values of the coefficients of each polynomial involved.
First, we note that the sum of the absolute values of th... | 29,826 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here where I need to find the sum of the absolute values of the coefficients of the polynomial Q(x), which is defined as the product of several P(x^{k}) terms. Let me break this down step by step.
First, the polynomial P(x) is given as 1 - (1/3)x + (1/6)x². Then Q(x) is the product... | [
{
"from": "human",
"value": "For polynomial $P(x)=1-\\dfrac{1}{3}x+\\dfrac{1}{6}x^{2}$, define\n\\[Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\\sum_{i=0}^{50} a_ix^{i}.\\]Find $\\sum_{i=0}^{50} |a_i|.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here where I need to find ... |
How many prime numbers are between 30 and 40? | How many prime numbers are between 30 and 40? | Okay, so I need to figure out how many prime numbers are between 30 and 40. Hmm, let's start by recalling what a prime number is. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. So, numbers like 2, 3, 5, 7, etc., are primes because you can't divide them by anythi... | To determine how many prime numbers are between 30 and 40, we check each number in this range:
- **31**: Not divisible by 2, 3, or 5 (prime)
- **32**: Divisible by 2 (not prime)
- **33**: Divisible by 3 (not prime)
- **34**: Divisible by 2 (not prime)
- **35**: Divisible by 5 (not prime)
- **36**: Divisible by 2 (not ... | 22,318 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many prime numbers are between 30 and 40. Hmm, let's start by recalling what a prime number is. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. So, numbers like 2, 3, 5, 7, etc., are primes because you can't divide them b... | [
{
"from": "human",
"value": "How many prime numbers are between 30 and 40?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many prime numbers are between 30 and 40. Hmm, let's start by recalling what a prime number is. A prime number is a natural number greater than 1 t... |
How many positive multiples of $7$ that are less than $1000$ end with the digit $3$? | How many positive multiples of $7$ that are less than $1000$ end with the digit $3$? | Okay, so I need to figure out how many positive multiples of 7 that are less than 1000 end with the digit 3. Hmm, let's start by understanding the problem. A multiple of 7 is a number that can be expressed as 7 times some integer. We want these multiples to be less than 1000, so the largest multiple of 7 we'll consider... | To determine how many positive multiples of 7 less than 1000 end with the digit 3, we start by noting that a multiple of 7 can be written as \(7k\) where \(k\) is a positive integer. We need \(7k < 1000\), which gives \(k \leq 142\) since \(1000/7 \approx 142.857\).
We need \(7k\) to end with the digit 3, which means ... | 20,506 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many positive multiples of 7 that are less than 1000 end with the digit 3. Hmm, let's start by understanding the problem. A multiple of 7 is a number that can be expressed as 7 times some integer. We want these multiples to be less than 1000, so the largest multiple of 7 we'll ... | [
{
"from": "human",
"value": "How many positive multiples of $7$ that are less than $1000$ end with the digit $3$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many positive multiples of 7 that are less than 1000 end with the digit 3. Hmm, let's start by understanding... |
A sign painter paints individual numerals for a row of 50 houses. The houses are numbered with consecutive integers from 1 to 50. How many 9s are painted? | A sign painter paints individual numerals for a row of 50 houses. The houses are numbered with consecutive integers from 1 to 50. How many 9s are painted? | Okay, so I need to figure out how many times the painter has to paint the number 9 when numbering houses from 1 to 50. Let me think. The question is asking for the total number of 9s in all the numbers from 1 through 50. That means I have to check each number and count how many 9s there are in each, then add them all u... | To determine how many times the number 9 is painted on the house numbers from 1 to 50, we need to check each number for the presence of the digit 9 in both the tens and units places.
1. **Single-digit numbers (1-9):**
- Only the number 9 contains the digit 9.
- Count: 1
2. **Two-digit numbers (10-50):**
- **... | 19,036 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many times the painter has to paint the number 9 when numbering houses from 1 to 50. Let me think. The question is asking for the total number of 9s in all the numbers from 1 through 50. That means I have to check each number and count how many 9s there are in each, then add th... | [
{
"from": "human",
"value": "A sign painter paints individual numerals for a row of 50 houses. The houses are numbered with consecutive integers from 1 to 50. How many 9s are painted?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many times the painter has to paint th... |
Find the maximum value of
\[(\tan x + \cot x)^2 + (\sin x - \csc x)^2\]for $0 < x < \frac{\pi}{2}.$ | Find the maximum value of
\[(\tan x + \cot x)^2 + (\sin x - \csc x)^2\]for $0 < x < \frac{\pi}{2}.$ | Okay, so I need to find the maximum value of this expression: \[(\tan x + \cot x)^2 + (\sin x - \csc x)^2\] where \(0 < x < \frac{\pi}{2}\). Hmm, let's start by trying to simplify the expression step by step. Maybe if I can rewrite everything in terms of sine and cosine, it'll be easier to handle.
First, let's recall ... | To find the maximum value of the expression \((\tan x + \cot x)^2 + (\sin x - \csc x)^2\) for \(0 < x < \frac{\pi}{2}\), we start by simplifying the expression using trigonometric identities.
First, we rewrite the terms in terms of sine and cosine:
- \(\tan x = \frac{\sin x}{\cos x}\)
- \(\cot x = \frac{\cos x}{\sin x... | 33,249 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the maximum value of this expression: \[(\tan x + \cot x)^2 + (\sin x - \csc x)^2\] where \(0 < x < \frac{\pi}{2}\). Hmm, let's start by trying to simplify the expression step by step. Maybe if I can rewrite everything in terms of sine and cosine, it'll be easier to handle.
First, let's... | [
{
"from": "human",
"value": "Find the maximum value of\n\\[(\\tan x + \\cot x)^2 + (\\sin x - \\csc x)^2\\]for $0 < x < \\frac{\\pi}{2}.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the maximum value of this expression: \\[(\\tan x + \\cot x)^2 + (\\sin x - \\csc x)^2\\] wher... |
A rectangular garden measures 12 meters by 15 meters. A path that is 0.8 meters wide is built around the garden. What is the area of the path? | A rectangular garden measures 12 meters by 15 meters. A path that is 0.8 meters wide is built around the garden. What is the area of the path? | Okay, so there's this rectangular garden that's 12 meters by 15 meters, and they built a path around it that's 0.8 meters wide. I need to find the area of the path. Hmm, let's see. I think the way to do this is to calculate the area of the garden plus the path and then subtract the area of just the garden. That should ... | To find the area of the path around the rectangular garden, we first calculate the area of the garden and then the area of the garden including the path. Subtracting the original garden area from the total area gives the area of the path.
1. **Area of the original garden**:
\[
12 \, \text{m} \times 15 \, \text{m... | 6,165 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this rectangular garden that's 12 meters by 15 meters, and they built a path around it that's 0.8 meters wide. I need to find the area of the path. Hmm, let's see. I think the way to do this is to calculate the area of the garden plus the path and then subtract the area of just the garden. That... | [
{
"from": "human",
"value": "A rectangular garden measures 12 meters by 15 meters. A path that is 0.8 meters wide is built around the garden. What is the area of the path?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so there's this rectangular garden that's 12 meters by 15 meters, and they bui... |
How many positive factors of 120 are also multiples of 15? | How many positive factors of 120 are also multiples of 15? | Okay, so I need to figure out how many positive factors of 120 are also multiples of 15. Hmm, let's start by recalling what factors and multiples mean. A factor of a number is another number that divides into it without leaving a remainder. A multiple of a number is the product of that number and an integer. So, in thi... | To determine how many positive factors of 120 are also multiples of 15, we start by prime factorizing 120: \(120 = 2^3 \times 3^1 \times 5^1\).
A multiple of 15 must include the prime factors 3 and 5. Therefore, any factor of 120 that is a multiple of 15 must be of the form \(3^1 \times 5^1 \times 2^a\), where \(a\) c... | 28,522 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many positive factors of 120 are also multiples of 15. Hmm, let's start by recalling what factors and multiples mean. A factor of a number is another number that divides into it without leaving a remainder. A multiple of a number is the product of that number and an integer. So... | [
{
"from": "human",
"value": "How many positive factors of 120 are also multiples of 15?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many positive factors of 120 are also multiples of 15. Hmm, let's start by recalling what factors and multiples mean. A factor of a nu... |
If $\displaystyle\frac{a}{b} = 4$, $\displaystyle\frac{b}{c} = \frac{1}{3}$, and $\displaystyle \frac{c}{d} = 6$, then what is $\displaystyle\frac{d}{a}$? | If $\displaystyle\frac{a}{b} = 4$, $\displaystyle\frac{b}{c} = \frac{1}{3}$, and $\displaystyle \frac{c}{d} = 6$, then what is $\displaystyle\frac{d}{a}$? | Alright, let's see. I need to find the value of d/a given the ratios of a/b, b/c, and c/d. Hmm, okay, so ratios. Maybe I can express all variables in terms of one another and then find the relationship between d and a. Let me start writing down what I know.
First, the problem states that a/b = 4. That means a is four ... | Given the ratios \(\frac{a}{b} = 4\), \(\frac{b}{c} = \frac{1}{3}\), and \(\frac{c}{d} = 6\), we need to find the value of \(\frac{d}{a}\).
First, express each variable in terms of the others:
1. From \(\frac{a}{b} = 4\), we have \(a = 4b\).
2. From \(\frac{b}{c} = \frac{1}{3}\), we have \(c = 3b\).
3. From \(\frac{c}... | 36,769 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's see. I need to find the value of d/a given the ratios of a/b, b/c, and c/d. Hmm, okay, so ratios. Maybe I can express all variables in terms of one another and then find the relationship between d and a. Let me start writing down what I know.
First, the problem states that a/b = 4. That means a ... | [
{
"from": "human",
"value": "If $\\displaystyle\\frac{a}{b} = 4$, $\\displaystyle\\frac{b}{c} = \\frac{1}{3}$, and $\\displaystyle \\frac{c}{d} = 6$, then what is $\\displaystyle\\frac{d}{a}$?"
},
{
"from": "gpt",
"value": "<think>\nAlright, let's see. I need to find the value of d/a given the r... |
Given an isosceles triangle with base angles $30^\circ$ and the vertex angle $120^\circ$, find the length of the arc intercepted by its two congruent sides if it is inscribed in a circle of radius $24$. Express your answer in terms of $\pi$. | Given an isosceles triangle with base angles $30^\circ$ and the vertex angle $120^\circ$, find the length of the arc intercepted by its two congruent sides if it is inscribed in a circle of radius $24$. Express your answer in terms of $\pi$. | Okay, so I need to find the length of the arc intercepted by the two congruent sides of an isosceles triangle that's inscribed in a circle with radius 24. The triangle has base angles of 30 degrees and a vertex angle of 120 degrees. Hmm, let me think step by step.
First, let me visualize the problem. There's a circle ... | Given an isosceles triangle with base angles \(30^\circ\) and a vertex angle \(120^\circ\) inscribed in a circle of radius 24, we need to find the length of the arc intercepted by its two congruent sides.
1. **Understanding the Inscribed Triangle**:
- The triangle has base angles of \(30^\circ\) each and a vertex a... | 20,943 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the length of the arc intercepted by the two congruent sides of an isosceles triangle that's inscribed in a circle with radius 24. The triangle has base angles of 30 degrees and a vertex angle of 120 degrees. Hmm, let me think step by step.
First, let me visualize the problem. There's a... | [
{
"from": "human",
"value": "Given an isosceles triangle with base angles $30^\\circ$ and the vertex angle $120^\\circ$, find the length of the arc intercepted by its two congruent sides if it is inscribed in a circle of radius $24$. Express your answer in terms of $\\pi$."
},
{
"from": "gpt",
"... |
Find the sum of the squares of the roots of the equation \[x^4 - 4x^3 + 7x^2 + 2x - 3 = 0.\] | Find the sum of the squares of the roots of the equation \[x^4 - 4x^3 + 7x^2 + 2x - 3 = 0.\] | Okay, so I need to find the sum of the squares of the roots of this equation: x⁴ - 4x³ + 7x² + 2x - 3 = 0. Hmm, let me think. I remember there's something called Vieta's formulas that relates the coefficients of a polynomial to sums and products of its roots. Maybe I can use that here.
First, let me recall Vieta's for... | To find the sum of the squares of the roots of the equation \(x^4 - 4x^3 + 7x^2 + 2x - 3 = 0\), we use Vieta's formulas and the relationship between the coefficients of the polynomial and the sums of the roots.
Given the polynomial \(x^4 - 4x^3 + 7x^2 + 2x - 3 = 0\), we compare it to the standard form \(x^4 + px^3 + q... | 19,389 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the sum of the squares of the roots of this equation: x⁴ - 4x³ + 7x² + 2x - 3 = 0. Hmm, let me think. I remember there's something called Vieta's formulas that relates the coefficients of a polynomial to sums and products of its roots. Maybe I can use that here.
First, let me recall Vie... | [
{
"from": "human",
"value": "Find the sum of the squares of the roots of the equation \\[x^4 - 4x^3 + 7x^2 + 2x - 3 = 0.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of the squares of the roots of this equation: x⁴ - 4x³ + 7x² + 2x - 3 = 0. Hmm, let me think. I reme... |
Compute the number of intersection points of the graphs of
\[(x - \lfloor x \rfloor)^3 + y^2 = 2(x - \lfloor x \rfloor)\]and $y = \frac{1}{7} x.$ | Compute the number of intersection points of the graphs of
\[(x - \lfloor x \rfloor)^3 + y^2 = 2(x - \lfloor x \rfloor)\]and $y = \frac{1}{7} x.$ | Okay, so I need to find the number of intersection points between the two graphs given:
The first one is \[(x - \lfloor x \rfloor)^3 + y^2 = 2(x - \lfloor x \rfloor)\] and the second one is \(y = \frac{1}{7} x\).
Let me start by understanding each equation.
First, the term \(x - \lfloor x \rfloor\) is the fractiona... | To find the number of intersection points between the graphs of \((x - \lfloor x \rfloor)^3 + y^2 = 2(x - \lfloor x \rfloor)\) and \(y = \frac{1}{7} x\), we start by analyzing the first equation, which involves the fractional part of \(x\), denoted \(\{x\}\). This equation is periodic with period 1 in the \(x\)-directi... | 11,096 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the number of intersection points between the two graphs given:
The first one is \[(x - \lfloor x \rfloor)^3 + y^2 = 2(x - \lfloor x \rfloor)\] and the second one is \(y = \frac{1}{7} x\).
Let me start by understanding each equation.
First, the term \(x - \lfloor x \rfloor\) is the f... | [
{
"from": "human",
"value": "Compute the number of intersection points of the graphs of \n\\[(x - \\lfloor x \\rfloor)^3 + y^2 = 2(x - \\lfloor x \\rfloor)\\]and $y = \\frac{1}{7} x.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the number of intersection points between the tw... |
Let $f(x)$ be a quadratic function passing through the points $(0,1), (2,-2),$ and $(4,7)$. Determine $a+b+c$ if $f(x)=ax^2+bx+c$. | Let $f(x)$ be a quadratic function passing through the points $(0,1), (2,-2),$ and $(4,7)$. Determine $a+b+c$ if $f(x)=ax^2+bx+c$. | Okay, so I need to find the quadratic function f(x) = ax² + bx + c that passes through the three points (0,1), (2,-2), and (4,7). Then, once I find a, b, and c, I have to add them together to get a + b + c. Alright, let's start.
First, since it's a quadratic function, it has the general form f(x) = ax² + bx + c. There... | Given the quadratic function \( f(x) = ax^2 + bx + c \) passing through the points \((0, 1)\), \((2, -2)\), and \((4, 7)\), we need to determine \(a + b + c\).
1. Using the point \((0, 1)\):
\[
f(0) = c = 1 \implies c = 1
\]
2. Using the point \((2, -2)\):
\[
f(2) = 4a + 2b + 1 = -2 \implies 4a + 2b = ... | 33,817 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the quadratic function f(x) = ax² + bx + c that passes through the three points (0,1), (2,-2), and (4,7). Then, once I find a, b, and c, I have to add them together to get a + b + c. Alright, let's start.
First, since it's a quadratic function, it has the general form f(x) = ax² + bx + ... | [
{
"from": "human",
"value": "Let $f(x)$ be a quadratic function passing through the points $(0,1), (2,-2),$ and $(4,7)$. Determine $a+b+c$ if $f(x)=ax^2+bx+c$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the quadratic function f(x) = ax² + bx + c that passes through the three... |
In a certain country, each province has its own special numerical code, which is a positive integer. The provinces are arranged in a circle, and the numerical codes of any two adjacent provinces add up to 13. If the sum of the numerical codes of all the provinces is 364, how many provinces are there in the country? | In a certain country, each province has its own special numerical code, which is a positive integer. The provinces are arranged in a circle, and the numerical codes of any two adjacent provinces add up to 13. If the sum of the numerical codes of all the provinces is 364, how many provinces are there in the country? | Okay, so there's this country where each province has a special numerical code, which is a positive integer. The provinces are arranged in a circle, and the numerical codes of any two adjacent provinces add up to 13. The total sum of all the numerical codes is 364. The question is: how many provinces are there in the c... | Given that each province has a numerical code which is a positive integer, and the sum of the codes of any two adjacent provinces is 13. The provinces are arranged in a circle, and the total sum of all codes is 364. We need to determine the number of provinces.
1. **Adjacent Pair Sum Condition**: Each pair of adjacent... | 23,483 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this country where each province has a special numerical code, which is a positive integer. The provinces are arranged in a circle, and the numerical codes of any two adjacent provinces add up to 13. The total sum of all the numerical codes is 364. The question is: how many provinces are there ... | [
{
"from": "human",
"value": "In a certain country, each province has its own special numerical code, which is a positive integer. The provinces are arranged in a circle, and the numerical codes of any two adjacent provinces add up to 13. If the sum of the numerical codes of all the provinces is 364, how man... |
Problem:
Find the value of $k$ for which the equation $x^2 + 8x + y^2 - 4y + k = 0$ represents a circle that passes through the point $(2, 2)$ and has a radius of $2$. | Problem:
Find the value of $k$ for which the equation $x^2 + 8x + y^2 - 4y + k = 0$ represents a circle that passes through the point $(2, 2)$ and has a radius of $2$. | Okay, so I need to find the value of k for which the equation x² + 8x + y² - 4y + k = 0 represents a circle that passes through the point (2, 2) and has a radius of 2. Hmm, let me think about how to approach this.
First, I remember that the general equation of a circle is x² + y² + Dx + Ey + F = 0, and it can be rewri... | To find the value of \( k \) for which the equation \( x^2 + 8x + y^2 - 4y + k = 0 \) represents a circle that passes through the point \( (2, 2) \) and has a radius of 2, we start by rewriting the equation in standard form by completing the square.
1. **Completing the square:**
- For \( x \): \( x^2 + 8x \) become... | 26,755 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of k for which the equation x² + 8x + y² - 4y + k = 0 represents a circle that passes through the point (2, 2) and has a radius of 2. Hmm, let me think about how to approach this.
First, I remember that the general equation of a circle is x² + y² + Dx + Ey + F = 0, and it can ... | [
{
"from": "human",
"value": "Problem:\nFind the value of $k$ for which the equation $x^2 + 8x + y^2 - 4y + k = 0$ represents a circle that passes through the point $(2, 2)$ and has a radius of $2$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of k for which the equat... |
The quadratic $x^2-4x-14=3x+16$ has two solutions. What is the sum of these solutions? | The quadratic $x^2-4x-14=3x+16$ has two solutions. What is the sum of these solutions? | Okay, so I need to solve this quadratic equation: x² - 4x - 14 = 3x + 16. The question is asking for the sum of the two solutions. Hmm, quadratic equations... Let me remember. I think there are a couple of ways to approach this. Either I can use the quadratic formula after rearranging the equation, or maybe factor it i... | To find the sum of the solutions of the quadratic equation \(x^2 - 4x - 14 = 3x + 16\), we first rearrange it into standard form.
Subtract \(3x\) and 16 from both sides:
\[
x^2 - 4x - 14 - 3x - 16 = 0
\]
Combine like terms:
\[
x^2 - 7x - 30 = 0
\]
Using Vieta's formulas, the sum of the solutions of a quadratic equat... | 10,798 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this quadratic equation: x² - 4x - 14 = 3x + 16. The question is asking for the sum of the two solutions. Hmm, quadratic equations... Let me remember. I think there are a couple of ways to approach this. Either I can use the quadratic formula after rearranging the equation, or maybe fac... | [
{
"from": "human",
"value": "The quadratic $x^2-4x-14=3x+16$ has two solutions. What is the sum of these solutions?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this quadratic equation: x² - 4x - 14 = 3x + 16. The question is asking for the sum of the two solutions. Hmm, qua... |
A rectangular prism-shaped tank is being filled with a mixture of water and antifreeze. The ratio of water to antifreeze in the mixture is 3:1. If the tank has a length of 10 feet, a width of 4 feet, and a height of 6 feet, and it is being filled at a rate of 0.5 cubic feet per minute, how many minutes will it take to ... | A rectangular prism-shaped tank is being filled with a mixture of water and antifreeze. The ratio of water to antifreeze in the mixture is 3:1. If the tank has a length of 10 feet, a width of 4 feet, and a height of 6 feet, and it is being filled at a rate of 0.5 cubic feet per minute, how many minutes will it take to ... | Okay, let me try to figure out this problem. So, we have a rectangular prism tank, which is basically a box shape. The dimensions are length 10 feet, width 4 feet, and height 6 feet. They are filling it with a mixture of water and antifreeze, and the ratio of water to antifreeze is 3:1. The filling rate is 0.5 cubic fe... | To determine how many minutes it will take to fill the tank with 12 cubic feet of antifreeze, we start by noting the ratio of water to antifreeze in the mixture is 3:1. This means that antifreeze constitutes 1 part of the mixture, while water constitutes 3 parts. Therefore, the total mixture is 4 parts, with antifreeze... | 5,335 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, we have a rectangular prism tank, which is basically a box shape. The dimensions are length 10 feet, width 4 feet, and height 6 feet. They are filling it with a mixture of water and antifreeze, and the ratio of water to antifreeze is 3:1. The filling rate is 0.5 ... | [
{
"from": "human",
"value": "A rectangular prism-shaped tank is being filled with a mixture of water and antifreeze. The ratio of water to antifreeze in the mixture is 3:1. If the tank has a length of 10 feet, a width of 4 feet, and a height of 6 feet, and it is being filled at a rate of 0.5 cubic feet per ... |
What is $\frac{3421_{12}}{121_{5}}-1768_{6}+8012_{11}$? Express your answer in base 10. | What is $\frac{3421_{12}}{121_{5}}-1768_{6}+8012_{11}$? Express your answer in base 10. | Okay, let's see. I need to calculate this expression: 3421 base 12 divided by 121 base 5, minus 1768 base 6, plus 8012 base 11. Then express the answer in base 10. Hmm, right. So first, I need to convert all these numbers from their respective bases to base 10, perform the arithmetic operations, and then combine them. ... | To solve the expression \(\frac{3421_{12}}{121_{5}} - 1768_{6} + 8012_{11}\) and express the answer in base 10, we need to convert each number from its respective base to base 10, perform the arithmetic operations, and combine the results.
1. **Convert \(3421_{12}\) to base 10:**
\[
3 \cdot 12^3 + 4 \cdot 12^2 +... | 25,010 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to calculate this expression: 3421 base 12 divided by 121 base 5, minus 1768 base 6, plus 8012 base 11. Then express the answer in base 10. Hmm, right. So first, I need to convert all these numbers from their respective bases to base 10, perform the arithmetic operations, and then combin... | [
{
"from": "human",
"value": "What is $\\frac{3421_{12}}{121_{5}}-1768_{6}+8012_{11}$? Express your answer in base 10."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to calculate this expression: 3421 base 12 divided by 121 base 5, minus 1768 base 6, plus 8012 base 11. Then expre... |
Find the value of $c$ if $-2x^2+cx-3<0$ only when $x\in (-\infty, 2)\cup(4,\infty)$. | Find the value of $c$ if $-2x^2+cx-3<0$ only when $x\in (-\infty, 2)\cup(4,\infty)$. | Okay, so I need to find the value of \( c \) such that the quadratic inequality \( -2x^2 + cx - 3 < 0 \) holds only when \( x \in (-\infty, 2) \cup (4, \infty) \). Hmm, let's start by understanding what this means. The inequality is negative outside the interval [2,4], which suggests that the quadratic opens upwards be... | To find the value of \( c \) such that the quadratic inequality \( -2x^2 + cx - 3 < 0 \) holds only when \( x \in (-\infty, 2) \cup (4, \infty) \), we need to analyze the quadratic equation and its roots.
1. **Understanding the Inequality**:
- The inequality \( -2x^2 + cx - 3 < 0 \) is negative outside the interval... | 1,652 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of \( c \) such that the quadratic inequality \( -2x^2 + cx - 3 < 0 \) holds only when \( x \in (-\infty, 2) \cup (4, \infty) \). Hmm, let's start by understanding what this means. The inequality is negative outside the interval [2,4], which suggests that the quadratic opens up... | [
{
"from": "human",
"value": "Find the value of $c$ if $-2x^2+cx-3<0$ only when $x\\in (-\\infty, 2)\\cup(4,\\infty)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of \\( c \\) such that the quadratic inequality \\( -2x^2 + cx - 3 < 0 \\) holds only when \\( x \\in (-... |
What is the greatest common divisor of the sum of the interior angles of a regular hexagon and the sum of the interior angles of a regular pentagon? Express your answer as an integer. | What is the greatest common divisor of the sum of the interior angles of a regular hexagon and the sum of the interior angles of a regular pentagon? Express your answer as an integer. | Okay, so I need to find the greatest common divisor (GCD) of the sum of the interior angles of a regular hexagon and the sum of the interior angles of a regular pentagon. Hmm, let me break this down step by step.
First, I remember that the formula for the sum of the interior angles of a polygon with n sides is given b... | To find the greatest common divisor (GCD) of the sum of the interior angles of a regular hexagon and a regular pentagon, we start by using the formula for the sum of the interior angles of a polygon with \( n \) sides, which is \((n - 2) \times 180^\circ\).
For a regular hexagon (6 sides):
\[
(6 - 2) \times 180^\circ ... | 23,492 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the greatest common divisor (GCD) of the sum of the interior angles of a regular hexagon and the sum of the interior angles of a regular pentagon. Hmm, let me break this down step by step.
First, I remember that the formula for the sum of the interior angles of a polygon with n sides is... | [
{
"from": "human",
"value": "What is the greatest common divisor of the sum of the interior angles of a regular hexagon and the sum of the interior angles of a regular pentagon? Express your answer as an integer."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the greatest common... |
Let $0 \le a,$ $b,$ $c \le 1.$ Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\] | Let $0 \le a,$ $b,$ $c \le 1.$ Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\] | Okay, so I need to find the maximum value of the expression √(abc) + √((1 - a)(1 - b)(1 - c)) where a, b, c are all between 0 and 1 inclusive. Hmm, interesting. Let me start by understanding the problem better.
First, all variables a, b, c are in the interval [0,1]. The expression has two terms: one is the square root... | To find the maximum value of the expression \(\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}\) where \(0 \le a, b, c \le 1\), we start by considering boundary cases and symmetry.
1. **Boundary Cases**:
- When \(a = b = c = 0\): \(\sqrt{0 \cdot 0 \cdot 0} + \sqrt{1 \cdot 1 \cdot 1} = 0 + 1 = 1\).
- When \(a = b = c = ... | 10,150 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the maximum value of the expression √(abc) + √((1 - a)(1 - b)(1 - c)) where a, b, c are all between 0 and 1 inclusive. Hmm, interesting. Let me start by understanding the problem better.
First, all variables a, b, c are in the interval [0,1]. The expression has two terms: one is the squ... | [
{
"from": "human",
"value": "Let $0 \\le a,$ $b,$ $c \\le 1.$ Find the maximum value of\n\\[\\sqrt{abc} + \\sqrt{(1 - a)(1 - b)(1 - c)}.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the maximum value of the expression √(abc) + √((1 - a)(1 - b)(1 - c)) where a, b, c are all... |
A circle of radius 4 is inscribed in the ellipse \[\frac{x^2}{9} + \frac{y^2}{16} = 1,\] so that it is tangent to the ellipse at four points and its center coincides with the center of the ellipse. Find the area of the circle. | A circle of radius 4 is inscribed in the ellipse \[\frac{x^2}{9} + \frac{y^2}{16} = 1,\] so that it is tangent to the ellipse at four points and its center coincides with the center of the ellipse. Find the area of the circle. | Okay, so I need to find the area of a circle that's inscribed in the given ellipse. The ellipse equation is x²/9 + y²/16 = 1, and the circle has a radius of 4. Wait, hold on, the problem says the circle is inscribed in the ellipse, tangent at four points, and their centers coincide. The question is to find the area of ... | The problem states that a circle of radius 4 is inscribed in the ellipse \(\frac{x^2}{9} + \frac{y^2}{16} = 1\), tangent at four points, with the same center as the ellipse. We need to find the area of the circle.
1. **Ellipse and Circle Equations**:
- The ellipse equation is \(\frac{x^2}{9} + \frac{y^2}{16} = 1\).... | 37,261 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the area of a circle that's inscribed in the given ellipse. The ellipse equation is x²/9 + y²/16 = 1, and the circle has a radius of 4. Wait, hold on, the problem says the circle is inscribed in the ellipse, tangent at four points, and their centers coincide. The question is to find the ... | [
{
"from": "human",
"value": "A circle of radius 4 is inscribed in the ellipse \\[\\frac{x^2}{9} + \\frac{y^2}{16} = 1,\\] so that it is tangent to the ellipse at four points and its center coincides with the center of the ellipse. Find the area of the circle."
},
{
"from": "gpt",
"value": "<thin... |
Find the volume of the solid defined by the inequalities $|x| + |y| + |z| \le 2,$ $|x| + |y| \le 1,$ $|y| + |z| \le 1,$ and $|x| + |z| \le 1.$ | Find the volume of the solid defined by the inequalities $|x| + |y| + |z| \le 2,$ $|x| + |y| \le 1,$ $|y| + |z| \le 1,$ and $|x| + |z| \le 1.$ | Okay, so I need to find the volume of a solid defined by four inequalities: |x| + |y| + |z| ≤ 2, and then three more inequalities: |x| + |y| ≤ 1, |y| + |z| ≤ 1, and |x| + |z| ≤ 1. Hmm, let me try to visualize this.
First, the main solid is defined by |x| + |y| + |z| ≤ 2, which is an octahedron with vertices at (±2, 0... | The volume of the solid defined by the inequalities \( |x| + |y| + |z| \le 2 \), \( |x| + |y| \le 1 \), \( |y| + |z| \le 1 \), and \( |x| + |z| \le 1 \) can be found by analyzing the intersection of these regions.
1. **Symmetry Consideration**: The problem is symmetric in \( x \), \( y \), and \( z \). Therefore, we ... | 24,349 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the volume of a solid defined by four inequalities: |x| + |y| + |z| ≤ 2, and then three more inequalities: |x| + |y| ≤ 1, |y| + |z| ≤ 1, and |x| + |z| ≤ 1. Hmm, let me try to visualize this.
First, the main solid is defined by |x| + |y| + |z| ≤ 2, which is an octahedron with vertices a... | [
{
"from": "human",
"value": "Find the volume of the solid defined by the inequalities $|x| + |y| + |z| \\le 2,$ $|x| + |y| \\le 1,$ $|y| + |z| \\le 1,$ and $|x| + |z| \\le 1.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the volume of a solid defined by four inequalities: |x| ... |
In a triangle with angles A, B, and C, the sine of angle A is $\frac{1}{2}$, the sine of angle B is $\frac{\sqrt{3}}{2}$, and the sine of angle C is $\frac{\sqrt{2}}{2}$. Find the length of the side opposite angle A.
Note: This problem requires a different approach, using trigonometric identities and relationships bet... | In a triangle with angles A, B, and C, the sine of angle A is $\frac{1}{2}$, the sine of angle B is $\frac{\sqrt{3}}{2}$, and the sine of angle C is $\frac{\sqrt{2}}{2}$. Find the length of the side opposite angle A.
Note: This problem requires a different approach, using trigonometric identities and relationships bet... | Okay, so I need to find the length of the side opposite angle A in a triangle where the sines of angles A, B, and C are given as 1/2, √3/2, and √2/2 respectively. Hmm, the problem mentions using trigonometric identities and relationships between sine, cosine, and tangent. Let me think about how to approach this.
First... | Given a triangle with angles \(A\), \(B\), and \(C\) where \(\sin A = \frac{1}{2}\), \(\sin B = \frac{\sqrt{3}}{2}\), and \(\sin C = \frac{\sqrt{2}}{2}\), we need to find the length of the side opposite angle \(A\).
1. **Determine the angles using sine values**:
- \(\sin A = \frac{1}{2}\) implies \(A = 30^\circ\) o... | 36,436 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the length of the side opposite angle A in a triangle where the sines of angles A, B, and C are given as 1/2, √3/2, and √2/2 respectively. Hmm, the problem mentions using trigonometric identities and relationships between sine, cosine, and tangent. Let me think about how to approach this... | [
{
"from": "human",
"value": "In a triangle with angles A, B, and C, the sine of angle A is $\\frac{1}{2}$, the sine of angle B is $\\frac{\\sqrt{3}}{2}$, and the sine of angle C is $\\frac{\\sqrt{2}}{2}$. Find the length of the side opposite angle A.\n\nNote: This problem requires a different approach, usin... |
Point $P$ is the intersection of the diagonals of quadrilateral $ABCD$, where $A(0,0)$, $B(6,2)$, $C(4,4)$, and $D(2,6)$. If point $P$ is twice as far from vertex $A$ as it is from vertex $C$, find the coordinates of point $P$. | Point $P$ is the intersection of the diagonals of quadrilateral $ABCD$, where $A(0,0)$, $B(6,2)$, $C(4,4)$, and $D(2,6)$. If point $P$ is twice as far from vertex $A$ as it is from vertex $C$, find the coordinates of point $P$. | Okay, so I need to find the coordinates of point P, which is the intersection of the diagonals of quadrilateral ABCD. The vertices are given as A(0,0), B(6,2), C(4,4), and D(2,6). Also, it's mentioned that P is twice as far from vertex A as it is from vertex C. Hmm, let me think about how to approach this.
First, mayb... | To find the coordinates of point \( P \), the intersection of the diagonals of quadrilateral \( ABCD \) with vertices \( A(0,0) \), \( B(6,2) \), \( C(4,4) \), and \( D(2,6) \), where \( P \) is twice as far from vertex \( A \) as it is from vertex \( C \):
1. **Parametrize the diagonals**:
- Diagonal \( AC \) is p... | 20,726 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the coordinates of point P, which is the intersection of the diagonals of quadrilateral ABCD. The vertices are given as A(0,0), B(6,2), C(4,4), and D(2,6). Also, it's mentioned that P is twice as far from vertex A as it is from vertex C. Hmm, let me think about how to approach this.
Fir... | [
{
"from": "human",
"value": "Point $P$ is the intersection of the diagonals of quadrilateral $ABCD$, where $A(0,0)$, $B(6,2)$, $C(4,4)$, and $D(2,6)$. If point $P$ is twice as far from vertex $A$ as it is from vertex $C$, find the coordinates of point $P$."
},
{
"from": "gpt",
"value": "<think>... |
The six zeros of the polynomial $x^6 + ax^4 + bx^3 + cx^2 + dx + e$ are distinct real numbers such that the sum of any three consecutive zeros is equal to zero. Compute the value of $a.$ | The six zeros of the polynomial $x^6 + ax^4 + bx^3 + cx^2 + dx + e$ are distinct real numbers such that the sum of any three consecutive zeros is equal to zero. Compute the value of $a.$ | Okay, let me try to figure this problem out. So, we have a polynomial \( x^6 + ax^4 + bx^3 + cx^2 + dx + e \) with six distinct real zeros. The condition given is that the sum of any three consecutive zeros is equal to zero. We need to find the value of \( a \).
First, let me parse the problem. The polynomial is a six... | Given the polynomial \( x^6 + ax^4 + bx^3 + cx^2 + dx + e \) with six distinct real zeros such that the sum of any three consecutive zeros is zero, we need to find the value of \( a \).
1. **Root Relationships**:
- Let the roots be \( r_1, r_2, r_3, r_4, r_5, r_6 \).
- The conditions given are:
\[
\beg... | 36,604 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure this problem out. So, we have a polynomial \( x^6 + ax^4 + bx^3 + cx^2 + dx + e \) with six distinct real zeros. The condition given is that the sum of any three consecutive zeros is equal to zero. We need to find the value of \( a \).
First, let me parse the problem. The polynomial ... | [
{
"from": "human",
"value": "The six zeros of the polynomial $x^6 + ax^4 + bx^3 + cx^2 + dx + e$ are distinct real numbers such that the sum of any three consecutive zeros is equal to zero. Compute the value of $a.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure this problem... |
A positive integer is divisible by both $70$ and $21$. Determine the smallest possible value of the quotient obtained by dividing the number by the product of its distinct prime factors. | A positive integer is divisible by both $70$ and $21$. Determine the smallest possible value of the quotient obtained by dividing the number by the product of its distinct prime factors. | Okay, so I need to find the smallest possible value of the quotient when a positive integer divisible by both 70 and 21 is divided by the product of its distinct prime factors. Let me break this down step by step.
First, the number has to be divisible by both 70 and 21. That means it has to be a common multiple of 70 ... | To determine the smallest possible value of the quotient obtained by dividing a positive integer divisible by both 70 and 21 by the product of its distinct prime factors, we start by finding the least common multiple (LCM) of 70 and 21.
First, we compute the greatest common divisor (GCD) of 70 and 21. The prime facto... | 2,291 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest possible value of the quotient when a positive integer divisible by both 70 and 21 is divided by the product of its distinct prime factors. Let me break this down step by step.
First, the number has to be divisible by both 70 and 21. That means it has to be a common multipl... | [
{
"from": "human",
"value": "A positive integer is divisible by both $70$ and $21$. Determine the smallest possible value of the quotient obtained by dividing the number by the product of its distinct prime factors."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest pos... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.