instruction_seed stringlengths 15 7.54k | response_seed stringlengths 15 7.54k ⌀ | reasoning stringlengths 0 107k | deepseek_solution stringlengths 0 13.3k | __original_row_idx int64 0 40k ⌀ | source stringclasses 2
values | final_reasoning_trace stringlengths 19 108k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
What is the maximum possible value for $z$ in the equation $z = 3x^2 - 8x - 10$? | What is the maximum possible value for $z$ in the equation $z = 3x^2 - 8x - 10$? | Okay, so I need to find the maximum possible value for z in the equation z = 3x² - 8x - 10. Hmm, let me think. I remember that quadratic equations graph as parabolas. Since the coefficient of x² here is 3, which is positive, that means the parabola opens upwards. Wait, if it opens upwards, then the vertex would be the ... | To find the maximum possible value for \( z \) in the equation \( z = 3x^2 - 8x - 10 \), we first recognize that this is a quadratic equation. The coefficient of \( x^2 \) is 3, which is positive, indicating that the parabola opens upwards.
Since the parabola opens upwards, it has a minimum value at its vertex and no... | 25,122 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the maximum possible value for z in the equation z = 3x² - 8x - 10. Hmm, let me think. I remember that quadratic equations graph as parabolas. Since the coefficient of x² here is 3, which is positive, that means the parabola opens upwards. Wait, if it opens upwards, then the vertex would... | [
{
"from": "human",
"value": "What is the maximum possible value for $z$ in the equation $z = 3x^2 - 8x - 10$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the maximum possible value for z in the equation z = 3x² - 8x - 10. Hmm, let me think. I remember that quadratic equations... |
[ Carnot's Theorem ]
Prove that the altitudes of a triangle intersect at one point.
# | null | Okay, so I need to prove that the altitudes of a triangle intersect at one point. Hmm, I remember that the point where the altitudes meet is called the orthocenter. But how do I show that all three altitudes actually intersect at a single point? Let me think.
First, let me recall what an altitude is. In a triangle, an... | To prove that the altitudes of a triangle intersect at one point, we can use Ceva's Theorem or coordinate geometry. Here, we present both approaches:
### Using Ceva's Theorem
1. **Ceva's Theorem**: For three cevians \(AD\), \(BE\), and \(CF\) of a triangle \(ABC\) to be concurrent, the product of the ratios \(\frac{BD... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that the altitudes of a triangle intersect at one point. Hmm, I remember that the point where the altitudes meet is called the orthocenter. But how do I show that all three altitudes actually intersect at a single point? Let me think.
First, let me recall what an altitude is. In a tria... | [
{
"from": "human",
"value": "[ Carnot's Theorem ]\n\nProve that the altitudes of a triangle intersect at one point.\n\n#"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to prove that the altitudes of a triangle intersect at one point. Hmm, I remember that the point where the altitudes me... |
Find the average of the odd numbers from 1 to 10 that are less than 6. | null | Okay, so I need to find the average of the odd numbers from 1 to 10 that are less than 6. Hmm, let me break this down step by step. First, let's make sure I understand the problem correctly. The task is to average the odd numbers within the range 1 to 10, but only those that are less than 6. Alright, so two conditions ... | To find the average of the odd numbers from 1 to 10 that are less than 6, we first identify the odd numbers in the range 1 to 10. These are 1, 3, 5, 7, and 9. Next, we filter these numbers to include only those less than 6, which are 1, 3, and 5.
We then calculate the sum of these numbers:
\[
1 + 3 + 5 = 9
\]
There a... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the average of the odd numbers from 1 to 10 that are less than 6. Hmm, let me break this down step by step. First, let's make sure I understand the problem correctly. The task is to average the odd numbers within the range 1 to 10, but only those that are less than 6. Alright, so two con... | [
{
"from": "human",
"value": "Find the average of the odd numbers from 1 to 10 that are less than 6."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the average of the odd numbers from 1 to 10 that are less than 6. Hmm, let me break this down step by step. First, let's make sure I... |
Consider the binary operation defined by $(a \bullet b) = \frac{a}{b}$. Find the value of $(3 \bullet 4) \bullet (2 \bullet 6)$. | Consider the binary operation defined by $(a \bullet b) = \frac{a}{b}$. Find the value of $(3 \bullet 4) \bullet (2 \bullet 6)$. | Okay, so I have this problem here: Consider the binary operation defined by (a • b) = a/b. Find the value of (3 • 4) • (2 • 6). Hmm, let me see. Alright, first, I need to remember what a binary operation is. If I recall correctly, a binary operation is just a rule that combines two elements from a set to produce anothe... | To solve the problem, we need to evaluate the expression \((3 \bullet 4) \bullet (2 \bullet 6)\) where the binary operation \(\bullet\) is defined by \(a \bullet b = \frac{a}{b}\).
First, we compute the inner operations:
1. \(3 \bullet 4 = \frac{3}{4}\)
2. \(2 \bullet 6 = \frac{2}{6} = \frac{1}{3}\)
Next, we use thes... | 5,860 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: Consider the binary operation defined by (a • b) = a/b. Find the value of (3 • 4) • (2 • 6). Hmm, let me see. Alright, first, I need to remember what a binary operation is. If I recall correctly, a binary operation is just a rule that combines two elements from a set to produc... | [
{
"from": "human",
"value": "Consider the binary operation defined by $(a \\bullet b) = \\frac{a}{b}$. Find the value of $(3 \\bullet 4) \\bullet (2 \\bullet 6)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: Consider the binary operation defined by (a • b) = a/b. Fi... |
Subtract $111.11$ from $333.33.$ Express the result as a decimal to the nearest hundredth. | Subtract $111.11$ from $333.33.$ Express the result as a decimal to the nearest hundredth. | Okay, so I need to subtract 111.11 from 333.33 and then make sure the result is a decimal rounded to the nearest hundredth. Hmm, let's start by writing down the numbers to set up the subtraction.
First, 333.33 minus 111.11. Let me line them up by the decimal point to avoid mistakes:
```
333.33
-111.11
--------
```
... | To subtract \(111.11\) from \(333.33\), we start by aligning the numbers by their decimal points:
\[
\begin{array}{r}
333.33 \\
-111.11 \\
\hline
\end{array}
\]
Subtracting each place value from right to left:
- Hundredths place: \(3 - 1 = 2\)
- Tenths place: \(3 - 1 = 2\)
- Units place: \(3 - 1 = 2\)
- Tens place: ... | 28,290 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to subtract 111.11 from 333.33 and then make sure the result is a decimal rounded to the nearest hundredth. Hmm, let's start by writing down the numbers to set up the subtraction.
First, 333.33 minus 111.11. Let me line them up by the decimal point to avoid mistakes:
```
333.33
-111.11
-----... | [
{
"from": "human",
"value": "Subtract $111.11$ from $333.33.$ Express the result as a decimal to the nearest hundredth."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to subtract 111.11 from 333.33 and then make sure the result is a decimal rounded to the nearest hundredth. Hmm, let's s... |
the average age of 50 students in a class is 14 years . if the age of teacher is also included , the average becomes 15 years , find the age of the teacher . | null | First, I need to calculate the total age of the 50 students by multiplying the average age by the number of students. The average age is 14 years, so the total age for the students is 50 multiplied by 14, which equals 700 years.
Next, when the teacher's age is included, there are now 51 individuals in total. The new a... | **Solution:**
1. **Calculate the total age of the 50 students:**
\[
\text{Total age of students} = 50 \text{ students} \times 14 \text{ years/student} = 700 \text{ years}
\]
2. **Calculate the total age when the teacher is included:**
\[
\text{Total age with teacher} = 51 \text{ people} \times ... | null | AI-MO/NuminaMath-1.5 | <think>
First, I need to calculate the total age of the 50 students by multiplying the average age by the number of students. The average age is 14 years, so the total age for the students is 50 multiplied by 14, which equals 700 years.
Next, when the teacher's age is included, there are now 51 individuals in total. T... | [
{
"from": "human",
"value": "the average age of 50 students in a class is 14 years . if the age of teacher is also included , the average becomes 15 years , find the age of the teacher ."
},
{
"from": "gpt",
"value": "<think>\nFirst, I need to calculate the total age of the 50 students by multip... |
In an examination, there were some candidates, out of which 900 candidates were girls and the rest were boys. If 38% of the boys and 32% of the girls passed, then the total percentage of failed candidates is 64.7%. How many candidates were there in total? | null | Okay, let me try to figure this problem out. So, there's an examination with some candidates. 900 of them are girls, and the rest are boys. We need to find the total number of candidates. The information given is about the pass rates: 38% of boys passed, 32% of girls passed, and the total percentage of failed candidate... | Let \( T \) be the total number of candidates. The number of girls is 900, so the number of boys is \( T - 900 \).
Given that 38% of the boys and 32% of the girls passed, the percentage of failed candidates is 62% for boys and 68% for girls. The total percentage of failed candidates is 64.7%. We can set up the equatio... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure this problem out. So, there's an examination with some candidates. 900 of them are girls, and the rest are boys. We need to find the total number of candidates. The information given is about the pass rates: 38% of boys passed, 32% of girls passed, and the total percentage of failed c... | [
{
"from": "human",
"value": "In an examination, there were some candidates, out of which 900 candidates were girls and the rest were boys. If 38% of the boys and 32% of the girls passed, then the total percentage of failed candidates is 64.7%. How many candidates were there in total?"
},
{
"from": "... |
Prove that for any integer value of $n$, $10^{n}(9n-1)+1$ is always divisible by 81. | null | Okay, so I need to prove that for any integer value of n, the expression 10^n(9n - 1) + 1 is always divisible by 81. Hmm, let me start by understanding what the problem is asking. Divisible by 81 means that when you divide the expression by 81, there's no remainder. So, 81 is 9 squared, which is 3 to the fourth power. ... | To prove that for any integer value of \( n \), \( 10^n(9n - 1) + 1 \) is always divisible by 81, we can use mathematical induction and modular arithmetic.
### Base Case
For \( n = 0 \):
\[
10^0(9 \cdot 0 - 1) + 1 = 1(-1) + 1 = -1 + 1 = 0
\]
0 is divisible by 81.
### Inductive Step
Assume for some integer \( k \geq 0... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that for any integer value of n, the expression 10^n(9n - 1) + 1 is always divisible by 81. Hmm, let me start by understanding what the problem is asking. Divisible by 81 means that when you divide the expression by 81, there's no remainder. So, 81 is 9 squared, which is 3 to the fourth... | [
{
"from": "human",
"value": "Prove that for any integer value of $n$, $10^{n}(9n-1)+1$ is always divisible by 81."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to prove that for any integer value of n, the expression 10^n(9n - 1) + 1 is always divisible by 81. Hmm, let me start by unde... |
In a village, there are 10 houses, and each house has a different color. Each day, each villager visits exactly 3 houses of different colors. If no villager visits the same house twice, and no two villagers visit the same combination of houses, how many days can this go on? | In a village, there are 10 houses, and each house has a different color. Each day, each villager visits exactly 3 houses of different colors. If no villager visits the same house twice, and no two villagers visit the same combination of houses, how many days can this go on? | Okay, so I need to figure out how many days the villagers can keep visiting houses under these rules. Let me try to break down the problem step by step.
First, there are 10 houses, each with a different color. Each day, every villager visits exactly 3 houses of different colors. The constraints are that no villager v... | Given 10 houses, each with a different color, each villager visits exactly 3 different houses each day. The constraints are that no villager visits the same house twice and no two villagers visit the same combination of houses. We need to determine the maximum number of days this can continue.
1. **Total Combinations*... | 37,356 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many days the villagers can keep visiting houses under these rules. Let me try to break down the problem step by step.
First, there are 10 houses, each with a different color. Each day, every villager visits exactly 3 houses of different colors. The constraints are that no vi... | [
{
"from": "human",
"value": "In a village, there are 10 houses, and each house has a different color. Each day, each villager visits exactly 3 houses of different colors. If no villager visits the same house twice, and no two villagers visit the same combination of houses, how many days can this go on?"
}... |
Find the product of the roots of the equation $x^4 - 2x^3 + 5x^2 - x - 1 = 0$. Express your answer as a decimal to the nearest tenth. | Find the product of the roots of the equation $x^4 - 2x^3 + 5x^2 - x - 1 = 0$. Express your answer as a decimal to the nearest tenth. | Okay, so I need to find the product of the roots of the equation \( x^4 - 2x^3 + 5x^2 - x - 1 = 0 \), and then express the answer as a decimal to the nearest tenth. Hmm, let me recall what I know about polynomials and their roots.
First off, I remember that for any polynomial equation of the form \( ax^n + bx^{n-1} +... | To find the product of the roots of the equation \(x^4 - 2x^3 + 5x^2 - x - 1 = 0\), we use Vieta's formulas. For a polynomial \(a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\), the product of the roots is given by \((-1)^n \times \frac{a_0}{a_n}\).
Given the polynomial \(x^4 - 2x^3 + 5x^2 - x - 1 = 0\):
- The leadin... | 20,510 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the product of the roots of the equation \( x^4 - 2x^3 + 5x^2 - x - 1 = 0 \), and then express the answer as a decimal to the nearest tenth. Hmm, let me recall what I know about polynomials and their roots.
First off, I remember that for any polynomial equation of the form \( ax^n + bx... | [
{
"from": "human",
"value": "Find the product of the roots of the equation $x^4 - 2x^3 + 5x^2 - x - 1 = 0$. Express your answer as a decimal to the nearest tenth."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the product of the roots of the equation \\( x^4 - 2x^3 + 5x^2 - x - ... |
The range of the independent variable $x$ in the function $y=\frac{1}{\sqrt{x-3}}+\sqrt{5-x}$ is ______. | null | Okay, let me try to figure out the range of the independent variable x for this function: y = 1 over the square root of (x minus 3) plus the square root of (5 minus x). Hmm, so I need to find all the x values that make this function defined. That means I have to look at the domains of each part separately and then find... | To determine the range of the independent variable \( x \) for the function \( y = \frac{1}{\sqrt{x - 3}} + \sqrt{5 - x} \), we need to consider the domains of each part of the function separately and then find their intersection.
1. For the term \( \frac{1}{\sqrt{x - 3}} \) to be defined:
- The expression inside t... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out the range of the independent variable x for this function: y = 1 over the square root of (x minus 3) plus the square root of (5 minus x). Hmm, so I need to find all the x values that make this function defined. That means I have to look at the domains of each part separately and t... | [
{
"from": "human",
"value": "The range of the independent variable $x$ in the function $y=\\frac{1}{\\sqrt{x-3}}+\\sqrt{5-x}$ is ______."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out the range of the independent variable x for this function: y = 1 over the square root of... |
Let $a$ be a prime number and $n > 2$ an integer.
Find all integer solutions of the equation $x^n +ay^n = a^2z^n$
. | null | Alright, let's try to tackle this problem. The equation given is \( x^n + a y^n = a^2 z^n \), where \( a \) is a prime number and \( n > 2 \) is an integer. We need to find all integer solutions for this equation. Hmm, okay. Let's start by understanding what the equation is saying. It looks similar to some Diophantine ... | Given the equation \( x^n + a y^n = a^2 z^n \) where \( a \) is a prime number and \( n > 2 \) is an integer, we need to find all integer solutions.
1. **Trivial Solution Check**:
- If \( x = 0 \), \( y = 0 \), and \( z = 0 \), the equation holds as \( 0 + 0 = 0 \).
2. **Non-Trivial Solutions**:
- Suppose \( x ... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, let's try to tackle this problem. The equation given is \( x^n + a y^n = a^2 z^n \), where \( a \) is a prime number and \( n > 2 \) is an integer. We need to find all integer solutions for this equation. Hmm, okay. Let's start by understanding what the equation is saying. It looks similar to some Diop... | [
{
"from": "human",
"value": "Let $a$ be a prime number and $n > 2$ an integer. \nFind all integer solutions of the equation $x^n +ay^n = a^2z^n$\n."
},
{
"from": "gpt",
"value": "<think>\nAlright, let's try to tackle this problem. The equation given is \\( x^n + a y^n = a^2 z^n \\), where \\( a ... |
Given that $f(x)$ is a monotonically decreasing first-degree function on $R$ and $f[f(x)]=4x-1$.
(1) Find $f(x)$;
(2) Find the maximum and minimum values of the function $y=f(x)+x^{2}-x$ on $x∈[-1,2]$. | null | Okay, let's tackle this problem step by step. So, we have a function f(x) which is monotonically decreasing and a first-degree function, meaning it's linear. The first part asks us to find f(x) given that f(f(x)) = 4x -1. Then, the second part is about finding the maximum and minimum values of another function involvin... | Given that \( f(x) \) is a monotonically decreasing first-degree function on \( \mathbb{R} \) and \( f(f(x)) = 4x - 1 \):
1. **Finding \( f(x) \)**:
- Since \( f(x) \) is a linear function, we assume \( f(x) = ax + b \).
- Given \( f(f(x)) = 4x - 1 \), we substitute \( f(x) = ax + b \) into itself:
\[
... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's tackle this problem step by step. So, we have a function f(x) which is monotonically decreasing and a first-degree function, meaning it's linear. The first part asks us to find f(x) given that f(f(x)) = 4x -1. Then, the second part is about finding the maximum and minimum values of another function ... | [
{
"from": "human",
"value": "Given that $f(x)$ is a monotonically decreasing first-degree function on $R$ and $f[f(x)]=4x-1$.\n(1) Find $f(x)$;\n(2) Find the maximum and minimum values of the function $y=f(x)+x^{2}-x$ on $x∈[-1,2]$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's tackle this... |
The real numbers \(x\) and \(y\) satisfy
\[
\begin{pmatrix} 3 \\ x \\ -9 \end{pmatrix} \times \begin{pmatrix} 6 \\ 5 \\ y \end{pmatrix} = \mathbf{0}.
\]
Find the ordered pair \((x, y)\). | null | Alright, so I have this problem where two vectors are given, and their cross product is the zero vector. That means the vectors are parallel, right? Because if the cross product is zero, the vectors are either parallel or one of them is the zero vector. But in this case, neither of the vectors are zero vectors because ... | Given the vectors \(\begin{pmatrix} 3 \\ x \\ -9 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 5 \\ y \end{pmatrix}\), we know their cross product is the zero vector, implying they are parallel. Therefore, one vector must be a scalar multiple of the other.
We set up the equation:
\[
\begin{pmatrix} 3 \\ x \\ -9 \end{pma... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I have this problem where two vectors are given, and their cross product is the zero vector. That means the vectors are parallel, right? Because if the cross product is zero, the vectors are either parallel or one of them is the zero vector. But in this case, neither of the vectors are zero vectors ... | [
{
"from": "human",
"value": "The real numbers \\(x\\) and \\(y\\) satisfy\n\\[\n\\begin{pmatrix} 3 \\\\ x \\\\ -9 \\end{pmatrix} \\times \\begin{pmatrix} 6 \\\\ 5 \\\\ y \\end{pmatrix} = \\mathbf{0}.\n\\]\nFind the ordered pair \\((x, y)\\)."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I ... |
If $\log (xy^3)= 1$ and $\log (x^2y) = 1$, what is $\log (xy)$? | If $\log (xy^3)= 1$ and $\log (x^2y) = 1$, what is $\log (xy)$? | Okay, let's see. I need to find log(xy) given that log(xy³) = 1 and log(x²y) = 1. Hmm. Alright, let's start by recalling some logarithm properties. I remember that log(ab) = log a + log b, and log(a^n) = n log a. So maybe I can break down these equations into simpler parts.
First, let's take the first equation: log(xy... | Given the equations \(\log (xy^3) = 1\) and \(\log (x^2y) = 1\), we need to find \(\log (xy)\).
First, we use logarithm properties to rewrite the given equations. Let \(a = \log x\) and \(b = \log y\). Then the equations become:
1. \(\log (xy^3) = \log x + 3 \log y = a + 3b = 1\)
2. \(\log (x^2y) = 2 \log x + \log y ... | 32,205 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find log(xy) given that log(xy³) = 1 and log(x²y) = 1. Hmm. Alright, let's start by recalling some logarithm properties. I remember that log(ab) = log a + log b, and log(a^n) = n log a. So maybe I can break down these equations into simpler parts.
First, let's take the first equation... | [
{
"from": "human",
"value": "If $\\log (xy^3)= 1$ and $\\log (x^2y) = 1$, what is $\\log (xy)$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find log(xy) given that log(xy³) = 1 and log(x²y) = 1. Hmm. Alright, let's start by recalling some logarithm properties. I remember ... |
Two lines, parallel to the bases of the trapezoid, divide each of the lateral sides into three equal parts. The entire trapezoid is divided into three parts by them. Find the area of the midd... | null | Okay, so we have a trapezoid with two lines that are parallel to the bases. These lines divide each of the lateral sides into three equal parts. The trapezoid is then divided into three parts by these lines, and we know the areas of the extreme parts, S₁ and S₂. We need to find the area of the middle part.
First, let ... | Given a trapezoid divided into three parts by two lines parallel to the bases, which divide each lateral side into three equal parts, we need to find the area of the middle part given the areas of the extreme parts \( S_1 \) and \( S_2 \).
1. **Understanding the Division**:
- The lateral sides are divided into thre... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so we have a trapezoid with two lines that are parallel to the bases. These lines divide each of the lateral sides into three equal parts. The trapezoid is then divided into three parts by these lines, and we know the areas of the extreme parts, S₁ and S₂. We need to find the area of the middle part.
Fir... | [
{
"from": "human",
"value": "\n\nTwo lines, parallel to the bases of the trapezoid, divide each of the lateral sides into three equal parts. The entire trapezoid is divided into thr... |
Find the distance between the planes $2x - 4y + 4z = 10$ and $x - 2y + 2z = 2.$ | null | Okay, so I need to find the distance between two planes given by the equations 2x - 4y + 4z = 10 and x - 2y + 2z = 2. Hmm, let me recall how to do this. I think the formula for the distance between two parallel planes is the absolute value of the difference between the constants divided by the square root of the sum of... | To find the distance between the planes \(2x - 4y + 4z = 10\) and \(x - 2y + 2z = 2\), we first check if they are parallel by comparing their normal vectors. The normal vector of the first plane is \((2, -4, 4)\) and the normal vector of the second plane is \((1, -2, 2)\). Since the normal vectors are scalar multiples ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the distance between two planes given by the equations 2x - 4y + 4z = 10 and x - 2y + 2z = 2. Hmm, let me recall how to do this. I think the formula for the distance between two parallel planes is the absolute value of the difference between the constants divided by the square root of th... | [
{
"from": "human",
"value": "Find the distance between the planes $2x - 4y + 4z = 10$ and $x - 2y + 2z = 2.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the distance between two planes given by the equations 2x - 4y + 4z = 10 and x - 2y + 2z = 2. Hmm, let me recall how to do ... |
Find $A$ and $B$ such that
\[\frac{6x+3}{x^2-9x-36}=\frac{A}{x-12}+\frac{B}{x+3}.\]
Write your answer in the form $(A,B)$. | null | Okay, so I need to find constants A and B such that the equation (6x + 3)/(x² - 9x - 36) = A/(x - 12) + B/(x + 3). Hmm, this looks like a partial fraction decomposition problem. Let me recall how partial fractions work.
First, I need to make sure the denominator factors correctly. The denominator here is x² - 9x - 36... | To find constants \( A \) and \( B \) such that
\[
\frac{6x+3}{x^2-9x-36} = \frac{A}{x-12} + \frac{B}{x+3},
\]
we start by factoring the denominator:
\[
x^2 - 9x - 36 = (x - 12)(x + 3).
\]
This allows us to rewrite the equation as:
\[
\frac{6x + 3}{(x - 12)(x + 3)} = \frac{A}{x - 12} + \frac{B}{x + 3}.
\]
Next, we comb... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find constants A and B such that the equation (6x + 3)/(x² - 9x - 36) = A/(x - 12) + B/(x + 3). Hmm, this looks like a partial fraction decomposition problem. Let me recall how partial fractions work.
First, I need to make sure the denominator factors correctly. The denominator here is x² -... | [
{
"from": "human",
"value": "Find $A$ and $B$ such that\n\\[\\frac{6x+3}{x^2-9x-36}=\\frac{A}{x-12}+\\frac{B}{x+3}.\\]\nWrite your answer in the form $(A,B)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find constants A and B such that the equation (6x + 3)/(x² - 9x - 36) = A/(x -... |
A certain school held a speech contest with the theme "Preventing Drowning Starts with Me". Judges will score the contestants based on the content of the speech, the delivery ability, and the effectiveness of the speech. The scores for each aspect are calculated on a percentage basis, and then weighted as follows: cont... | null | Okay, let me see. So, the problem is about calculating Xiao Liang's comprehensive score for the speech contest based on the three criteria: content, delivery ability, and effectiveness. Each of these has a different weight: 50%, 40%, and 10% respectively. His scores for each category are 88, 95, and 90. Hmm, I need to ... | To calculate Xiao Liang's comprehensive score, we need to compute the weighted average of his scores in the three categories: content, delivery ability, and effectiveness. The weights for these categories are 50%, 40%, and 10% respectively.
The formula for the comprehensive score is:
\[
\text{Comprehensive score} = ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me see. So, the problem is about calculating Xiao Liang's comprehensive score for the speech contest based on the three criteria: content, delivery ability, and effectiveness. Each of these has a different weight: 50%, 40%, and 10% respectively. His scores for each category are 88, 95, and 90. Hmm, I ... | [
{
"from": "human",
"value": "A certain school held a speech contest with the theme \"Preventing Drowning Starts with Me\". Judges will score the contestants based on the content of the speech, the delivery ability, and the effectiveness of the speech. The scores for each aspect are calculated on a percentag... |
Pat's parents invested $7000 in three different long-term bonds, with principal amounts P1, P2, and P3, and interest rates r1, r2, and r3, respectively, compounded annually. The sum of these three principal amounts is equal to $7000, and the amount of each investment will double in approximately 70 / r1, 70 / r2, and 7... | null | Okay, so Pat's parents invested $7000 in three different bonds. The problem gives me the details for two of them, P1 and P2, and tells me the third one, P3, is the remaining amount. I need to calculate the total amount after 18 years using the compound interest formula. Let me start by understanding each part of the pr... | Pat's parents invested $7000 in three different bonds with the following principal amounts and interest rates:
- \( P1 = \$2000 \) at \( r1 = 6\% \)
- \( P2 = \$2500 \) at \( r2 = 8.5\% \)
- \( P3 = \$2500 \) at \( r3 = 7\% \)
Using the compound interest formula \( A = P(1 + r)^t \) (compounded annually), we calculate... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so Pat's parents invested $7000 in three different bonds. The problem gives me the details for two of them, P1 and P2, and tells me the third one, P3, is the remaining amount. I need to calculate the total amount after 18 years using the compound interest formula. Let me start by understanding each part o... | [
{
"from": "human",
"value": "Pat's parents invested $7000 in three different long-term bonds, with principal amounts P1, P2, and P3, and interest rates r1, r2, and r3, respectively, compounded annually. The sum of these three principal amounts is equal to $7000, and the amount of each investment will double... |
Let $S$ be a subset of the set of all integers such that for any two distinct elements $a$ and $b$ of $S$, the equation $\lfloor x^2 \rfloor = ab$ has exactly one solution for $x$ in the interval $(a,b)$. What is the maximum number of elements in $S$? | Let $S$ be a subset of the set of all integers such that for any two distinct elements $a$ and $b$ of $S$, the equation $\lfloor x^2 \rfloor = ab$ has exactly one solution for $x$ in the interval $(a,b)$. What is the maximum number of elements in $S$? | Okay, so I have this problem here: We need to find the maximum number of elements in a subset S of all integers such that for any two distinct elements a and b in S, the equation floor(x²) = ab has exactly one solution for x in the interval (a, b). Hmm, that's a bit tricky, but let's try to unpack it step by step.
Fir... | Given a subset \( S \) of integers such that for any two distinct elements \( a \) and \( b \), the equation \( \lfloor x^2 \rfloor = ab \) has exactly one solution for \( x \) in the interval \( (a, b) \), we need to determine the maximum number of elements in \( S \).
1. **Understanding the Problem**:
- The equat... | 14,131 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: We need to find the maximum number of elements in a subset S of all integers such that for any two distinct elements a and b in S, the equation floor(x²) = ab has exactly one solution for x in the interval (a, b). Hmm, that's a bit tricky, but let's try to unpack it step by st... | [
{
"from": "human",
"value": "Let $S$ be a subset of the set of all integers such that for any two distinct elements $a$ and $b$ of $S$, the equation $\\lfloor x^2 \\rfloor = ab$ has exactly one solution for $x$ in the interval $(a,b)$. What is the maximum number of elements in $S$?"
},
{
"from": "gp... |
How many integer Fahrenheit temperatures between 50 and 500 inclusive, when converted to Celsius using the formula $C = \frac{5}{9}(F-32)$, rounded down (instead of the nearest integer), returned back to Fahrenheit using $F' = \left \lfloor \frac{9}{5}C \right\rfloor + 32$, and rounded down, yield the same original Fah... | null | Okay, let me try to figure out this problem step by step. The question is asking how many integer Fahrenheit temperatures between 50 and 500 inclusive, when converted to Celsius using the formula C = (5/9)(F - 32), then rounded down (which is basically taking the floor of the Celsius temperature), and then converted ba... | To determine how many integer Fahrenheit temperatures between 50 and 500 inclusive, when converted to Celsius and rounded down, then converted back to Fahrenheit using the same rounding method, yield the original temperature, we analyze the following steps:
1. Convert Fahrenheit to Celsius using \( C = \frac{5}{9}(F -... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this problem step by step. The question is asking how many integer Fahrenheit temperatures between 50 and 500 inclusive, when converted to Celsius using the formula C = (5/9)(F - 32), then rounded down (which is basically taking the floor of the Celsius temperature), and then conv... | [
{
"from": "human",
"value": "How many integer Fahrenheit temperatures between 50 and 500 inclusive, when converted to Celsius using the formula $C = \\frac{5}{9}(F-32)$, rounded down (instead of the nearest integer), returned back to Fahrenheit using $F' = \\left \\lfloor \\frac{9}{5}C \\right\\rfloor + 32$... |
In a given region, the total area of wheat fields is 8.75 million hectares. It is known that 25% of these fields are used for wheat export, and the remaining 75% are used for domestic consumption, with 2/3 of the remaining fields being irrigated and the rest being non-irrigated. If the average wheat yield per hectare i... | In a given region, the total area of wheat fields is 8.75 million hectares. It is known that 25% of these fields are used for wheat export, and the remaining 75% are used for domestic consumption, with 2/3 of the remaining fields being irrigated and the rest being non-irrigated. If the average wheat yield per hectare i... | Okay, let's tackle this problem step by step. First, I need to figure out the total wheat production for domestic consumption. The given total area of wheat fields is 8.75 million hectares. Out of these, 25% are used for export, which leaves 75% for domestic consumption. So, I should start by calculating the area alloc... | The total area of wheat fields is 8.75 million hectares. 25% of these fields are used for export, leaving 75% for domestic consumption.
First, calculate the area for domestic consumption:
\[
8.75 \times 0.75 = 6.5625 \text{ million hectares}
\]
Next, determine the irrigated and non-irrigated areas for domestic consu... | 18,225 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's tackle this problem step by step. First, I need to figure out the total wheat production for domestic consumption. The given total area of wheat fields is 8.75 million hectares. Out of these, 25% are used for export, which leaves 75% for domestic consumption. So, I should start by calculating the ar... | [
{
"from": "human",
"value": "In a given region, the total area of wheat fields is 8.75 million hectares. It is known that 25% of these fields are used for wheat export, and the remaining 75% are used for domestic consumption, with 2/3 of the remaining fields being irrigated and the rest being non-irrigated.... |
Tom and Alex went to the bakery together to buy some cookies. Tom bought $36$ cookies and Alex bought $27$. If each pack of cookies sold at the bakery contains the same number of cookies, what is the largest possible number of cookies in a pack? | Tom and Alex went to the bakery together to buy some cookies. Tom bought $36$ cookies and Alex bought $27$. If each pack of cookies sold at the bakery contains the same number of cookies, what is the largest possible number of cookies in a pack? | Okay, so Tom and Alex went to the bakery and bought some cookies. Tom bought 36 cookies, and Alex bought 27. The question is asking for the largest possible number of cookies in a pack, given that each pack has the same number of cookies. Hmm, I think this might be a greatest common divisor problem. Let me see.
First,... | Tom and Alex bought 36 and 27 cookies respectively, each pack containing the same number of cookies. To find the largest possible number of cookies in a pack, we need the greatest common divisor (GCD) of 36 and 27.
First, we use prime factorization:
- The prime factorization of 36 is \(2^2 \times 3^2\).
- The prime fa... | 30,930 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Tom and Alex went to the bakery and bought some cookies. Tom bought 36 cookies, and Alex bought 27. The question is asking for the largest possible number of cookies in a pack, given that each pack has the same number of cookies. Hmm, I think this might be a greatest common divisor problem. Let me see.... | [
{
"from": "human",
"value": "Tom and Alex went to the bakery together to buy some cookies. Tom bought $36$ cookies and Alex bought $27$. If each pack of cookies sold at the bakery contains the same number of cookies, what is the largest possible number of cookies in a pack?"
},
{
"from": "gpt",
... |
[ Geometric Inequalities (Miscellaneous) ]
A circle is inscribed in a triangle. A square is circumscribed around it. Prove that outside the triangle lies less than half the perimeter of the square.
# | null | Okay, so I need to prove that when a circle is inscribed in a triangle and a square is circumscribed around that circle, the part of the square's perimeter that lies outside the triangle is less than half of the square's total perimeter. Hmm, let me try to visualize this.
First, let's recall that an inscribed circle ... | To prove that less than half the perimeter of a square circumscribed around a circle inscribed in a triangle lies outside the triangle, we start by noting the following key points:
1. **Inradius and Square Dimensions**: The circle inscribed in the triangle has a radius \( r \). The square circumscribed around this cir... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that when a circle is inscribed in a triangle and a square is circumscribed around that circle, the part of the square's perimeter that lies outside the triangle is less than half of the square's total perimeter. Hmm, let me try to visualize this.
First, let's recall that an inscribed... | [
{
"from": "human",
"value": "[ Geometric Inequalities (Miscellaneous) ]\n\nA circle is inscribed in a triangle. A square is circumscribed around it. Prove that outside the triangle lies less than half the perimeter of the square.\n\n#"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to pr... |
Let $x \in \mathbb{R}$. Then $|x+1| < 1$ is a ( ) condition for $|x| < 2$ to hold.
A. Sufficient but not necessary condition B. Necessary but not sufficient condition
C. Necessary and sufficient condition D. Neither sufficient nor necessary condition | null | Alright, let me try to figure out this problem. So, the question is about determining whether the condition |x + 1| < 1 is sufficient, necessary, both, or neither for |x| < 2 to hold. The options are A to D as given.
First, let me recall what sufficient and necessary conditions mean. A sufficient condition means that... | To determine whether \(|x + 1| < 1\) is a sufficient, necessary, both, or neither condition for \(|x| < 2\), we analyze each inequality separately.
First, solving \(|x + 1| < 1\):
\[
-1 < x + 1 < 1 \implies -2 < x < 0
\]
This means the solutions are all real numbers between \(-2\) and \(0\).
Next, solving \(|x| < 2\)... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, let me try to figure out this problem. So, the question is about determining whether the condition |x + 1| < 1 is sufficient, necessary, both, or neither for |x| < 2 to hold. The options are A to D as given.
First, let me recall what sufficient and necessary conditions mean. A sufficient condition me... | [
{
"from": "human",
"value": "Let $x \\in \\mathbb{R}$. Then $|x+1| < 1$ is a ( ) condition for $|x| < 2$ to hold. \n A. Sufficient but not necessary condition B. Necessary but not sufficient condition \nC. Necessary and sufficient condition D. Neither sufficient nor necessary con... |
A box contains 10 balls, each with a different number from 1 to 10. If you randomly select 3 balls from the box without replacement, what is the probability that the sum of the numbers on the selected balls is an even number? Express your answer as a common fraction. | A box contains 10 balls, each with a different number from 1 to 10. If you randomly select 3 balls from the box without replacement, what is the probability that the sum of the numbers on the selected balls is an even number? Express your answer as a common fraction. | Okay, so I need to find the probability that the sum of three randomly selected balls from a box of 10 (numbered 1 to 10) is even. Hmm. Let me start by recalling that probability is the number of favorable outcomes divided by the total number of possible outcomes.
First, the total number of ways to choose 3 balls out... | To find the probability that the sum of the numbers on three randomly selected balls from a box of 10 balls (each numbered 1 to 10) is even, we start by calculating the total number of ways to choose 3 balls out of 10. This is given by the combination formula \( \binom{10}{3} \):
\[
\binom{10}{3} = \frac{10!}{3!(10-3)... | 35,358 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that the sum of three randomly selected balls from a box of 10 (numbered 1 to 10) is even. Hmm. Let me start by recalling that probability is the number of favorable outcomes divided by the total number of possible outcomes.
First, the total number of ways to choose 3 b... | [
{
"from": "human",
"value": "A box contains 10 balls, each with a different number from 1 to 10. If you randomly select 3 balls from the box without replacement, what is the probability that the sum of the numbers on the selected balls is an even number? Express your answer as a common fraction."
},
{
... |
Given that $F\_1$ and $F\_2$ are the common foci of an ellipse and a hyperbola, $P$ is their common point, and $\angle{F_1 P F_2}= \frac{2\pi}{3}$, the eccentricity of the ellipse is $e\_1$ and the eccentricity of the hyperbola is $e\_2$. Find the value of $\frac{3}{e\_1^{2}}+ \frac{1}{e\_2^{2}}$. | null | Okay, so I need to find the value of \(\frac{3}{e_1^{2}} + \frac{1}{e_2^{2}}\) where \(e_1\) is the eccentricity of an ellipse and \(e_2\) is the eccentricity of a hyperbola that share the same foci \(F_1\) and \(F_2\), and a common point \(P\) where the angle \(\angle F_1 P F_2 = \frac{2\pi}{3}\) (which is 120 degrees... | Given that \(F_1\) and \(F_2\) are the common foci of an ellipse and a hyperbola, and \(P\) is their common point with \(\angle F_1 P F_2 = \frac{2\pi}{3}\), we need to find the value of \(\frac{3}{e_1^2} + \frac{1}{e_2^2}\) where \(e_1\) is the eccentricity of the ellipse and \(e_2\) is the eccentricity of the hyperbo... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the value of \(\frac{3}{e_1^{2}} + \frac{1}{e_2^{2}}\) where \(e_1\) is the eccentricity of an ellipse and \(e_2\) is the eccentricity of a hyperbola that share the same foci \(F_1\) and \(F_2\), and a common point \(P\) where the angle \(\angle F_1 P F_2 = \frac{2\pi}{3}\) (which is 120... | [
{
"from": "human",
"value": "Given that $F\\_1$ and $F\\_2$ are the common foci of an ellipse and a hyperbola, $P$ is their common point, and $\\angle{F_1 P F_2}= \\frac{2\\pi}{3}$, the eccentricity of the ellipse is $e\\_1$ and the eccentricity of the hyperbola is $e\\_2$. Find the value of $\\frac{3}{e\\_... |
Given the ellipse $\frac{x^{2}}{t^{2}} + \frac{y^{2}}{5t} = 1$ with a focal length of $2\sqrt{6}$, find the real number value of $t$. | null | Okay, so I have this problem here about an ellipse. The equation given is \(\frac{x^{2}}{t^{2}} + \frac{y^{2}}{5t} = 1\), and they tell me the focal length is \(2\sqrt{6}\). I need to find the real number value of \(t\). Hmm, let me recall what I know about ellipses.
First, the standard form of an ellipse is \(\frac{x... | Given the ellipse \(\frac{x^{2}}{t^{2}} + \frac{y^{2}}{5t} = 1\) with a focal length of \(2\sqrt{6}\), we need to find the real number value of \(t\).
First, we note that the focal length of an ellipse is given by \(2c\), where \(c\) is the distance from the center to each focus. For the given ellipse, the focal lengt... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem here about an ellipse. The equation given is \(\frac{x^{2}}{t^{2}} + \frac{y^{2}}{5t} = 1\), and they tell me the focal length is \(2\sqrt{6}\). I need to find the real number value of \(t\). Hmm, let me recall what I know about ellipses.
First, the standard form of an ellipse is \... | [
{
"from": "human",
"value": "Given the ellipse $\\frac{x^{2}}{t^{2}} + \\frac{y^{2}}{5t} = 1$ with a focal length of $2\\sqrt{6}$, find the real number value of $t$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here about an ellipse. The equation given is \\(\\frac{x^{2}}... |
Sandy grew 6 carrots. Sam grew some carrots. They grew a total of 9 carrots. How many carrots did Sam grow? | null | Okay, so Sandy grew 6 carrots, and Sam grew some carrots. Together, they have a total of 9 carrots. The question is asking how many carrots Sam grew. Alright, let's break this down.
First, I know that Sandy has 6 carrots. Let me write that down: Sandy = 6 carrots. Then Sam has an unknown number of carrots, which we ca... | Sandy grew 6 carrots, and together with Sam, they have a total of 9 carrots. To find out how many carrots Sam grew, subtract Sandy's carrots from the total:
\[
\text{Sam's carrots} = \text{Total carrots} - \text{Sandy's carrots} = 9 - 6 = 3
\]
**Answer:** Sam grew \(\boxed{3}\) carrots. | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so Sandy grew 6 carrots, and Sam grew some carrots. Together, they have a total of 9 carrots. The question is asking how many carrots Sam grew. Alright, let's break this down.
First, I know that Sandy has 6 carrots. Let me write that down: Sandy = 6 carrots. Then Sam has an unknown number of carrots, whi... | [
{
"from": "human",
"value": "Sandy grew 6 carrots. Sam grew some carrots. They grew a total of 9 carrots. How many carrots did Sam grow?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so Sandy grew 6 carrots, and Sam grew some carrots. Together, they have a total of 9 carrots. The question is ask... |
The positional relationship between the line $3x+y+1=0$ and the line $6x+2y+1=0$ is ( )
A: Coincident
B: Parallel
C: Perpendicular
D: Intersecting but not perpendicular | null | Okay, so I need to figure out the positional relationship between the two lines given by the equations 3x + y + 1 = 0 and 6x + 2y + 1 = 0. The options are coincident, parallel, perpendicular, or intersecting but not perpendicular. Hmm, let's start by recalling some basics about lines and their relationships.
First, I ... | To determine the positional relationship between the lines \(3x + y + 1 = 0\) and \(6x + 2y + 1 = 0\), we convert both equations to slope-intercept form \(y = mx + b\).
For the first line \(3x + y + 1 = 0\):
\[
y = -3x - 1
\]
This line has a slope \(m_1 = -3\) and a y-intercept \(b_1 = -1\).
For the second line \(6x ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out the positional relationship between the two lines given by the equations 3x + y + 1 = 0 and 6x + 2y + 1 = 0. The options are coincident, parallel, perpendicular, or intersecting but not perpendicular. Hmm, let's start by recalling some basics about lines and their relationships.
F... | [
{
"from": "human",
"value": "The positional relationship between the line $3x+y+1=0$ and the line $6x+2y+1=0$ is ( )\nA: Coincident\nB: Parallel\nC: Perpendicular\nD: Intersecting but not perpendicular"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the positional relati... |
Find the smallest possible area of a convex pentagon whose vertexes are lattice points in a plane. | null | Okay, so I need to find the smallest possible area of a convex pentagon with vertices on lattice points. Hmm, lattice points are points with integer coordinates, right? So all the vertices of the pentagon must be at (x, y) where x and y are integers. And the pentagon has to be convex, meaning all its interior angles ar... | To find the smallest possible area of a convex pentagon with vertices on lattice points, we use Pick's Theorem, which states that the area \(A\) of a polygon with integer coordinates is given by \(A = I + \frac{B}{2} - 1\), where \(I\) is the number of interior lattice points and \(B\) is the number of boundary lattice... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the smallest possible area of a convex pentagon with vertices on lattice points. Hmm, lattice points are points with integer coordinates, right? So all the vertices of the pentagon must be at (x, y) where x and y are integers. And the pentagon has to be convex, meaning all its interior a... | [
{
"from": "human",
"value": "Find the smallest possible area of a convex pentagon whose vertexes are lattice points in a plane."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest possible area of a convex pentagon with vertices on lattice points. Hmm, lattice points are... |
11.8 A section was made on a rectangular parallelepiped, with the shape of a hexagon. It is known that this hexagon can fit into a rectangle $\Pi$. Prove: In the rectangle $\Pi$, a face of the rectangular parallelepiped can be placed. | null | Okay, so I need to prove that if a hexagon formed by a section of a rectangular parallelepiped can fit into a rectangle Π, then one of the faces of the parallelepiped can also fit into Π. Hmm, let's start by understanding the problem.
First, a rectangular parallelepiped is just a box with all sides being rectangles, r... | To prove that a face of the rectangular parallelepiped can be placed into the rectangle \(\Pi\) if the hexagon formed by a section of the parallelepiped can fit into \(\Pi\), we can use a proof by contradiction.
1. **Assumption**: Assume none of the faces of the parallelepiped can be placed into \(\Pi\). This means:
... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that if a hexagon formed by a section of a rectangular parallelepiped can fit into a rectangle Π, then one of the faces of the parallelepiped can also fit into Π. Hmm, let's start by understanding the problem.
First, a rectangular parallelepiped is just a box with all sides being recta... | [
{
"from": "human",
"value": "11.8 A section was made on a rectangular parallelepiped, with the shape of a hexagon. It is known that this hexagon can fit into a rectangle $\\Pi$. Prove: In the rectangle $\\Pi$, a face of the rectangular parallelepiped can be placed."
},
{
"from": "gpt",
"value": ... |
A man saves a certain portion of his income during a year and spends the remaining portion on his personal expenses. Next year his income increases by 30% but his savings increase by 100%. His total expenditure in 2 years is double his expenditure in the 1st year. What percentage of his income in the first year did he ... | null | Okay, let me try to work through this problem step by step. Hmm, so we have a man who saves a portion of his income each year and spends the rest. The next year, his income goes up by 30%, and his savings double. The total expenditure over the two years is double the expenditure of the first year. We need to find out w... | Let \( I \) be the man's income in the first year, and let \( s \) be the fraction of his income he saves in the first year. His expenditure in the first year is then \( (1 - s)I \).
In the second year, his income increases by 30%, making his income \( 1.3I \). His savings double, so his savings in the second year are... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to work through this problem step by step. Hmm, so we have a man who saves a portion of his income each year and spends the rest. The next year, his income goes up by 30%, and his savings double. The total expenditure over the two years is double the expenditure of the first year. We need to fi... | [
{
"from": "human",
"value": "A man saves a certain portion of his income during a year and spends the remaining portion on his personal expenses. Next year his income increases by 30% but his savings increase by 100%. His total expenditure in 2 years is double his expenditure in the 1st year. What percentag... |
Given \\( \overrightarrow{a}=(2,-3,5) \\) and \\( \overrightarrow{b}=(-3,1,2) \\), then \\( | \overrightarrow{a}-2 \overrightarrow{b}|= \)
A: \\(7 \sqrt {2}\\)
B: \\(5 \sqrt {2}\\)
C: \\(3 \sqrt {10}\\)
D: \\(6 \sqrt {3}\\) | null | Okay, let's see. I have this problem where I need to find the magnitude of the vector a minus twice vector b. The vectors are given as a = (2, -3, 5) and b = (-3, 1, 2). The options are A through D with different square roots. Hmm, I need to calculate |a - 2b|. Alright, step by step.
First, I remember that when you su... | Given vectors \( \overrightarrow{a} = (2, -3, 5) \) and \( \overrightarrow{b} = (-3, 1, 2) \), we need to find the magnitude of \( \overrightarrow{a} - 2 \overrightarrow{b} \).
First, compute \( 2 \overrightarrow{b} \):
\[
2 \overrightarrow{b} = 2 \times (-3, 1, 2) = (-6, 2, 4)
\]
Next, subtract \( 2 \overrightarrow{... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I have this problem where I need to find the magnitude of the vector a minus twice vector b. The vectors are given as a = (2, -3, 5) and b = (-3, 1, 2). The options are A through D with different square roots. Hmm, I need to calculate |a - 2b|. Alright, step by step.
First, I remember that whe... | [
{
"from": "human",
"value": "Given \\\\( \\overrightarrow{a}=(2,-3,5) \\\\) and \\\\( \\overrightarrow{b}=(-3,1,2) \\\\), then \\\\( | \\overrightarrow{a}-2 \\overrightarrow{b}|= \\) \nA: \\\\(7 \\sqrt {2}\\\\) \nB: \\\\(5 \\sqrt {2}\\\\) \nC: \\\\(3 \\sqrt {10}\\\\) \nD: \\\\(6 \\sqrt {3}\\\\)"
},
... |
A new operation is defined as follows: for any rational numbers $a$ and $b$, $a$☆$b=2a-b+1$. Please calculate the value of $1$☆$[2$☆$\left(-3\right)]$. | null | Okay, let's see. I need to calculate the value of 1☆[2☆(-3)] using this new operation defined as a☆b = 2a - b + 1. Hmm, operations inside operations. So, this is like nested operations. I remember in math, when there are parentheses, you should start with the innermost ones first. So, first, I need to compute the value... | To calculate the value of \(1 \mathbin{\bigstar} [2 \mathbin{\bigstar} (-3)]\) using the operation defined as \(a \mathbin{\bigstar} b = 2a - b + 1\), we start by evaluating the inner operation first.
First, compute \(2 \mathbin{\bigstar} (-3)\):
\[
2 \mathbin{\bigstar} (-3) = 2 \cdot 2 - (-3) + 1 = 4 + 3 + 1 = 8
\]
... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to calculate the value of 1☆[2☆(-3)] using this new operation defined as a☆b = 2a - b + 1. Hmm, operations inside operations. So, this is like nested operations. I remember in math, when there are parentheses, you should start with the innermost ones first. So, first, I need to compute t... | [
{
"from": "human",
"value": "A new operation is defined as follows: for any rational numbers $a$ and $b$, $a$☆$b=2a-b+1$. Please calculate the value of $1$☆$[2$☆$\\left(-3\\right)]$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to calculate the value of 1☆[2☆(-3)] using this n... |
At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner? | null | Alright, so I need to figure out the minimum number of moves a knight needs to go from one corner of a chessboard to the opposite corner. Let me start by recalling how a knight moves in chess. A knight moves in an L-shape: two squares in one direction and then one square perpendicular, right? So from any given position... | To determine the minimum number of moves a knight needs to travel from one corner of a chessboard to the opposite corner, we start by noting that a knight moves in an L-shape (two squares in one direction and one square perpendicular). The knight alternates between light and dark squares with each move. Since both the ... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I need to figure out the minimum number of moves a knight needs to go from one corner of a chessboard to the opposite corner. Let me start by recalling how a knight moves in chess. A knight moves in an L-shape: two squares in one direction and then one square perpendicular, right? So from any given ... | [
{
"from": "human",
"value": "At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to figure out the minimum number of moves a knight needs to go from one corner of a chessboard to the ... |
When $0.73\overline{264}$ is expressed as a fraction in the form $\frac{y}{999900}$, what is the value of $y$? | null | Alright, so I need to convert the repeating decimal 0.73\overline{264} into a fraction and express it as y/999900, then find the value of y. Hmm, okay. Let me think about how to approach this. I remember that to convert a repeating decimal to a fraction, there's a standard method involving setting up an equation and us... | To convert the repeating decimal \(0.73\overline{264}\) into a fraction and express it as \(\frac{y}{999900}\), we follow these steps:
1. Let \(x = 0.73\overline{264}\). This decimal has two non-repeating digits (73) followed by three repeating digits (264).
2. Multiply \(x\) by 100 to move the decimal point past the ... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I need to convert the repeating decimal 0.73\overline{264} into a fraction and express it as y/999900, then find the value of y. Hmm, okay. Let me think about how to approach this. I remember that to convert a repeating decimal to a fraction, there's a standard method involving setting up an equatio... | [
{
"from": "human",
"value": "When $0.73\\overline{264}$ is expressed as a fraction in the form $\\frac{y}{999900}$, what is the value of $y$?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to convert the repeating decimal 0.73\\overline{264} into a fraction and express it as y/999900... |
Dr. Fu Manchu has a bank account that has an annual interest rate of 6 percent, but it compounds monthly. If this is equivalent to a bank account that compounds annually at a rate of $r$ percent, then what is $r$? (Give your answer to the nearest hundredth.) | Dr. Fu Manchu has a bank account that has an annual interest rate of 6 percent, but it compounds monthly. If this is equivalent to a bank account that compounds annually at a rate of $r$ percent, then what is $r$? (Give your answer to the nearest hundredth.) | Okay, so I need to figure out the equivalent annual interest rate for Dr. Fu Manchu's bank account. The problem says his account has an annual interest rate of 6 percent, but it compounds monthly. Then, we need to find what the equivalent annual rate would be if it were compounded annually instead. That rate is called ... | To find the equivalent annual interest rate \( r \) for Dr. Fu Manchu's bank account that compounds monthly at an annual interest rate of 6%, we use the formula for effective annual rate (EAR):
\[
\text{EAR} = \left(1 + \frac{r_{\text{nominal}}}{n}\right)^n - 1
\]
where \( r_{\text{nominal}} = 0.06 \) (6% annual rate... | 3,641 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the equivalent annual interest rate for Dr. Fu Manchu's bank account. The problem says his account has an annual interest rate of 6 percent, but it compounds monthly. Then, we need to find what the equivalent annual rate would be if it were compounded annually instead. That rate is... | [
{
"from": "human",
"value": "Dr. Fu Manchu has a bank account that has an annual interest rate of 6 percent, but it compounds monthly. If this is equivalent to a bank account that compounds annually at a rate of $r$ percent, then what is $r$? (Give your answer to the nearest hundredth.)"
},
{
"fro... |
Points \(A\), \(B\), \(C\), and \(T\) are in space such that each of \(\overline{TA}\), \(\overline{TB}\), and \(\overline{TC}\) is perpendicular to the other two. If \(TA = TB = 12\) and \(TC = 10\), then what is the distance from \(T\) to face \(ABC\)? | null | Okay, so I have this geometry problem here. Points A, B, C, and T are in space, and each of the line segments TA, TB, and TC is perpendicular to the other two. That means TA is perpendicular to TB and TC, TB is perpendicular to TA and TC, and TC is perpendicular to TA and TB. They also give the lengths: TA and TB are b... | Given points \(A\), \(B\), \(C\), and \(T\) in space where each of \(\overline{TA}\), \(\overline{TB}\), and \(\overline{TC}\) is perpendicular to the other two, with \(TA = TB = 12\) and \(TC = 10\), we need to find the distance from \(T\) to the face \(ABC\).
We model the problem using a coordinate system with \(T\)... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this geometry problem here. Points A, B, C, and T are in space, and each of the line segments TA, TB, and TC is perpendicular to the other two. That means TA is perpendicular to TB and TC, TB is perpendicular to TA and TC, and TC is perpendicular to TA and TB. They also give the lengths: TA and ... | [
{
"from": "human",
"value": "Points \\(A\\), \\(B\\), \\(C\\), and \\(T\\) are in space such that each of \\(\\overline{TA}\\), \\(\\overline{TB}\\), and \\(\\overline{TC}\\) is perpendicular to the other two. If \\(TA = TB = 12\\) and \\(TC = 10\\), then what is the distance from \\(T\\) to face \\(ABC\\)?... |
Four points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle’s center? | null | Okay, so the problem is: Four points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle’s center?
Hmm, let me try to understand this. First, we have a circle, and we randomly pick four points on its circumference. We need to find the ... | To determine the probability that four points chosen uniformly at random on a circle form no obtuse triangles with the circle’s center, we need to ensure that all central angles between any two points are at most 90 degrees. This is equivalent to all four points lying within a 90-degree arc.
Key steps:
1. **Understand... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so the problem is: Four points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle’s center?
Hmm, let me try to understand this. First, we have a circle, and we randomly pick four points on its circumference. We need to f... | [
{
"from": "human",
"value": "Four points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle’s center?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem is: Four points are chosen uniformly at random o... |
Let $x$ be a positive real number. Define
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.
[i]Proposed by Evan Chen[/i] | null | Alright, let me try to figure out how to compute ABC given that equation A³ + B³ + C³ + 8ABC = 2014. Hmm, okay. So first, I need to recall what these A, B, and C are. They are sums of parts of the exponential series, split based on the exponent modulo 3. That is, A is the sum of x^{3k}/(3k)!, B is x^{3k+1}/(3k+1)!, and... | Given the problem, we need to compute \(ABC\) given that \(A^3 + B^3 + C^3 + 8ABC = 2014\), where \(A\), \(B\), and \(C\) are defined as follows:
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad \text{and} \quad C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, let me try to figure out how to compute ABC given that equation A³ + B³ + C³ + 8ABC = 2014. Hmm, okay. So first, I need to recall what these A, B, and C are. They are sums of parts of the exponential series, split based on the exponent modulo 3. That is, A is the sum of x^{3k}/(3k)!, B is x^{3k+1}/(3k+... | [
{
"from": "human",
"value": "Let $x$ be a positive real number. Define\n\\[\n A = \\sum_{k=0}^{\\infty} \\frac{x^{3k}}{(3k)!}, \\quad\n B = \\sum_{k=0}^{\\infty} \\frac{x^{3k+1}}{(3k+1)!}, \\quad\\text{and}\\quad\n C = \\sum_{k=0}^{\\infty} \\frac{x^{3k+2}}{(3k+2)!}.\n\\] Given that $A^3+B^3+C^3 + 8ABC =... |
A bag contains 4 red balls, 6 blue balls, and 10 green balls. Three balls are drawn at random from the bag without replacement. What is the probability that the first ball is red, the second ball is green, and the third ball is either blue or red? | A bag contains 4 red balls, 6 blue balls, and 10 green balls. Three balls are drawn at random from the bag without replacement. What is the probability that the first ball is red, the second ball is green, and the third ball is either blue or red? | Okay, so I need to find the probability that when drawing three balls from a bag without replacement, the first ball is red, the second is green, and the third is either blue or red. Let me break this down step by step.
First, let's note the total number of balls in the bag. There are 4 red, 6 blue, and 10 green balls... | To find the probability that the first ball drawn is red, the second is green, and the third is either blue or red, we use the multiplication rule for probability, considering the changing composition of the bag after each draw.
1. **First draw (Red):**
- There are 4 red balls out of 20 total.
- Probability: \( ... | 5,431 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that when drawing three balls from a bag without replacement, the first ball is red, the second is green, and the third is either blue or red. Let me break this down step by step.
First, let's note the total number of balls in the bag. There are 4 red, 6 blue, and 10 gre... | [
{
"from": "human",
"value": "A bag contains 4 red balls, 6 blue balls, and 10 green balls. Three balls are drawn at random from the bag without replacement. What is the probability that the first ball is red, the second ball is green, and the third ball is either blue or red?"
},
{
"from": "gpt",
... |
Find the value of $\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}$ expressed as a simplified fraction.
Use the concept of rationalizing the denominator to solve this problem. Multiply the numerator and denominator of each fraction by the conjugate of the denominator, then simplify and add the resulting fractions.
$$\frac{1... | Find the value of $\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}$ expressed as a simplified fraction.
Use the concept of rationalizing the denominator to solve this problem. Multiply the numerator and denominator of each fraction by the conjugate of the denominator, then simplify and add the resulting fractions.
$$\frac{1... | Okay, so I need to find the value of 1 over square root of 2 plus 1 over square root of 3 and express it as a simplified fraction. The problem mentions using rationalizing the denominator, which I remember is when you get rid of the square root in the denominator by multiplying by the conjugate or something. Let me thi... | To find the value of \(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}\) expressed as a simplified fraction, we start by rationalizing the denominators of each term.
For \(\frac{1}{\sqrt{2}}\), we multiply the numerator and denominator by \(\sqrt{2}\):
\[
\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2... | 5,341 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of 1 over square root of 2 plus 1 over square root of 3 and express it as a simplified fraction. The problem mentions using rationalizing the denominator, which I remember is when you get rid of the square root in the denominator by multiplying by the conjugate or something. Le... | [
{
"from": "human",
"value": "Find the value of $\\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}}$ expressed as a simplified fraction.\n\nUse the concept of rationalizing the denominator to solve this problem. Multiply the numerator and denominator of each fraction by the conjugate of the denominator, then simpli... |
A school is organizing a field trip for 8 students, who will be accompanied by 2 chaperones. When they arrive at their destination, the chaperones decide to divide the students into two groups, with each chaperone leading one group. However, to ensure each group is manageable, each chaperone must have at least 2 studen... | A school is organizing a field trip for 8 students, who will be accompanied by 2 chaperones. When they arrive at their destination, the chaperones decide to divide the students into two groups, with each chaperone leading one group. However, to ensure each group is manageable, each chaperone must have at least 2 studen... | Okay, so there's this problem where a school is organizing a field trip with 8 students and 2 chaperones. When they arrive, the chaperones need to split the students into two groups, each led by one chaperone. The catch is that each group must have at least 2 students. The question is asking how many different ways the... | To determine the number of ways to divide 8 students into two groups with each chaperone leading one group and each group having at least 2 students, we start by considering the total number of ways to assign students to the two chaperones without restrictions. This is \(2^8 = 256\). We then subtract the invalid cases ... | 30,355 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this problem where a school is organizing a field trip with 8 students and 2 chaperones. When they arrive, the chaperones need to split the students into two groups, each led by one chaperone. The catch is that each group must have at least 2 students. The question is asking how many different ... | [
{
"from": "human",
"value": "A school is organizing a field trip for 8 students, who will be accompanied by 2 chaperones. When they arrive at their destination, the chaperones decide to divide the students into two groups, with each chaperone leading one group. However, to ensure each group is manageable, e... |
There are 12 students in a class, consisting of 4 boys and 8 girls. They are assigned to stand in a line for a class photo, with the boys wearing identical blue shirts and the girls wearing identical yellow shirts. How many distinct ways are there to arrange the 4 blue shirts and 8 yellow shirts in the line? | There are 12 students in a class, consisting of 4 boys and 8 girls. They are assigned to stand in a line for a class photo, with the boys wearing identical blue shirts and the girls wearing identical yellow shirts. How many distinct ways are there to arrange the 4 blue shirts and 8 yellow shirts in the line? | Okay, so there's this problem about arranging 12 students in a line for a class photo. The class has 4 boys and 8 girls. The boys all wear identical blue shirts, and the girls wear identical yellow shirts. The question is asking how many distinct ways there are to arrange these shirts in the line. Hmm, let me think.
F... | To determine the number of distinct ways to arrange the 4 blue shirts and 8 yellow shirts in a line, we need to consider the permutations of these shirts. Since the boys and girls are wearing identical shirts, the problem reduces to finding the number of ways to choose positions for the blue shirts (or yellow shirts) a... | 34,941 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this problem about arranging 12 students in a line for a class photo. The class has 4 boys and 8 girls. The boys all wear identical blue shirts, and the girls wear identical yellow shirts. The question is asking how many distinct ways there are to arrange these shirts in the line. Hmm, let me t... | [
{
"from": "human",
"value": "There are 12 students in a class, consisting of 4 boys and 8 girls. They are assigned to stand in a line for a class photo, with the boys wearing identical blue shirts and the girls wearing identical yellow shirts. How many distinct ways are there to arrange the 4 blue shirts an... |
$26 \cdot 12$ The function $f(x)=x^{2}+2 b x+1$ and $g(x)=2 a(x+b)$, where the variable $x$ and constants $a, b$ are real numbers. Each pair of constants $a, b$ can be considered as a point $(a, b)$ in the $ab$-plane. Let $S$ be the set of these points $(a, b)$ for which the curves $y=$ $f(x)$ and $y=g(x)$ do not inter... | null | Okay, so I have this problem here about the functions f(x) and g(x). The question is asking about the set S of points (a, b) in the ab-plane such that the curves y = f(x) and y = g(x) don't intersect. Then we need to find the area of S. The answer choices are given as numbers 1, π, 4, 4π, or infinite. Hmm, interesting.... | To determine the set \( S \) of points \((a, b)\) for which the curves \( y = f(x) \) and \( y = g(x) \) do not intersect, we start by setting the functions equal to each other and analyzing the resulting quadratic equation.
Given:
\[ f(x) = x^2 + 2bx + 1 \]
\[ g(x) = 2a(x + b) \]
Setting \( f(x) = g(x) \):
\[ x^2 +... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem here about the functions f(x) and g(x). The question is asking about the set S of points (a, b) in the ab-plane such that the curves y = f(x) and y = g(x) don't intersect. Then we need to find the area of S. The answer choices are given as numbers 1, π, 4, 4π, or infinite. Hmm, inte... | [
{
"from": "human",
"value": "$26 \\cdot 12$ The function $f(x)=x^{2}+2 b x+1$ and $g(x)=2 a(x+b)$, where the variable $x$ and constants $a, b$ are real numbers. Each pair of constants $a, b$ can be considered as a point $(a, b)$ in the $ab$-plane. Let $S$ be the set of these points $(a, b)$ for which the cu... |
3. Given $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right)$ are points on and outside the line $l$ respectively, if the equation of the line $l$ is $f(x, y)=$ 0, then the equation $f(x, y)-f\left(x_{1}, y_{1}\right)-f\left(x_{2}, y_{2}\right)=0$ represents
A. a line coinciding with $l$
B. a line passing t... | null | Okay, let's see. The problem is about two points, P₁ and P₂, where P₁ is on the line l and P₂ is outside of it. The equation of line l is given by f(x, y) = 0. Then we have this new equation: f(x, y) - f(x₁, y₁) - f(x₂, y₂) = 0. The question is asking what this new equation represents, with options A to D provided.
Fi... | Given points \( P_1(x_1, y_1) \) on the line \( l \) and \( P_2(x_2, y_2) \) outside the line \( l \), where the equation of the line \( l \) is \( f(x, y) = 0 \). We need to determine what the equation \( f(x, y) - f(x_1, y_1) - f(x_2, y_2) = 0 \) represents.
1. Since \( P_1 \) is on the line \( l \), substituting \(... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. The problem is about two points, P₁ and P₂, where P₁ is on the line l and P₂ is outside of it. The equation of line l is given by f(x, y) = 0. Then we have this new equation: f(x, y) - f(x₁, y₁) - f(x₂, y₂) = 0. The question is asking what this new equation represents, with options A to D provi... | [
{
"from": "human",
"value": "3. Given $P_{1}\\left(x_{1}, y_{1}\\right), P_{2}\\left(x_{2}, y_{2}\\right)$ are points on and outside the line $l$ respectively, if the equation of the line $l$ is $f(x, y)=$ 0, then the equation $f(x, y)-f\\left(x_{1}, y_{1}\\right)-f\\left(x_{2}, y_{2}\\right)=0$ represents\... |
Let $S(n)$ denote the sum of the digits of the number $n$ written in the decimal system, and let $U_{k}=11 \ldots 1$ be the number written with $k$ ones. Find those values of $k$ for which $S\left(U_{k}^{2}\right)=\left(S\left(U_{k}\right)\right)^{2}$. | null | Alright, so I need to figure out for which values of k the sum of the digits of U_k squared equals the square of the sum of the digits of U_k. Here, U_k is the number made up of k ones, like 1, 11, 111, and so on. Let me break this down step by step.
First, let me understand what S(n) means. It's the sum of the digits... | To solve the problem, we need to find values of \( k \) such that \( S(U_k^2) = (S(U_k))^2 \), where \( U_k \) is the number consisting of \( k \) ones.
1. **Understanding \( S(U_k) \)**:
- \( U_k \) is a number with \( k \) ones, so the sum of its digits \( S(U_k) = k \).
- Therefore, \( (S(U_k))^2 = k^2 \).
... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I need to figure out for which values of k the sum of the digits of U_k squared equals the square of the sum of the digits of U_k. Here, U_k is the number made up of k ones, like 1, 11, 111, and so on. Let me break this down step by step.
First, let me understand what S(n) means. It's the sum of th... | [
{
"from": "human",
"value": "Let $S(n)$ denote the sum of the digits of the number $n$ written in the decimal system, and let $U_{k}=11 \\ldots 1$ be the number written with $k$ ones. Find those values of $k$ for which $S\\left(U_{k}^{2}\\right)=\\left(S\\left(U_{k}\\right)\\right)^{2}$."
},
{
"from... |
Let point $M$ be the center of the circle defined by the equation $x^2 + y^2 = 41$, let $P$ be $P(4,2)$, and let $Q$ be a point such that $M$, $P$, and $Q$ are collinear. Find the sum of the squares of the coordinates of point $Q$ given that $M$ is the midpoint of $\overline{PQ}$. | Let point $M$ be the center of the circle defined by the equation $x^2 + y^2 = 41$, let $P$ be $P(4,2)$, and let $Q$ be a point such that $M$, $P$, and $Q$ are collinear. Find the sum of the squares of the coordinates of point $Q$ given that $M$ is the midpoint of $\overline{PQ}$. | Okay, let's see. The problem says that point M is the center of the circle defined by the equation x² + y² = 41. Hmm, so the center of that circle should be at the origin, right? Because the standard equation of a circle is (x - h)² + (y - k)² = r², so if it's x² + y² = 41, then h and k are both 0. So M is (0, 0). Got ... | Point \( M \) is the center of the circle defined by the equation \( x^2 + y^2 = 41 \), which is the origin \((0, 0)\). Point \( P \) is given as \((4, 2)\). We need to find the coordinates of point \( Q \) such that \( M \) is the midpoint of \(\overline{PQ}\) and then compute the sum of the squares of the coordinates... | 4,883 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem says that point M is the center of the circle defined by the equation x² + y² = 41. Hmm, so the center of that circle should be at the origin, right? Because the standard equation of a circle is (x - h)² + (y - k)² = r², so if it's x² + y² = 41, then h and k are both 0. So M is (0, ... | [
{
"from": "human",
"value": "Let point $M$ be the center of the circle defined by the equation $x^2 + y^2 = 41$, let $P$ be $P(4,2)$, and let $Q$ be a point such that $M$, $P$, and $Q$ are collinear. Find the sum of the squares of the coordinates of point $Q$ given that $M$ is the midpoint of $\\overline{PQ... |
What is the smallest base-8 number that is a multiple of 3 and has exactly three digits? Express your answer in base 10. | What is the smallest base-8 number that is a multiple of 3 and has exactly three digits? Express your answer in base 10. | Okay, so I need to find the smallest base-8 (octal) number that is a multiple of 3 and has exactly three digits. Then I have to express the answer in base 10. Hmm, let me break this down step by step.
First, understanding the problem. A three-digit number in base-8. That means it ranges from 100₈ to 777₈. But I need t... | To find the smallest base-8 number that is a multiple of 3 and has exactly three digits, we need to convert the base-8 numbers to base-10 and check for divisibility by 3.
A three-digit number in base-8 ranges from \(100_8\) to \(777_8\). The smallest three-digit number in base-8 is \(100_8\), which converts to \(1 \t... | 11,189 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest base-8 (octal) number that is a multiple of 3 and has exactly three digits. Then I have to express the answer in base 10. Hmm, let me break this down step by step.
First, understanding the problem. A three-digit number in base-8. That means it ranges from 100₈ to 777₈. But ... | [
{
"from": "human",
"value": "What is the smallest base-8 number that is a multiple of 3 and has exactly three digits? Express your answer in base 10."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest base-8 (octal) number that is a multiple of 3 and has exactly three d... |
Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that
$f(x) + f(-x) = f(x^2)$. | Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that
$f(x) + f(-x) = f(x^2)$. | Okay, so I need to find the number of cubic functions f(x) = ax³ + bx² + cx + d that satisfy the equation f(x) + f(-x) = f(x²). Hmm, let me start by understanding what this condition means.
First, let me compute f(-x). If f(x) is ax³ + bx² + cx + d, then replacing x with -x gives f(-x) = a(-x)³ + b(-x)² + c(-x) + d. ... | To find the number of functions of the form \( f(x) = ax^3 + bx^2 + cx + d \) such that \( f(x) + f(-x) = f(x^2) \), we start by computing \( f(-x) \) and adding it to \( f(x) \):
\[
f(-x) = -ax^3 + bx^2 - cx + d
\]
Adding \( f(x) \) and \( f(-x) \):
\[
f(x) + f(-x) = (ax^3 + bx^2 + cx + d) + (-ax^3 + bx^2 - cx + d)... | 4,291 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the number of cubic functions f(x) = ax³ + bx² + cx + d that satisfy the equation f(x) + f(-x) = f(x²). Hmm, let me start by understanding what this condition means.
First, let me compute f(-x). If f(x) is ax³ + bx² + cx + d, then replacing x with -x gives f(-x) = a(-x)³ + b(-x)² + c(-... | [
{
"from": "human",
"value": "Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that\n$f(x) + f(-x) = f(x^2)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the number of cubic functions f(x) = ax³ + bx² + cx + d that satisfy the equation f(x) + f(-x) =... |
540. The knowledge of ten students was tested in two tests: $A$ and $B$. The scores on a hundred-point scale were as follows (the first row shows the scores for test $A$, and the second row shows the scores for test $B$):
| 95 | 90 | 86 | 84 | 75 | 70 | 62 | 60 | 57 | 50 | (*) |
| :--- | :--- | :--- | :--- | :--- | :-... | null | Okay, so I need to find the Spearman's rank correlation coefficient between the scores of test A and test B for these ten students. Hmm, let me remember how Spearman's rank correlation works. From what I recall, it's a non-parametric measure that assesses how well the relationship between two variables can be described... | To find the sample Spearman's rank correlation coefficient between the scores of test A and test B for the ten students, we follow these steps:
1. **Assign ranks to each student's scores in both tests:**
- **Test A Scores:** 95, 90, 86, 84, 75, 70, 62, 60, 57, 50
- Ranks: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
- **T... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the Spearman's rank correlation coefficient between the scores of test A and test B for these ten students. Hmm, let me remember how Spearman's rank correlation works. From what I recall, it's a non-parametric measure that assesses how well the relationship between two variables can be d... | [
{
"from": "human",
"value": "540. The knowledge of ten students was tested in two tests: $A$ and $B$. The scores on a hundred-point scale were as follows (the first row shows the scores for test $A$, and the second row shows the scores for test $B$):\n\n| 95 | 90 | 86 | 84 | 75 | 70 | 62 | 60 | 57 | 50 | (*... |
A gambler is playing a game where they can either win $1, lose $1, or draw with equal probability. They play the game 5 times. What is the probability that the gambler ends up with a net gain or a net loss, but not a net draw? | A gambler is playing a game where they can either win $1, lose $1, or draw with equal probability. They play the game 5 times. What is the probability that the gambler ends up with a net gain or a net loss, but not a net draw? | Okay, so there's this gambler who plays a game five times. Each time they play, they can either win $1, lose $1, or draw, and each of these outcomes has the same probability. The question is asking for the probability that after five games, the gambler has either a net gain or a net loss, but not a net draw. So, basica... | The gambler plays a game 5 times, with each game resulting in a win, loss, or draw, each with equal probability. We need to find the probability that the gambler ends up with a net gain or a net loss, but not a net draw.
First, we calculate the total number of possible outcomes for 5 games, which is \(3^5 = 243\).
Ne... | 8,358 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this gambler who plays a game five times. Each time they play, they can either win $1, lose $1, or draw, and each of these outcomes has the same probability. The question is asking for the probability that after five games, the gambler has either a net gain or a net loss, but not a net draw. So... | [
{
"from": "human",
"value": "A gambler is playing a game where they can either win $1, lose $1, or draw with equal probability. They play the game 5 times. What is the probability that the gambler ends up with a net gain or a net loss, but not a net draw?"
},
{
"from": "gpt",
"value": "<think>\n... |
VIII OM - I - Problem 9
Prove that if $ x $, $ y $, $ z $, and $ \sqrt{x} + \sqrt{y} + \sqrt{z} $ are rational numbers, then $ \sqrt{x} $, $ \sqrt{y} $, and $ \sqrt{z} $ are also rational numbers. | null | Alright, so I have this problem here: Prove that if \( x \), \( y \), \( z \), and \( \sqrt{x} + \sqrt{y} + \sqrt{z} \) are rational numbers, then \( \sqrt{x} \), \( \sqrt{y} \), and \( \sqrt{z} \) are also rational numbers. Hmm, okay. Let me try to wrap my head around this.
First, let's parse the problem. We have thr... | Given that \( x \), \( y \), \( z \), and \( \sqrt{x} + \sqrt{y} + \sqrt{z} \) are rational numbers, we need to prove that \( \sqrt{x} \), \( \sqrt{y} \), and \( \sqrt{z} \) are also rational.
1. **Denote the square roots**: Let \( a = \sqrt{x} \), \( b = \sqrt{y} \), and \( c = \sqrt{z} \). We know \( a + b + c \) is... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I have this problem here: Prove that if \( x \), \( y \), \( z \), and \( \sqrt{x} + \sqrt{y} + \sqrt{z} \) are rational numbers, then \( \sqrt{x} \), \( \sqrt{y} \), and \( \sqrt{z} \) are also rational numbers. Hmm, okay. Let me try to wrap my head around this.
First, let's parse the problem. We ... | [
{
"from": "human",
"value": "VIII OM - I - Problem 9\n\nProve that if $ x $, $ y $, $ z $, and $ \\sqrt{x} + \\sqrt{y} + \\sqrt{z} $ are rational numbers, then $ \\sqrt{x} $, $ \\sqrt{y} $, and $ \\sqrt{z} $ are also rational numbers."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have th... |
Given the ellipse $C: \frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1(a>b>0)$, the equation of the "related circle" is defined as $x^{2}+y^{2}= \frac{a^{2}b^{2}}{a^{2}+b^{2}}$. If the focus of the parabola $x^2=4y$ coincides with one focus of the ellipse $C$, and one endpoint of the minor axis of the ellipse $C$ and its two... | null | Okay, let's try to tackle this problem step by step. It's about an ellipse and its related circle, with some conditions involving a parabola and a right triangle. Then there's a second part about a line intersecting the ellipse and proving something about the distance from the origin. Hmm, sounds a bit involved, but le... | Given the ellipse \( C: \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \) with \( a > b > 0 \), and the related circle defined as \( x^2 + y^2 = \frac{a^2 b^2}{a^2 + b^2} \):
1. **Equation of the Ellipse and Related Circle**:
- The focus of the parabola \( x^2 = 4y \) is at \( (0, 1) \), which coincides with one focus of th... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to tackle this problem step by step. It's about an ellipse and its related circle, with some conditions involving a parabola and a right triangle. Then there's a second part about a line intersecting the ellipse and proving something about the distance from the origin. Hmm, sounds a bit involved... | [
{
"from": "human",
"value": "Given the ellipse $C: \\frac{y^{2}}{a^{2}}+ \\frac{x^{2}}{b^{2}}=1(a>b>0)$, the equation of the \"related circle\" is defined as $x^{2}+y^{2}= \\frac{a^{2}b^{2}}{a^{2}+b^{2}}$. If the focus of the parabola $x^2=4y$ coincides with one focus of the ellipse $C$, and one endpoint of... |
$PQRST$ is inscribed in a circle with $PQ = QR = RS = ST = 3$ and $PT = 2.$ Compute $(1 - \cos \angle Q)(1 - \cos \angle PST).$ | $PQRST$ is inscribed in a circle with $PQ = QR = RS = ST = 3$ and $PT = 2.$ Compute $(1 - \cos \angle Q)(1 - \cos \angle PST).$ | Okay, so I need to solve this geometry problem where a pentagon PQRST is inscribed in a circle. The sides PQ, QR, RS, and ST are all 3 units long, and PT is 2 units. I have to compute the product (1 - cos ∠Q)(1 - cos ∠PST). Hmm, let's break this down step by step.
First, since the pentagon is inscribed in a circle, it... | Given a cyclic pentagon \(PQRST\) with \(PQ = QR = RS = ST = 3\) and \(PT = 2\), we need to compute \((1 - \cos \angle Q)(1 - \cos \angle PST)\).
1. **Identify Central Angles**:
- The sides \(PQ, QR, RS, ST\) each subtend a central angle \(\theta\).
- The side \(PT\) subtends a central angle \(\varepsilon = 360^... | 6,908 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this geometry problem where a pentagon PQRST is inscribed in a circle. The sides PQ, QR, RS, and ST are all 3 units long, and PT is 2 units. I have to compute the product (1 - cos ∠Q)(1 - cos ∠PST). Hmm, let's break this down step by step.
First, since the pentagon is inscribed in a ci... | [
{
"from": "human",
"value": "$PQRST$ is inscribed in a circle with $PQ = QR = RS = ST = 3$ and $PT = 2.$ Compute $(1 - \\cos \\angle Q)(1 - \\cos \\angle PST).$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this geometry problem where a pentagon PQRST is inscribed in a circle... |
What is the value of $\dfrac{3 \times 4}{6}?$ | What is the value of $\dfrac{3 \times 4}{6}?$ | Okay, so I need to figure out the value of this fraction: (3 times 4) divided by 6. Let me write that down step by step. First, the numerator is 3 multiplied by 4, and then the denominator is 6. So, the expression is (3 × 4)/6. Hmm, right. Let me compute the numerator first. 3 times 4, that's straightforward. 3 times 4... | To find the value of \(\dfrac{3 \times 4}{6}\), we can approach it in two ways:
1. **Multiplying first and then dividing:**
- Calculate the numerator: \(3 \times 4 = 12\).
- Divide by the denominator: \(\dfrac{12}{6} = 2\).
2. **Simplifying before multiplying:**
- Simplify the fraction by canceling common fa... | 5,822 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the value of this fraction: (3 times 4) divided by 6. Let me write that down step by step. First, the numerator is 3 multiplied by 4, and then the denominator is 6. So, the expression is (3 × 4)/6. Hmm, right. Let me compute the numerator first. 3 times 4, that's straightforward. 3... | [
{
"from": "human",
"value": "What is the value of $\\dfrac{3 \\times 4}{6}?$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the value of this fraction: (3 times 4) divided by 6. Let me write that down step by step. First, the numerator is 3 multiplied by 4, and then the de... |
Stu has 9 books and Albert has a certain number of times more books than Stu. Together, they have 45 books in total. How many times more books does Albert have than Stu? | null | Okay, let's see. Stu has 9 books. Albert has a certain number of times more books than Stu. Together, they have 45 books in total. I need to find how many times more books Albert has than Stu. Hmm.
Alright, so let me start by understanding the problem. Stu's books are given as 9. Albert has "a certain number of times ... | Stu has 9 books. Albert has a certain number of times more books than Stu. Together, they have 45 books in total. We need to determine how many times more books Albert has than Stu.
1. Let \( x \) be the number of times more books Albert has than Stu.
2. If Albert has \( x \) times as many books as Stu, then Albert ha... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. Stu has 9 books. Albert has a certain number of times more books than Stu. Together, they have 45 books in total. I need to find how many times more books Albert has than Stu. Hmm.
Alright, so let me start by understanding the problem. Stu's books are given as 9. Albert has "a certain number o... | [
{
"from": "human",
"value": "Stu has 9 books and Albert has a certain number of times more books than Stu. Together, they have 45 books in total. How many times more books does Albert have than Stu?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. Stu has 9 books. Albert has a certain nu... |
Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AC$ is 6 cm. What is the area of triangle $ABC$, in square centimeters? | Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AC$ is 6 cm. What is the area of triangle $ABC$, in square centimeters? | Okay, so I need to find the area of triangle ABC, which is an isosceles right triangle with angle A being 90 degrees. The length of segment AC is 6 cm. Hmm, let me think about how to approach this.
First, let me recall what an isosceles right triangle is. An isosceles right triangle has two legs of equal length and a ... | Triangle \(ABC\) is an isosceles right triangle with \(\angle A = 90^\circ\). The length of segment \(AC\) is 6 cm.
In an isosceles right triangle with a right angle at \(A\), the legs \(AB\) and \(AC\) are equal. Since \(AC = 6\) cm, it follows that \(AB = 6\) cm as well.
The area of a right triangle is given by \... | 2,408 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the area of triangle ABC, which is an isosceles right triangle with angle A being 90 degrees. The length of segment AC is 6 cm. Hmm, let me think about how to approach this.
First, let me recall what an isosceles right triangle is. An isosceles right triangle has two legs of equal lengt... | [
{
"from": "human",
"value": "Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AC$ is 6 cm. What is the area of triangle $ABC$, in square centimeters?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the area of... |
Given a sequence $\{a_n\}$ satisfies $a_1=2$, $a_n=1- \frac{1}{a_{n-1}}$ for $n=2, 3, 4, \ldots$, find the value of $a_{12}$. | null | Okay, so I need to find the value of a_{12} in this sequence where a_1 is 2, and each term after that is defined by a_n = 1 - 1/a_{n-1}. Hmm, let's see. Maybe I can compute the first few terms to see if there's a pattern or if it repeats periodically. That might save me from having to calculate all the way up to the 12... | Given the sequence \(\{a_n\}\) where \(a_1 = 2\) and \(a_n = 1 - \frac{1}{a_{n-1}}\) for \(n \geq 2\), we need to find the value of \(a_{12}\).
First, we compute the initial terms of the sequence to identify any patterns:
\[
\begin{aligned}
a_1 &= 2, \\
a_2 &= 1 - \frac{1}{a_1} = 1 - \frac{1}{2} = \frac{1}{2}, \\
a_3... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the value of a_{12} in this sequence where a_1 is 2, and each term after that is defined by a_n = 1 - 1/a_{n-1}. Hmm, let's see. Maybe I can compute the first few terms to see if there's a pattern or if it repeats periodically. That might save me from having to calculate all the way up t... | [
{
"from": "human",
"value": "Given a sequence $\\{a_n\\}$ satisfies $a_1=2$, $a_n=1- \\frac{1}{a_{n-1}}$ for $n=2, 3, 4, \\ldots$, find the value of $a_{12}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of a_{12} in this sequence where a_1 is 2, and each term after ... |
A student got twice as many sums wrong as he got right. If he attempted 36 sums in all, how many did he solve correctly? | null | Okay, so the problem says: A student got twice as many sums wrong as he got right. If he attempted 36 sums in all, how many did he solve correctly?
Hmm, let me think. Let's break this down. The student attempted a total of 36 sums. Out of these, some are correct and some are wrong. The number of wrong answers is twice... | Let \( x \) be the number of sums the student solved correctly. According to the problem, the number of sums he got wrong is twice the number he got right, which is \( 2x \). The total number of sums he attempted is 36. Therefore, we can write the equation:
\[
x + 2x = 36
\]
Combining like terms, we get:
\[
3x = 36
... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so the problem says: A student got twice as many sums wrong as he got right. If he attempted 36 sums in all, how many did he solve correctly?
Hmm, let me think. Let's break this down. The student attempted a total of 36 sums. Out of these, some are correct and some are wrong. The number of wrong answers ... | [
{
"from": "human",
"value": "A student got twice as many sums wrong as he got right. If he attempted 36 sums in all, how many did he solve correctly?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem says: A student got twice as many sums wrong as he got right. If he attempted 36 sums... |
If $y\%$ of five-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is $y$? Express your answer as a decimal to the nearest tenth. | If $y\%$ of five-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is $y$? Express your answer as a decimal to the nearest tenth. | Okay, so the problem is asking: If y% of five-digit numbers have a repeated digit, what is y? We need to express the answer as a decimal to the nearest tenth. Hmm, let's break this down.
First, let's clarify what exactly is being asked. We need to find the percentage of five-digit numbers that have at least one repeat... | To find the percentage \( y \) of five-digit numbers that have a repeated digit, we start by calculating the total number of five-digit numbers, which is from 10000 to 99999 inclusive. This gives:
\[
99999 - 10000 + 1 = 90000
\]
Next, we calculate the number of five-digit numbers with all unique digits. The first dig... | 15,228 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so the problem is asking: If y% of five-digit numbers have a repeated digit, what is y? We need to express the answer as a decimal to the nearest tenth. Hmm, let's break this down.
First, let's clarify what exactly is being asked. We need to find the percentage of five-digit numbers that have at least on... | [
{
"from": "human",
"value": "If $y\\%$ of five-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is $y$? Express your answer as a decimal to the nearest tenth."
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem is asking: If y% of five-digi... |
The Screamers are coached by Coach Yellsalot. The Screamers have 12 players, but two of them, Bob and Yogi, refuse to play together. How many starting lineups (of 5 players) can Coach Yellsalot make, if the starting lineup can't contain both Bob and Yogi? (The order of the 5 players in the lineup does not matter; th... | The Screamers are coached by Coach Yellsalot. The Screamers have 12 players, but two of them, Bob and Yogi, refuse to play together. How many starting lineups (of 5 players) can Coach Yellsalot make, if the starting lineup can't contain both Bob and Yogi? (The order of the 5 players in the lineup does not matter; th... | Okay, so the problem is about the Screamers basketball team. They have 12 players, but two of them, Bob and Yogi, don't want to play together. The coach needs to choose a starting lineup of 5 players, but he can't have both Bob and Yogi in the lineup. We need to find out how many possible starting lineups there are und... | The Screamers have 12 players, but Bob and Yogi refuse to play together. We need to find the number of starting lineups of 5 players that do not include both Bob and Yogi.
First, calculate the total number of unrestricted starting lineups using combinations:
\[
\binom{12}{5} = \frac{12!}{5!(12-5)!} = \frac{12 \times 1... | 35,186 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so the problem is about the Screamers basketball team. They have 12 players, but two of them, Bob and Yogi, don't want to play together. The coach needs to choose a starting lineup of 5 players, but he can't have both Bob and Yogi in the lineup. We need to find out how many possible starting lineups there... | [
{
"from": "human",
"value": "The Screamers are coached by Coach Yellsalot. The Screamers have 12 players, but two of them, Bob and Yogi, refuse to play together. How many starting lineups (of 5 players) can Coach Yellsalot make, if the starting lineup can't contain both Bob and Yogi? (The order of the 5 ... |
A complex number $z$ is such that the sum of its real part and the imaginary part is $-3$, and the product of the real part and the imaginary part is $-4$. Find $|z|^2$, where $|z|$ denotes the modulus of $z$. | A complex number $z$ is such that the sum of its real part and the imaginary part is $-3$, and the product of the real part and the imaginary part is $-4$. Find $|z|^2$, where $|z|$ denotes the modulus of $z$. | Okay, let me try to solve this problem. So, we have a complex number z. Let me recall that a complex number can be written as z = a + bi, where a is the real part and b is the imaginary part. The problem says that the sum of its real part and the imaginary part is -3. That means a + b = -3. Also, the product of the rea... | Given a complex number \( z = a + bi \) where the real part \( a \) and the imaginary part \( b \) satisfy the conditions \( a + b = -3 \) and \( ab = -4 \), we need to find \( |z|^2 \).
The modulus squared of a complex number \( z \) is given by \( |z|^2 = a^2 + b^2 \). To find \( a^2 + b^2 \), we use the identity:
\... | 8,145 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to solve this problem. So, we have a complex number z. Let me recall that a complex number can be written as z = a + bi, where a is the real part and b is the imaginary part. The problem says that the sum of its real part and the imaginary part is -3. That means a + b = -3. Also, the product of... | [
{
"from": "human",
"value": "A complex number $z$ is such that the sum of its real part and the imaginary part is $-3$, and the product of the real part and the imaginary part is $-4$. Find $|z|^2$, where $|z|$ denotes the modulus of $z$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try ... |
In the figure below, if the area of $\triangle ABC$ is 40, what is the value of $p$? [asy]
size(5cm);defaultpen(fontsize(9));
pair o = (0, 0); pair q = (0, 15); pair b = (15, 0);
pair a = (3, 15); pair t = (3, 0); pair c = (0, 9);
draw((-2, 0)--(18, 0), Arrow);
draw((0, -2)--(0, 18), Arrow);
draw(q--a--b);
draw(a--c--... | null | Okay, so I need to find the value of \( p \) such that the area of triangle \( ABC \) is 40. The figure is drawn on a coordinate plane with points \( A(3, 15) \), \( B(15, 0) \), and \( C(0, p) \). Let me recall how to calculate the area of a triangle given three coordinates.
One method is to use the formula for the a... | To find the value of \( p \) such that the area of \(\triangle ABC\) is 40, we use the coordinates of points \( A(3, 15) \), \( B(15, 0) \), and \( C(0, p) \). The area of the triangle can be calculated using the formula for the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[
\t... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the value of \( p \) such that the area of triangle \( ABC \) is 40. The figure is drawn on a coordinate plane with points \( A(3, 15) \), \( B(15, 0) \), and \( C(0, p) \). Let me recall how to calculate the area of a triangle given three coordinates.
One method is to use the formula f... | [
{
"from": "human",
"value": "In the figure below, if the area of $\\triangle ABC$ is 40, what is the value of $p$? [asy]\nsize(5cm);defaultpen(fontsize(9));\npair o = (0, 0); pair q = (0, 15); pair b = (15, 0);\npair a = (3, 15); pair t = (3, 0); pair c = (0, 9);\n\ndraw((-2, 0)--(18, 0), Arrow);\ndraw((0, ... |
A rectangular prism container has a base with dimensions 4 inches by 6 inches and is 8 inches tall. The container is filled to 3/4 of its height with a mixture of water and juice in a ratio of 3:5. If the total volume of the mixture is 48 cubic inches, what is the volume of the juice in the container? Express your answ... | A rectangular prism container has a base with dimensions 4 inches by 6 inches and is 8 inches tall. The container is filled to 3/4 of its height with a mixture of water and juice in a ratio of 3:5. If the total volume of the mixture is 48 cubic inches, what is the volume of the juice in the container? Express your answ... | Okay, let me try to figure this problem out. So, we have a rectangular prism container. The base is 4 inches by 6 inches, and it's 8 inches tall. The container is filled to 3/4 of its height with a mixture of water and juice in a ratio of 3:5. The total volume of the mixture is 48 cubic inches. We need to find the volu... | The problem involves a rectangular prism container with a base of 4 inches by 6 inches and a height of 8 inches. The container is filled to 3/4 of its height with a mixture of water and juice in a ratio of 3:5. The total volume of the mixture is given as 48 cubic inches, and we need to find the volume of the juice.
1.... | 16,420 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure this problem out. So, we have a rectangular prism container. The base is 4 inches by 6 inches, and it's 8 inches tall. The container is filled to 3/4 of its height with a mixture of water and juice in a ratio of 3:5. The total volume of the mixture is 48 cubic inches. We need to find ... | [
{
"from": "human",
"value": "A rectangular prism container has a base with dimensions 4 inches by 6 inches and is 8 inches tall. The container is filled to 3/4 of its height with a mixture of water and juice in a ratio of 3:5. If the total volume of the mixture is 48 cubic inches, what is the volume of the ... |
Given that $2+i$ is a root of the equation $x^3 + ax^2 + bx + 5 = 0$ and that $a$ and $b$ are rational numbers, compute $a$. | Given that $2+i$ is a root of the equation $x^3 + ax^2 + bx + 5 = 0$ and that $a$ and $b$ are rational numbers, compute $a$. | Okay, so I need to find the value of 'a' in the equation x³ + a x² + b x + 5 = 0, given that 2 + i is a root and that a and b are rational numbers. Hmm, let's start by recalling some algebra concepts. Since the coefficients a and b are rational, and 2 + i is a root, then its complex conjugate should also be a root, rig... | Given that \(2 + i\) is a root of the equation \(x^3 + ax^2 + bx + 5 = 0\) and that \(a\) and \(b\) are rational numbers, we need to compute \(a\).
Since the coefficients \(a\) and \(b\) are rational, the complex conjugate \(2 - i\) must also be a root. Let the third root be \(r\), which must be rational. Using Vieta'... | 17,648 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of 'a' in the equation x³ + a x² + b x + 5 = 0, given that 2 + i is a root and that a and b are rational numbers. Hmm, let's start by recalling some algebra concepts. Since the coefficients a and b are rational, and 2 + i is a root, then its complex conjugate should also be a r... | [
{
"from": "human",
"value": "Given that $2+i$ is a root of the equation $x^3 + ax^2 + bx + 5 = 0$ and that $a$ and $b$ are rational numbers, compute $a$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of 'a' in the equation x³ + a x² + b x + 5 = 0, given that 2 + i is ... |
Calculate: $\cos (-\frac{11\pi}{6})=\_\_\_\_\_\_$. | null | Okay, so I need to find the value of cos(-11π/6). Hmm, let me think. First, I remember that cosine is an even function. What does that mean again? Oh, right! Even functions satisfy the condition f(-x) = f(x). So, that would mean cos(-θ) = cos(θ). Therefore, cos(-11π/6) should be the same as cos(11π/6). Cool, so I can j... | To find the value of $\cos\left(-\frac{11\pi}{6}\right)$, we start by noting that cosine is an even function, which means $\cos(-\theta) = \cos(\theta)$. Therefore, $\cos\left(-\frac{11\pi}{6}\right) = \cos\left(\frac{11\pi}{6}\right)$.
Next, we convert $\frac{11\pi}{6}$ radians to degrees to better understand its pos... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the value of cos(-11π/6). Hmm, let me think. First, I remember that cosine is an even function. What does that mean again? Oh, right! Even functions satisfy the condition f(-x) = f(x). So, that would mean cos(-θ) = cos(θ). Therefore, cos(-11π/6) should be the same as cos(11π/6). Cool, so... | [
{
"from": "human",
"value": "Calculate: $\\cos (-\\frac{11\\pi}{6})=\\_\\_\\_\\_\\_\\_$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of cos(-11π/6). Hmm, let me think. First, I remember that cosine is an even function. What does that mean again? Oh, right! Even func... |
Define an operation $a \otimes b = \begin{cases} a & \text{if } a \leqslant b \\ b & \text{if } a > b \end{cases}$. Let $f(x) = (3x^2 + 6) \otimes (23 - x^2)$, then the maximum value of the function $f(x)$ is \_\_\_\_\_\_. | null | Okay, so I need to find the maximum value of the function f(x) which is defined as the operation between 3x² + 6 and 23 - x², where the operation ⊗ is defined such that if a ≤ b, then a ⊗ b = a, otherwise it's b. Hmm, let me parse that.
First, let me make sure I understand the operation. If I have two numbers, a and b... | To find the maximum value of the function \( f(x) = (3x^2 + 6) \otimes (23 - x^2) \), we need to determine where the two expressions \( 3x^2 + 6 \) and \( 23 - x^2 \) intersect, as the operation \( \otimes \) returns the minimum of the two values. The maximum value of \( f(x) \) will occur at the intersection point of ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the maximum value of the function f(x) which is defined as the operation between 3x² + 6 and 23 - x², where the operation ⊗ is defined such that if a ≤ b, then a ⊗ b = a, otherwise it's b. Hmm, let me parse that.
First, let me make sure I understand the operation. If I have two numbers,... | [
{
"from": "human",
"value": "Define an operation $a \\otimes b = \\begin{cases} a & \\text{if } a \\leqslant b \\\\ b & \\text{if } a > b \\end{cases}$. Let $f(x) = (3x^2 + 6) \\otimes (23 - x^2)$, then the maximum value of the function $f(x)$ is \\_\\_\\_\\_\\_\\_."
},
{
"from": "gpt",
"value":... |
To promote national spirit and inherit traditional culture, a certain high school held a zongzi-making competition with the theme "Celebrating the Dragon Boat Festival, Fragrant Zongzi Leaves." It is known that the probabilities of student A, B, and C successfully making a zongzi in the competition are $\frac{1}{2}, \f... | null | Okay, so we have this problem about three students, A, B, and C, making zongzi. The question is asking for the probability that at least one of them succeeds. Their individual probabilities of success are given: A has a 1/2 chance, B has 3/4, and C has 2/5. The events are independent, meaning whether one succeeds doesn... | To find the probability that at least one of the students A, B, or C succeeds in making a zongzi, we use the complement rule. First, we calculate the probability that all three fail, and then subtract this from 1.
The probabilities of failure for each student are:
- Student A: \(1 - \frac{1}{2} = \frac{1}{2}\)
- Stude... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so we have this problem about three students, A, B, and C, making zongzi. The question is asking for the probability that at least one of them succeeds. Their individual probabilities of success are given: A has a 1/2 chance, B has 3/4, and C has 2/5. The events are independent, meaning whether one succee... | [
{
"from": "human",
"value": "To promote national spirit and inherit traditional culture, a certain high school held a zongzi-making competition with the theme \"Celebrating the Dragon Boat Festival, Fragrant Zongzi Leaves.\" It is known that the probabilities of student A, B, and C successfully making a zon... |
Find all values of \( k \) so that
\[ x^2 - (k + 4)x + k - 3 < 0 \]
for all \( x \). | null | Okay, so I need to find all values of \( k \) such that the quadratic inequality \( x^2 - (k + 4)x + k - 3 < 0 \) holds for all \( x \). Hmm, let's start by recalling what I know about quadratic inequalities.
First, a quadratic equation \( ax^2 + bx + c < 0 \) will be true for all \( x \) only if the quadratic never ... | To determine all values of \( k \) such that the quadratic inequality \( x^2 - (k + 4)x + k - 3 < 0 \) holds for all \( x \), we start by analyzing the quadratic expression.
The quadratic \( x^2 - (k + 4)x + k - 3 \) has a leading coefficient of 1, which is positive. This means the parabola opens upwards. For a quadr... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find all values of \( k \) such that the quadratic inequality \( x^2 - (k + 4)x + k - 3 < 0 \) holds for all \( x \). Hmm, let's start by recalling what I know about quadratic inequalities.
First, a quadratic equation \( ax^2 + bx + c < 0 \) will be true for all \( x \) only if the quadrati... | [
{
"from": "human",
"value": "Find all values of \\( k \\) so that \n\\[ x^2 - (k + 4)x + k - 3 < 0 \\] \nfor all \\( x \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find all values of \\( k \\) such that the quadratic inequality \\( x^2 - (k + 4)x + k - 3 < 0 \\) holds for all ... |
Carter can read half as many pages as Lucy in x hour. Lucy can read 20 more pages than Oliver in 1 hour. Oliver can read 40 pages. Carter can read 30 pages in 1 hour. What is the value of unknown variable x? What is the value of unknown variable x? | null | Alright, let's try to tackle this problem step by step. Hmm, okay, let's see. The problem is about how many pages Carter, Lucy, and Oliver can read in a certain amount of time, and we need to find the value of x, which is the time in hours. Let me parse each sentence and write down what I know.
First sentence: "Carter... | Given the problem, we need to determine the value of \( x \) based on the reading rates of Carter, Lucy, and Oliver.
1. **Lucy's Reading Rate**: Lucy can read 20 more pages than Oliver in 1 hour. Oliver can read 40 pages in 1 hour. Therefore, Lucy's reading rate is:
\[
40 + 20 = 60 \text{ pages per hour}
\]
... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, let's try to tackle this problem step by step. Hmm, okay, let's see. The problem is about how many pages Carter, Lucy, and Oliver can read in a certain amount of time, and we need to find the value of x, which is the time in hours. Let me parse each sentence and write down what I know.
First sentence:... | [
{
"from": "human",
"value": "Carter can read half as many pages as Lucy in x hour. Lucy can read 20 more pages than Oliver in 1 hour. Oliver can read 40 pages. Carter can read 30 pages in 1 hour. What is the value of unknown variable x? What is the value of unknown variable x?"
},
{
"from": "gpt",
... |
A hotel charges $\$20$ per day plus an additional $\$0.50$ per minute for any phone calls made from the room. If a guest stays for 3 days and makes a total of 120 minutes of phone calls, what is the total cost of their stay? | A hotel charges $\$20$ per day plus an additional $\$0.50$ per minute for any phone calls made from the room. If a guest stays for 3 days and makes a total of 120 minutes of phone calls, what is the total cost of their stay? | Okay, so I need to figure out the total cost for a guest's stay at this hotel. Let me read the problem again to make sure I get all the details.
The hotel charges $20 per day. Plus, there's an extra $0.50 per minute for any phone calls made from the room. The guest stayed for 3 days and made a total of 120 minutes of... | The hotel charges $20 per day for the room and an additional $0.50 per minute for phone calls. The guest stayed for 3 days and made 120 minutes of phone calls.
1. Calculate the cost for the room:
\[
3 \text{ days} \times \$20/\text{day} = \$60
\]
2. Calculate the cost for the phone calls:
\[
120 \text{... | 5,915 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the total cost for a guest's stay at this hotel. Let me read the problem again to make sure I get all the details.
The hotel charges $20 per day. Plus, there's an extra $0.50 per minute for any phone calls made from the room. The guest stayed for 3 days and made a total of 120 mi... | [
{
"from": "human",
"value": "A hotel charges $\\$20$ per day plus an additional $\\$0.50$ per minute for any phone calls made from the room. If a guest stays for 3 days and makes a total of 120 minutes of phone calls, what is the total cost of their stay?"
},
{
"from": "gpt",
"value": "<think>\n... |
Liz is preparing for a huge family Thanksgiving dinner and she decides to roast 4 turkeys. The turkeys weigh 16 pounds, 18 pounds, 20 pounds, and 22 pounds respectively. She can only roast one turkey at a time due to the size of her oven. The first two turkeys need to be roasted at a higher temperature, requiring an ad... | null | Alright, so Liz has to roast four turkeys for Thanksgiving dinner, and she needs to figure out the latest time she can start preheating the oven to have everything ready by 6:00 pm. Let me break this down step by step.
First, let's list out the turkeys and their weights: 16, 18, 20, and 22 pounds. Each turkey takes 15... | Liz needs to roast four turkeys weighing 16, 18, 20, and 22 pounds respectively. The first two turkeys (16 and 18 pounds) require an additional 20 minutes of preheating time. Each turkey is roasted for 15 minutes per pound.
1. **Calculate roasting times for each turkey:**
- 16 pounds: \(16 \times 15 = 240\) minute... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so Liz has to roast four turkeys for Thanksgiving dinner, and she needs to figure out the latest time she can start preheating the oven to have everything ready by 6:00 pm. Let me break this down step by step.
First, let's list out the turkeys and their weights: 16, 18, 20, and 22 pounds. Each turkey ... | [
{
"from": "human",
"value": "Liz is preparing for a huge family Thanksgiving dinner and she decides to roast 4 turkeys. The turkeys weigh 16 pounds, 18 pounds, 20 pounds, and 22 pounds respectively. She can only roast one turkey at a time due to the size of her oven. The first two turkeys need to be roasted... |
The number of four-digit even numbers formed by the digits \\(1,2,3,4,5\\) without repetition is \\((\ )\\)
A: \\(8\\)
B: \\(24\\)
C: \\(48\\)
D: \\(120\\) | null | Okay, let's see. The problem is asking for the number of four-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5 without repeating any digits. The options are A: 8, B: 24, C: 48, D: 120. Hmm, I need to figure out how to approach this. Let me start by recalling some basic combinatorics principles.
Fir... | To determine the number of four-digit even numbers formed by the digits \(1, 2, 3, 4, 5\) without repetition, we start by noting that the last digit (units place) must be even. The even digits available are 2 and 4, giving us 2 choices for the units place.
For each choice of the units digit, we need to count the numbe... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. The problem is asking for the number of four-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5 without repeating any digits. The options are A: 8, B: 24, C: 48, D: 120. Hmm, I need to figure out how to approach this. Let me start by recalling some basic combinatorics principl... | [
{
"from": "human",
"value": "The number of four-digit even numbers formed by the digits \\\\(1,2,3,4,5\\\\) without repetition is \\\\((\\ )\\\\)\nA: \\\\(8\\\\)\nB: \\\\(24\\\\)\nC: \\\\(48\\\\)\nD: \\\\(120\\\\)"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem is asking for... |
A bag contains 10 balls, each labeled with a unique integer from 1 to 10. Five of the balls have an even number and five have an odd number. You randomly select one ball from the bag and observe its number. Given that the number you see is greater than 5, what is the probability that the number is also prime?
Note: I'... | A bag contains 10 balls, each labeled with a unique integer from 1 to 10. Five of the balls have an even number and five have an odd number. You randomly select one ball from the bag and observe its number. Given that the number you see is greater than 5, what is the probability that the number is also prime?
Note: I'... | Okay, let's see. I need to find the probability that the number on the ball is prime, given that it's greater than 5. Hmm, so this is a conditional probability problem. Alright, let me recall the formula for conditional probability. It's P(A|B) = P(A ∩ B) / P(B), right? So, in this case, event A is "the number is prime... | Given a bag containing 10 balls labeled with unique integers from 1 to 10, we need to find the probability that a randomly selected ball, which shows a number greater than 5, is also a prime number.
First, we identify the numbers greater than 5: these are 6, 7, 8, 9, and 10. This gives us a sample space of 5 numbers.
... | 28,858 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the probability that the number on the ball is prime, given that it's greater than 5. Hmm, so this is a conditional probability problem. Alright, let me recall the formula for conditional probability. It's P(A|B) = P(A ∩ B) / P(B), right? So, in this case, event A is "the number ... | [
{
"from": "human",
"value": "A bag contains 10 balls, each labeled with a unique integer from 1 to 10. Five of the balls have an even number and five have an odd number. You randomly select one ball from the bag and observe its number. Given that the number you see is greater than 5, what is the probability... |
Simplify $5y + 7y + 8y$. | null | Okay, let's see. I need to simplify the expression 5y + 7y + 8y. Hmm, simplifying expressions usually means combining like terms. Since all these terms have the variable y, they are like terms and can be combined together.
First, I'll check each term: 5y, 7y, and 8y. They all have the same variable part, which is y, ... | To simplify the expression \(5y + 7y + 8y\), we need to combine the like terms. All terms have the variable \(y\), so we can combine their coefficients.
First, we add the coefficients:
\[
5 + 7 + 8
\]
Adding these step-by-step:
\[
5 + 7 = 12
\]
\[
12 + 8 = 20
\]
Thus, combining the coefficients gives us 20, and the ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to simplify the expression 5y + 7y + 8y. Hmm, simplifying expressions usually means combining like terms. Since all these terms have the variable y, they are like terms and can be combined together.
First, I'll check each term: 5y, 7y, and 8y. They all have the same variable part, whic... | [
{
"from": "human",
"value": "Simplify $5y + 7y + 8y$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to simplify the expression 5y + 7y + 8y. Hmm, simplifying expressions usually means combining like terms. Since all these terms have the variable y, they are like terms and can b... |
Compute $2 \begin{pmatrix} 3 \\ -2 \end{pmatrix} + 4 \begin{pmatrix} 1 \\ 5 \end{pmatrix}$. | Compute $2 \begin{pmatrix} 3 \\ -2 \end{pmatrix} + 4 \begin{pmatrix} 1 \\ 5 \end{pmatrix}$. | Okay, so I need to compute this vector addition problem. Let me see. It's 2 times the vector (3, -2) plus 4 times the vector (1, 5). Hmm, vector addition and scalar multiplication, right? Alright, I think I can handle this step by step. Let me remember how these operations work.
First, scalar multiplication. If I have... | To compute \(2 \begin{pmatrix} 3 \\ -2 \end{pmatrix} + 4 \begin{pmatrix} 1 \\ 5 \end{pmatrix}\), we start by performing the scalar multiplications:
1. Multiply 2 by the vector \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\):
\[
2 \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 2 \cdot 3 \\ 2 \cdot (-2) \end{pm... | 31,524 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to compute this vector addition problem. Let me see. It's 2 times the vector (3, -2) plus 4 times the vector (1, 5). Hmm, vector addition and scalar multiplication, right? Alright, I think I can handle this step by step. Let me remember how these operations work.
First, scalar multiplication. I... | [
{
"from": "human",
"value": "Compute $2 \\begin{pmatrix} 3 \\\\ -2 \\end{pmatrix} + 4 \\begin{pmatrix} 1 \\\\ 5 \\end{pmatrix}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to compute this vector addition problem. Let me see. It's 2 times the vector (3, -2) plus 4 times the vector (1... |
The Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri want it to be sunny one of those days for a World War III preenactment, but if it's sunny for more than one day they won't know what to do wi... | The Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri want it to be sunny one of those days for a World War III preenactment, but if it's sunny for more than one day they won't know what to do wi... | Okay, so Paul and Yuri want to have a World War III preenactment, and they need it to be sunny on one day out of the four-day Labor Day weekend. But if there's more than one sunny day, they won't know what to do with themselves. So, we need to find the probability that there's exactly one sunny day out of the four. The... | Paul and Yuri want exactly one sunny day out of the four-day Labor Day weekend. Each day has a 75% chance of rain and a 25% chance of sun. We use the binomial probability formula to find the probability of having exactly one sunny day.
The binomial probability formula is:
\[ P(k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{... | 9,746 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Paul and Yuri want to have a World War III preenactment, and they need it to be sunny on one day out of the four-day Labor Day weekend. But if there's more than one sunny day, they won't know what to do with themselves. So, we need to find the probability that there's exactly one sunny day out of the f... | [
{
"from": "human",
"value": "The Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri want it to be sunny one of those days for a World War III preenactment, but if it's sunny for more t... |
At Dario's Deli, seven different sandwich ingredients are available. Calculate:
1. How many combinations of three different ingredients can be made?
2. How many combinations of four different ingredients can be made? | null | Okay, so I need to figure out how many combinations of three and four different ingredients can be made from seven available ingredients at Dario's Deli. Hmm, let's start with the first question: combinations of three different ingredients.
I remember that when the order doesn't matter, like in sandwiches where the a... | To determine the number of combinations of three and four different ingredients from seven available ingredients at Dario's Deli, we use the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \).
1. For combinations of three different ingredients:
\[
C(7, 3) = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!} = \frac{7 \t... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out how many combinations of three and four different ingredients can be made from seven available ingredients at Dario's Deli. Hmm, let's start with the first question: combinations of three different ingredients.
I remember that when the order doesn't matter, like in sandwiches whe... | [
{
"from": "human",
"value": "At Dario's Deli, seven different sandwich ingredients are available. Calculate:\n1. How many combinations of three different ingredients can be made?\n2. How many combinations of four different ingredients can be made?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so... |
Given the function $f(x)= \begin{cases} & x^{2}-x+3, x\leqslant 1, \\ & x+\frac{2}{x}, x > 1. \\ \end{cases}$
(I) Solve the inequality $f(x)\geqslant 3$;
(II) Suppose $a\in R$, and the inequality $f(x)\geqslant |\frac{x}{2}+a|$ holds for all $x\in R$. Find the range of $a$. | null | Alright, so I need to solve this problem about the function f(x) which is defined piecewise. Let me first make sure I understand the function correctly.
The function f(x) is equal to x² - x + 3 when x is less than or equal to 1, and it's equal to x + 2/x when x is greater than 1. Got that. The problem has two parts: p... | Given the function \( f(x)= \begin{cases} x^2 - x + 3 & \text{if } x \leq 1, \\ x + \frac{2}{x} & \text{if } x > 1, \end{cases} \)
(I) To solve the inequality \( f(x) \geq 3 \):
1. For \( x \leq 1 \):
\[
x^2 - x + 3 \geq 3 \implies x^2 - x \geq 0 \implies x(x - 1) \geq 0.
\]
The solutions are \( x \leq 0 ... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I need to solve this problem about the function f(x) which is defined piecewise. Let me first make sure I understand the function correctly.
The function f(x) is equal to x² - x + 3 when x is less than or equal to 1, and it's equal to x + 2/x when x is greater than 1. Got that. The problem has two ... | [
{
"from": "human",
"value": "Given the function $f(x)= \\begin{cases} & x^{2}-x+3, x\\leqslant 1, \\\\ & x+\\frac{2}{x}, x > 1. \\\\ \\end{cases}$\n\n(I) Solve the inequality $f(x)\\geqslant 3$;\n\n(II) Suppose $a\\in R$, and the inequality $f(x)\\geqslant |\\frac{x}{2}+a|$ holds for all $x\\in R$. Find the... |
## Aufgabe 1
| $a$ | $b$ | $c$ | $d$ |
| :---: | :---: | :---: | :---: |
| | $2 a$ | $b+3$ | $53-c$ |
| 9 | | | |
| 6 | | | |
| 12 | | | |
| null | Okay, let's see. I need to solve this table puzzle. The table has columns labeled a, b, c, d, and four rows. The first row after the headers has some expressions: 2a, b+3, 53-c. Then there are three more rows with some numbers filled in: the first column has 9, 6, 12, and the rest are empty. Hmm. Wait, actually, lookin... | To solve the table, we apply the given formulas to each row independently using the value of \( a \) provided:
1. **First Row (\( a = 9 \))**
- \( b = 2a = 2 \times 9 = 18 \)
- \( c = b + 3 = 18 + 3 = 21 \)
- \( d = 53 - c = 53 - 21 = 32 \)
2. **Second Row (\( a = 6 \))**
- \( b = 2a = 2 \times 6 ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to solve this table puzzle. The table has columns labeled a, b, c, d, and four rows. The first row after the headers has some expressions: 2a, b+3, 53-c. Then there are three more rows with some numbers filled in: the first column has 9, 6, 12, and the rest are empty. Hmm. Wait, actually... | [
{
"from": "human",
"value": "## Aufgabe 1\n\n| $a$ | $b$ | $c$ | $d$ |\n| :---: | :---: | :---: | :---: |\n| | $2 a$ | $b+3$ | $53-c$ |\n| 9 | | | |\n| 6 | | | |\n| 12 | | | |\n\n"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to solve this table puzzle. The table has ... |
In a regular hexagon with side length 8, an isosceles triangle is drawn so that its base lies on one of the sides of the hexagon and its vertex lies on the opposite side. Find the minimum possible area of such a triangle. | In a regular hexagon with side length 8, an isosceles triangle is drawn so that its base lies on one of the sides of the hexagon and its vertex lies on the opposite side. Find the minimum possible area of such a triangle. | Okay, so I have this problem about a regular hexagon with side length 8. The question is asking for the minimum possible area of an isosceles triangle whose base lies on one side of the hexagon and whose vertex lies on the opposite side. Hmm, let me visualize this. A regular hexagon has all sides equal and all internal... | To solve the problem of finding the minimum possible area of an isosceles triangle in a regular hexagon with side length 8, where the base lies on one side and the vertex lies on the opposite side, we need to consider the geometric constraints and properties of the hexagon.
1. **Hexagon Properties**: A regular hexagon... | 32,760 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem about a regular hexagon with side length 8. The question is asking for the minimum possible area of an isosceles triangle whose base lies on one side of the hexagon and whose vertex lies on the opposite side. Hmm, let me visualize this. A regular hexagon has all sides equal and all ... | [
{
"from": "human",
"value": "In a regular hexagon with side length 8, an isosceles triangle is drawn so that its base lies on one of the sides of the hexagon and its vertex lies on the opposite side. Find the minimum possible area of such a triangle."
},
{
"from": "gpt",
"value": "<think>\nOkay,... |
## 230. Math Puzzle $7 / 84$
In a passenger airplane, there are 130 passengers. 25 of them do not speak either German or Russian. 70 passengers speak German and 80 travelers speak Russian.
How many of the passengers speak both German and Russian? | null | Okay, let's see. I need to figure out how many passengers speak both German and Russian on this airplane. The problem says there are 130 passengers total. Out of these, 25 don't speak either German or Russian. So that means the number of people who speak at least one of the languages, German or Russian, is 130 minus 25... | Given there are 130 passengers on an airplane, where 25 do not speak either German or Russian, we need to find how many passengers speak both German and Russian.
First, we calculate the number of passengers who speak at least one of the languages:
\[
130 - 25 = 105
\]
Next, we use the principle of inclusion-exclusio... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to figure out how many passengers speak both German and Russian on this airplane. The problem says there are 130 passengers total. Out of these, 25 don't speak either German or Russian. So that means the number of people who speak at least one of the languages, German or Russian, is 130 ... | [
{
"from": "human",
"value": "## 230. Math Puzzle $7 / 84$\n\nIn a passenger airplane, there are 130 passengers. 25 of them do not speak either German or Russian. 70 passengers speak German and 80 travelers speak Russian.\n\nHow many of the passengers speak both German and Russian?"
},
{
"from": "gpt... |
There is a spinner with three congruent sectors colored red, blue, and green. Whenever it lands on red, you earn $\$4$. Whenever it lands on blue, you lose $\$1$. Whenever it lands on green, you earn nothing. If you spin the spinner once, what is the expected value of the amount of money you earn? Express your answer a... | There is a spinner with three congruent sectors colored red, blue, and green. Whenever it lands on red, you earn $\$4$. Whenever it lands on blue, you lose $\$1$. Whenever it lands on green, you earn nothing. If you spin the spinner once, what is the expected value of the amount of money you earn? Express your answer a... | Okay, so there's this spinner with three equal sections: red, blue, and green. When it lands on red, you get $4. If it lands on blue, you lose $1, and green gives you nothing. The question is asking for the expected value of the amount of money you earn after one spin. They want the answer as a dollar value to the near... | The spinner has three congruent sectors: red, blue, and green, each with a probability of \(\frac{1}{3}\). The earnings for each outcome are as follows:
- Red: \$4
- Blue: -\$1
- Green: \$0
To find the expected value, we multiply each outcome by its probability and sum the results:
\[
\begin{align*}
\text{Expected Val... | 6,638 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this spinner with three equal sections: red, blue, and green. When it lands on red, you get $4. If it lands on blue, you lose $1, and green gives you nothing. The question is asking for the expected value of the amount of money you earn after one spin. They want the answer as a dollar value to ... | [
{
"from": "human",
"value": "There is a spinner with three congruent sectors colored red, blue, and green. Whenever it lands on red, you earn $\\$4$. Whenever it lands on blue, you lose $\\$1$. Whenever it lands on green, you earn nothing. If you spin the spinner once, what is the expected value of the amou... |
Given the polynomial $\left(x+2\right)\left(x-1\right)^{4}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}$, then $a_{2}=$____, $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=$____. | null | Okay, so I need to solve this polynomial problem. The question is about expanding (x + 2)(x - 1)^4 and finding two things: the coefficient a₂ of x², and the sum of the coefficients from a₁ to a₅. Let me take it step by step.
First, let me recall that when you multiply polynomials, you can use the distributive property... | To find the coefficient \(a_2\) and the sum \(a_1 + a_2 + a_3 + a_4 + a_5\) for the polynomial \((x + 2)(x - 1)^4\), we start by expanding \((x - 1)^4\) using the binomial theorem:
\[
(x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1
\]
Next, we multiply this result by \((x + 2)\):
\[
(x + 2)(x^4 - 4x^3 + 6x^2 - 4x + 1)
\]
Di... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to solve this polynomial problem. The question is about expanding (x + 2)(x - 1)^4 and finding two things: the coefficient a₂ of x², and the sum of the coefficients from a₁ to a₅. Let me take it step by step.
First, let me recall that when you multiply polynomials, you can use the distributive ... | [
{
"from": "human",
"value": "Given the polynomial $\\left(x+2\\right)\\left(x-1\\right)^{4}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}$, then $a_{2}=$____, $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=$____."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this polynomial problem... |
Given the function $f(x)=\ln |x|+x^{2}$, which of the following statements must be true?
A: $f(-e) f(2)$
C: $f(-1) > f(3)$
D: $f(-7) < f(6)$ | null | Okay, let's try to figure out which of these statements about the function f(x) = ln|x| + x² must be true. The options are A, B, C, D. Hmm, first, let me understand the function better.
So, f(x) is defined as the natural logarithm of the absolute value of x plus x squared. Since the logarithm is ln|x|, that means the ... | Given the function \( f(x) = \ln |x| + x^2 \), we need to determine which of the statements A, C, and D must be true.
1. **Statement A: \( f(-e) \cdot f(2) < 0 \)**
- Compute \( f(-e) \):
\[
f(-e) = \ln | -e | + (-e)^2 = \ln(e) + e^2 = 1 + e^2 \approx 1 + 7.389 \approx 8.389
\]
- Compute \( f(2) \... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to figure out which of these statements about the function f(x) = ln|x| + x² must be true. The options are A, B, C, D. Hmm, first, let me understand the function better.
So, f(x) is defined as the natural logarithm of the absolute value of x plus x squared. Since the logarithm is ln|x|, that me... | [
{
"from": "human",
"value": "Given the function $f(x)=\\ln |x|+x^{2}$, which of the following statements must be true?\nA: $f(-e) f(2)$\nC: $f(-1) > f(3)$\nD: $f(-7) < f(6)$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to figure out which of these statements about the function f(x) =... |
The product $M$ of three positive integers is $8$ times their sum, and one of the integers is the sum of the other two. Additionally, one of the integers is twice another. Find the sum of all possible values of $M$. | null | Okay, let's see. I need to solve this problem where the product of three positive integers is 8 times their sum, one of the integers is the sum of the other two, and one of the integers is twice another. Then find the sum of all possible values of M. Hmm. Let me break this down step by step.
First, let's denote the th... | Given three positive integers \(a\), \(b\), and \(c\) such that their product \(M\) is 8 times their sum, one of the integers is the sum of the other two, and one of the integers is twice another, we need to find the sum of all possible values of \(M\).
1. **Assume one integer is the sum of the other two**: Without lo... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to solve this problem where the product of three positive integers is 8 times their sum, one of the integers is the sum of the other two, and one of the integers is twice another. Then find the sum of all possible values of M. Hmm. Let me break this down step by step.
First, let's denot... | [
{
"from": "human",
"value": "The product $M$ of three positive integers is $8$ times their sum, and one of the integers is the sum of the other two. Additionally, one of the integers is twice another. Find the sum of all possible values of $M$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's... |
In triangle $ABC$, we have $\angle A = 45^\circ$, $BC = 15$, and $AB = 20$. What is the length of median from vertex $C$? | In triangle $ABC$, we have $\angle A = 45^\circ$, $BC = 15$, and $AB = 20$. What is the length of median from vertex $C$? | Okay, so I need to find the length of the median from vertex C in triangle ABC. The given information is angle A is 45 degrees, BC is 15 units long, and AB is 20 units long. Hmm, let me try to visualize this triangle first.
Alright, triangle ABC with angle at A being 45 degrees. The sides opposite to angles A, B, and... | Given triangle \(ABC\) with \(\angle A = 45^\circ\), \(BC = 15\), and \(AB = 20\), we need to find the length of the median from vertex \(C\).
1. **Identify the sides and angle:**
- \(\angle A = 45^\circ\)
- \(BC = 15\) (side \(a\))
- \(AB = 20\) (side \(c\))
- \(AC = b\) (unknown)
2. **Use the Law of Cos... | 13,921 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the length of the median from vertex C in triangle ABC. The given information is angle A is 45 degrees, BC is 15 units long, and AB is 20 units long. Hmm, let me try to visualize this triangle first.
Alright, triangle ABC with angle at A being 45 degrees. The sides opposite to angles A... | [
{
"from": "human",
"value": "In triangle $ABC$, we have $\\angle A = 45^\\circ$, $BC = 15$, and $AB = 20$. What is the length of median from vertex $C$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the length of the median from vertex C in triangle ABC. The given information i... |
Given the function $f(x)=|x-2|$.
(1) Find the solution set of the inequality $f(x)+x^{2}-4 > 0$;
(2) Let $g(x)=-|x+7|+3m$. If the solution set of the inequality $f(x) < g(x)$ with respect to $x$ is non-empty, find the range of values for the real number $m$. | null | Okay, let's tackle these two problems step by step. First, problem (1): solving the inequality f(x) + x² - 4 > 0, where f(x) is |x - 2|. Hmm. So that translates to |x - 2| + x² - 4 > 0. Alright, let's break this down.
First, absolute value functions can be tricky because they have different expressions depending on wh... | (1) To solve the inequality \( f(x) + x^2 - 4 > 0 \) where \( f(x) = |x - 2| \), we consider two cases based on the definition of the absolute value function.
**Case 1: \( x \geq 2 \)**
Here, \( |x - 2| = x - 2 \). Substituting into the inequality:
\[
(x - 2) + x^2 - 4 > 0 \implies x^2 + x - 6 > 0
\]
Factoring the qu... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's tackle these two problems step by step. First, problem (1): solving the inequality f(x) + x² - 4 > 0, where f(x) is |x - 2|. Hmm. So that translates to |x - 2| + x² - 4 > 0. Alright, let's break this down.
First, absolute value functions can be tricky because they have different expressions dependi... | [
{
"from": "human",
"value": "Given the function $f(x)=|x-2|$.\n(1) Find the solution set of the inequality $f(x)+x^{2}-4 > 0$;\n(2) Let $g(x)=-|x+7|+3m$. If the solution set of the inequality $f(x) < g(x)$ with respect to $x$ is non-empty, find the range of values for the real number $m$."
},
{
"fro... |
In a right triangle with hypotenuse 10, one of the acute angles is 30 degrees. Find the difference between the lengths of the two legs of the triangle, rounded to the nearest hundredth. | In a right triangle with hypotenuse 10, one of the acute angles is 30 degrees. Find the difference between the lengths of the two legs of the triangle, rounded to the nearest hundredth. | Okay, so I need to find the difference between the lengths of the two legs in a right triangle where the hypotenuse is 10 and one of the acute angles is 30 degrees. Let me think.
First, since it's a right triangle with a 30-degree angle, it must be a 30-60-90 triangle. Those have side length ratios of 1 : √3 : 2. Wai... | Given a right triangle with a hypotenuse of 10 and one of the acute angles being 30 degrees, we recognize it as a 30-60-90 triangle. The side ratios for such a triangle are 1 : √3 : 2, where the hypotenuse is twice the shorter leg.
1. The shorter leg (opposite the 30° angle) is calculated as:
\[
\text{Shorter l... | 12,159 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the difference between the lengths of the two legs in a right triangle where the hypotenuse is 10 and one of the acute angles is 30 degrees. Let me think.
First, since it's a right triangle with a 30-degree angle, it must be a 30-60-90 triangle. Those have side length ratios of 1 : √3 ... | [
{
"from": "human",
"value": "In a right triangle with hypotenuse 10, one of the acute angles is 30 degrees. Find the difference between the lengths of the two legs of the triangle, rounded to the nearest hundredth."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the difference be... |
Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{8!}-z^{7!}$ is a real number. | Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{8!}-z^{7!}$ is a real number. | Okay, let me try to figure out this problem. So, we need to find the number of different complex numbers z such that |z| = 1 and z^{8!} - z^{7!} is a real number. Hmm, okay. Let's break this down step by step.
First, since |z| = 1, we know that z lies on the unit circle in the complex plane. That means we can write z ... | Given the problem of finding the number of different complex numbers \( z \) such that \( |z| = 1 \) and \( z^{8!} - z^{7!} \) is a real number, we start by noting that \( z \) lies on the unit circle, so we can write \( z = e^{i\theta} \).
The condition \( z^{8!} - z^{7!} \) being real translates to the imaginary pa... | 28,493 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, we need to find the number of different complex numbers z such that |z| = 1 and z^{8!} - z^{7!} is a real number. Hmm, okay. Let's break this down step by step.
First, since |z| = 1, we know that z lies on the unit circle in the complex plane. That means we can ... | [
{
"from": "human",
"value": "Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{8!}-z^{7!}$ is a real number."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. So, we need to find the number of different complex numbers z ... |
The equation of line AB, where A and B are the intersection points of the circle $x^2+y^2-4x+6y=0$ and the circle $x^2+y^2-6x=0$, is: ______. | null | Okay, so I need to find the equation of line AB, where A and B are the intersection points of two circles. The circles given are:
First circle: \( x^2 + y^2 - 4x + 6y = 0 \)
Second circle: \( x^2 + y^2 - 6x = 0 \)
Hmm, I remember that when two circles intersect, the line connecting their intersection points is calle... | To find the equation of line AB, where A and B are the intersection points of the circles \(x^2 + y^2 - 4x + 6y = 0\) and \(x^2 + y^2 - 6x = 0\), we can use the concept of the radical axis. The radical axis of two circles is found by subtracting their equations.
Subtracting the second circle's equation from the first:... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the equation of line AB, where A and B are the intersection points of two circles. The circles given are:
First circle: \( x^2 + y^2 - 4x + 6y = 0 \)
Second circle: \( x^2 + y^2 - 6x = 0 \)
Hmm, I remember that when two circles intersect, the line connecting their intersection points ... | [
{
"from": "human",
"value": "The equation of line AB, where A and B are the intersection points of the circle $x^2+y^2-4x+6y=0$ and the circle $x^2+y^2-6x=0$, is: ______."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the equation of line AB, where A and B are the intersection p... |
Given that $n$ is an integer and $0 < 5n < 40$, what is the sum of all possible integer values of $n$? | null | Okay, let's see here. The problem says that n is an integer, and 0 is less than 5n, which is less than 40. I need to find the sum of all possible integer values of n. Hmm, let me break this down step by step.
First, let's parse the inequalities. The problem states 0 < 5n < 40. So there are two inequalities here: 0 is ... | Given that \( n \) is an integer and \( 0 < 5n < 40 \), we need to find the sum of all possible integer values of \( n \).
First, we solve the inequality \( 0 < 5n < 40 \). Dividing each part by 5, we get:
\[
0 < n < 8
\]
Since \( n \) is an integer, the possible values of \( n \) are 1, 2, 3, 4, 5, 6, and 7.
To fin... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see here. The problem says that n is an integer, and 0 is less than 5n, which is less than 40. I need to find the sum of all possible integer values of n. Hmm, let me break this down step by step.
First, let's parse the inequalities. The problem states 0 < 5n < 40. So there are two inequalities her... | [
{
"from": "human",
"value": "Given that $n$ is an integer and $0 < 5n < 40$, what is the sum of all possible integer values of $n$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see here. The problem says that n is an integer, and 0 is less than 5n, which is less than 40. I need to find th... |
Pipe A can fill a tank in 6 hours. Due to a leak at the bottom, it takes 8 hours for Pipe A to fill the tank. In what time can the leak alone empty the full tank? | null | Okay, so here's this problem: Pipe A can fill a tank in 6 hours normally, but because there's a leak at the bottom, it actually takes 8 hours to fill the tank. The question is, how long would it take for the leak alone to empty the full tank? Hmm, interesting. Let me think through this step by step.
First, I need to r... | Pipe A can fill a tank in 6 hours, so its filling rate is \( \frac{1}{6} \) of the tank per hour. When there is a leak, the effective filling rate is slower, taking 8 hours to fill the tank, resulting in a net rate of \( \frac{1}{8} \) of the tank per hour.
We denote the leak's emptying rate as \( R_L \). The combined... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so here's this problem: Pipe A can fill a tank in 6 hours normally, but because there's a leak at the bottom, it actually takes 8 hours to fill the tank. The question is, how long would it take for the leak alone to empty the full tank? Hmm, interesting. Let me think through this step by step.
First, I n... | [
{
"from": "human",
"value": "Pipe A can fill a tank in 6 hours. Due to a leak at the bottom, it takes 8 hours for Pipe A to fill the tank. In what time can the leak alone empty the full tank?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so here's this problem: Pipe A can fill a tank in 6 hours ... |
Find the maximum value of
\[\cos \frac{\theta}{2} \cdot (1 + \sin \theta)\]for $0 < \theta < \pi.$ | Find the maximum value of
\[\cos \frac{\theta}{2} \cdot (1 + \sin \theta)\]for $0 < \theta < \pi.$ | Okay, so I need to find the maximum value of the expression cos(θ/2) multiplied by (1 + sinθ) where θ is between 0 and π. Hmm, let's start by understanding the problem. The function given is a product of two trigonometric functions, cos(θ/2) and (1 + sinθ). My task is to find its maximum value in the specified interval... | To find the maximum value of the expression \(\cos \frac{\theta}{2} \cdot (1 + \sin \theta)\) for \(0 < \theta < \pi\), we start by defining the function \(f(\theta) = \cos \frac{\theta}{2} \cdot (1 + \sin \theta)\).
We use calculus techniques to find the critical points. The derivative of \(f(\theta)\) is computed u... | 5,697 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the maximum value of the expression cos(θ/2) multiplied by (1 + sinθ) where θ is between 0 and π. Hmm, let's start by understanding the problem. The function given is a product of two trigonometric functions, cos(θ/2) and (1 + sinθ). My task is to find its maximum value in the specified ... | [
{
"from": "human",
"value": "Find the maximum value of\n\\[\\cos \\frac{\\theta}{2} \\cdot (1 + \\sin \\theta)\\]for $0 < \\theta < \\pi.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the maximum value of the expression cos(θ/2) multiplied by (1 + sinθ) where θ is between 0 an... |
12.387. The base of the pyramid is an isosceles triangle with an angle $\alpha$ between the lateral sides. The pyramid is placed inside a certain cylinder such that its base is inscribed in the base of this cylinder, and the vertex coincides with the midpoint of one of the cylinder's generators. The volume of the cylin... | null | Okay, let's see. So, the problem is about a pyramid whose base is an isosceles triangle with angle α between the lateral sides. The pyramid is inside a cylinder, with its base inscribed in the base of the cylinder, and the vertex of the pyramid coincides with the midpoint of one of the cylinder's generators. The volume... | The base of the pyramid is an isosceles triangle inscribed in the base of a cylinder, which is a circle with radius \( r \). The vertex of the pyramid is at the midpoint of one of the cylinder's generators, meaning the height of the pyramid is half the height of the cylinder. The volume of the cylinder is given as \( V... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. So, the problem is about a pyramid whose base is an isosceles triangle with angle α between the lateral sides. The pyramid is inside a cylinder, with its base inscribed in the base of the cylinder, and the vertex of the pyramid coincides with the midpoint of one of the cylinder's generators. Th... | [
{
"from": "human",
"value": "12.387. The base of the pyramid is an isosceles triangle with an angle $\\alpha$ between the lateral sides. The pyramid is placed inside a certain cylinder such that its base is inscribed in the base of this cylinder, and the vertex coincides with the midpoint of one of the cyli... |
Three friends, Rachel, Mike, and Emma, are participating in a charity event where they have to fold paper cranes. Rachel folds a crane every 2 minutes, Mike folds a crane every 3 minutes, and Emma folds a crane every 5 minutes. If they start folding at the same time, what is the minimum number of cranes each person nee... | Three friends, Rachel, Mike, and Emma, are participating in a charity event where they have to fold paper cranes. Rachel folds a crane every 2 minutes, Mike folds a crane every 3 minutes, and Emma folds a crane every 5 minutes. If they start folding at the same time, what is the minimum number of cranes each person nee... | Okay, let's see. So Rachel, Mike, and Emma are folding paper cranes for a charity event. They each have different rates: Rachel folds one every 2 minutes, Mike every 3 minutes, and Emma every 5 minutes. The question is asking for the minimum number of cranes each person needs to fold so that they all finish folding a c... | To determine the minimum number of cranes each person needs to fold so that they all finish folding a crane at the same time, we need to find the least common multiple (LCM) of their folding times. Rachel folds a crane every 2 minutes, Mike every 3 minutes, and Emma every 5 minutes.
First, we calculate the LCM of 2, ... | 24,547 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. So Rachel, Mike, and Emma are folding paper cranes for a charity event. They each have different rates: Rachel folds one every 2 minutes, Mike every 3 minutes, and Emma every 5 minutes. The question is asking for the minimum number of cranes each person needs to fold so that they all finish fol... | [
{
"from": "human",
"value": "Three friends, Rachel, Mike, and Emma, are participating in a charity event where they have to fold paper cranes. Rachel folds a crane every 2 minutes, Mike folds a crane every 3 minutes, and Emma folds a crane every 5 minutes. If they start folding at the same time, what is the... |
Darren borrows $200$ clams from Ethan at a $12\%$ simple daily interest. Meanwhile, Fergie borrows $300$ clams from Gertie at a $7\%$ simple daily interest. If Darren already owed $50$ clams extra from a previous transaction before these borrowings, in how many days will Darren and Fergie owe the same amounts, assuming... | null | Okay, let me try to figure out this problem. So, Darren borrows 200 clams from Ethan at a 12% simple daily interest. Fergie borrows 300 clams from Gertie at a 7% simple daily interest. But Darren already owed 50 clams extra from before. We need to find out in how many days they will owe the same amount, assuming no rep... | Darren borrows 200 clams from Ethan at a 12% simple daily interest, while already owing 50 clams from a previous transaction. Fergie borrows 300 clams from Gertie at a 7% simple daily interest. We need to find the number of days after which their total debts will be equal.
Darren's total debt after \( t \) days consis... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this problem. So, Darren borrows 200 clams from Ethan at a 12% simple daily interest. Fergie borrows 300 clams from Gertie at a 7% simple daily interest. But Darren already owed 50 clams extra from before. We need to find out in how many days they will owe the same amount, assumin... | [
{
"from": "human",
"value": "Darren borrows $200$ clams from Ethan at a $12\\%$ simple daily interest. Meanwhile, Fergie borrows $300$ clams from Gertie at a $7\\%$ simple daily interest. If Darren already owed $50$ clams extra from a previous transaction before these borrowings, in how many days will Darre... |
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