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A physical education teacher has 9 identical footballs to be placed into three boxes labeled 1, 2, and 3. The requirement is that the number of footballs in each box must not be less than its own label number. Calculate the total number of different ways to distribute the footballs into the boxes. | null | Okay, so I need to figure out how many ways there are to distribute 9 identical footballs into three boxes labeled 1, 2, and 3. The catch is that each box must have at least as many footballs as its label number. Hmm, let's break this down.
First, let's restate the problem to make sure I understand it. We have three b... | To determine the number of ways to distribute 9 identical footballs into three boxes labeled 1, 2, and 3, where each box must contain at least as many footballs as its label number, we can follow these steps:
1. **Subtract the minimum number of footballs required for each box**:
- Box 1 needs at least 1 football.
... | null | SynthLabsAI/Big-Math-RL-Verified | <think>
Okay, so I need to figure out how many ways there are to distribute 9 identical footballs into three boxes labeled 1, 2, and 3. The catch is that each box must have at least as many footballs as its label number. Hmm, let's break this down.
First, let's restate the problem to make sure I understand it. We have... | [
{
"from": "human",
"value": "A physical education teacher has 9 identical footballs to be placed into three boxes labeled 1, 2, and 3. The requirement is that the number of footballs in each box must not be less than its own label number. Calculate the total number of different ways to distribute the footba... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
For which complex numbers \( x \) with real part greater than 1 does the equality \( \zeta(x) = \sum_{n=1}^\infty \frac{2.5}{x^{n^n}} \) hold, and what is the numerical value of \( \zeta(2) \) given this condition? | Alright, so I have this problem here: I need to figure out for which complex numbers \( x \) with real part greater than 1 the equality \( \zeta(x) = \sum_{n=1}^\infty \frac{2.5}{x^{n^n}} \) holds. Then, using this condition, I have to find the numerical value of \( \zeta(2) \). Hmm, okay. Let me start by recalling wha... | To determine for which complex numbers \( x \) with real part greater than 1 the equality \( \zeta(x) = \sum_{n=1}^\infty \frac{2.5}{x^{n^n}} \) holds, we start by comparing the given series with the definition of the Riemann zeta function \( \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \).
1. **Comparison of Series Ter... | 8,872 | ddrg/named_math_formulas | <think>
Alright, so I have this problem here: I need to figure out for which complex numbers \( x \) with real part greater than 1 the equality \( \zeta(x) = \sum_{n=1}^\infty \frac{2.5}{x^{n^n}} \) holds. Then, using this condition, I have to find the numerical value of \( \zeta(2) \). Hmm, okay. Let me start by recal... | [
{
"from": "human",
"value": "For which complex numbers \\( x \\) with real part greater than 1 does the equality \\( \\zeta(x) = \\sum_{n=1}^\\infty \\frac{2.5}{x^{n^n}} \\) hold, and what is the numerical value of \\( \\zeta(2) \\) given this condition?"
},
{
"from": "gpt",
"value": "<think>\nA... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | |
Given that the complex number $z$ satisfies $iz = 2 + 4i$, find the coordinates of the point corresponding to $z$ in the complex plane. Express your answer as an ordered pair (x, y), where x and y are integers. | null | Okay, so I have this problem here: Given that the complex number z satisfies iz = 2 + 4i, find the coordinates of the point corresponding to z in the complex plane. They want the answer as an ordered pair (x, y) where x and y are integers. Alright, let me think about how to approach this.
First, complex numbers. I rem... | Given that the complex number \( z \) satisfies \( iz = 2 + 4i \), we need to find the coordinates of the point corresponding to \( z \) in the complex plane.
First, we solve for \( z \) by dividing both sides of the equation by \( i \):
\[
z = \frac{2 + 4i}{i}
\]
To simplify this, we multiply the numerator and the d... | null | SynthLabsAI/Big-Math-RL-Verified | <think>
Okay, so I have this problem here: Given that the complex number z satisfies iz = 2 + 4i, find the coordinates of the point corresponding to z in the complex plane. They want the answer as an ordered pair (x, y) where x and y are integers. Alright, let me think about how to approach this.
First, complex number... | [
{
"from": "human",
"value": "Given that the complex number $z$ satisfies $iz = 2 + 4i$, find the coordinates of the point corresponding to $z$ in the complex plane. Express your answer as an ordered pair (x, y), where x and y are integers."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
Which positive numbers $x$ satisfy the equation $(\log_3x)(\log_x5)=\log_35$?
$\textbf{(A)}\ 3 \text{ and } 5 \text{ only} \qquad \textbf{(B)}\ 3, 5, \text{ and } 15 \text{ only} \qquad \\ \textbf{(C)}\ \text{only numbers of the form } 5^n \cdot 3^m, \text{ where } n \text{ and } m \text{ are positive integers} \qqua... | null | Okay, so I have this equation: (log base 3 of x) multiplied by (log base x of 5) equals log base 3 of 5. And I need to find which positive numbers x satisfy this equation. The answer choices are given, and I have to figure out which one is correct. Let me start by recalling some logarithm properties to see if I can man... | We start with the equation \((\log_3 x)(\log_x 5) = \log_3 5\). Using the property of logarithms that \(\log_a b \cdot \log_b c = \log_a c\), we can simplify the left-hand side of the equation.
Applying this property, we have:
\[
(\log_3 x)(\log_x 5) = \log_3 5
\]
This shows that the left-hand side is equal to the ri... | null | OpenMathInstruct-AIME | <think>
Okay, so I have this equation: (log base 3 of x) multiplied by (log base x of 5) equals log base 3 of 5. And I need to find which positive numbers x satisfy this equation. The answer choices are given, and I have to figure out which one is correct. Let me start by recalling some logarithm properties to see if I... | [
{
"from": "human",
"value": "Which positive numbers $x$ satisfy the equation $(\\log_3x)(\\log_x5)=\\log_35$? \n$\\textbf{(A)}\\ 3 \\text{ and } 5 \\text{ only} \\qquad \\textbf{(B)}\\ 3, 5, \\text{ and } 15 \\text{ only} \\qquad \\\\ \\textbf{(C)}\\ \\text{only numbers of the form } 5^n \\cdot 3^m, \\text... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
If Bob is building a vegetable garden that measures 220 feet in length and 120 feet in width, and half of the garden will be tilled land while a third of the remaining area will have trellises for vining plants, what will be the size of the raised bed section of the garden? | The area of the garden is calculated by multiplying the length by the width, so 220 feet x 120 feet = 26,400 square feet.
Half of the garden will be tilled land, so 26,400 square feet / 2 = 13,200 square feet.
A third of the remaining area will have trellises, so 13,200 square feet / 3 = 4,400 square feet.
The raised b... | Okay, let's see. Bob has a vegetable garden that's 220 feet long and 120 feet wide. So first, I should probably figure out the total area of the garden. To do that, I multiply the length by the width. Let me write that down: 220 feet times 120 feet. Hmm, 220 times 120. Let me calculate that.
220 multiplied by 100 is 2... | Bob's vegetable garden measures 220 feet in length and 120 feet in width. To find the total area of the garden, we multiply the length by the width:
\[
220 \, \text{ft} \times 120 \, \text{ft} = 26,400 \, \text{sq ft}
\]
Half of the garden will be tilled land, so we calculate half of the total area:
\[
\frac{26,400}... | 20,949 | TIGER-Lab/MATH-plus | <think>
Okay, let's see. Bob has a vegetable garden that's 220 feet long and 120 feet wide. So first, I should probably figure out the total area of the garden. To do that, I multiply the length by the width. Let me write that down: 220 feet times 120 feet. Hmm, 220 times 120. Let me calculate that.
220 multiplied by ... | [
{
"from": "human",
"value": "If Bob is building a vegetable garden that measures 220 feet in length and 120 feet in width, and half of the garden will be tilled land while a third of the remaining area will have trellises for vining plants, what will be the size of the raised bed section of the garden?"
}... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
"Given the divisor class formula for \\(\\overline{\\mathfrak{Quad}}_{g(t),n(t)}\\) as \n\n\\[\n\\ov(...TRUNCATED) | "Okay, let me try to figure out this problem. So, we have this divisor class formula for the moduli (...TRUNCATED) | "To determine the coefficient \\( b_{1:0}(2) \\) in the given divisor class formula for \\( \\overli(...TRUNCATED) | 26,466 | mlfoundations-dev/gair_mathpile | "<think>\nOkay, let me try to figure out this problem. So, we have this divisor class formula for th(...TRUNCATED) | [{"from":"human","value":"Given the divisor class formula for \\(\\overline{\\mathfrak{Quad}}_{g(t),(...TRUNCATED) | null | null | null | null | null | null | null | null | null | null | null | null | null | null | |
"Ethan owes Liam $900, which is three times what Ava owes Liam. Ethan decides to pay 50% of his debt(...TRUNCATED) | 970 | "Okay, let me try to figure out this problem step by step. So, Ethan owes Liam $900, and that's thre(...TRUNCATED) | "Ethan owes Liam $900, which is three times what Ava owes Liam. Therefore, Ava's debt is calculated (...TRUNCATED) | 29,680 | nvidia/OpenMathInstruct-2 | "<think>\nOkay, let me try to figure out this problem step by step. So, Ethan owes Liam $900, and th(...TRUNCATED) | [{"from":"human","value":"Ethan owes Liam $900, which is three times what Ava owes Liam. Ethan decid(...TRUNCATED) | null | "Let's first calculate how much Ethan and Ava owe Liam.\nEthan owes Liam $900. \nSince Ethan owes 3 (...TRUNCATED) | augmented_gsm8k | null | null | null | null | null | null | null | null | null | null | null |
"What is the maximum number of significant figures that can be accurately computed for the zeros of (...TRUNCATED) | "Okay, so I need to figure out the maximum number of significant figures that can be accurately comp(...TRUNCATED) | "To determine the maximum number of significant figures that can be accurately computed for the zero(...TRUNCATED) | 27,903 | math-ai/AutoMathText | "<think>\nOkay, so I need to figure out the maximum number of significant figures that can be accura(...TRUNCATED) | [{"from":"human","value":"What is the maximum number of significant figures that can be accurately c(...TRUNCATED) | "CoCalc Public Fileswww / talks / bernoulli / buzzard-project_improve.txt\nAuthor: William A. Stein\(...TRUNCATED) | null | null | null | null | null | null | null | null | null | null | null | null | null | |
9 - 3 ÷ 1/3 + 6 = ?
Answer Choices: (A) 1 (B) 3 (C) 6 (D) 9 (E) 12 | null | "Okay, let's see. I need to solve this math problem: 9 - 3 ÷ 1/3 + 6. The answer choices are A) 1, (...TRUNCATED) | "To solve the expression \\(9 - 3 \\div \\frac{1}{3} + 6\\), we follow the order of operations (PEMD(...TRUNCATED) | null | mlfoundations-dev/BAAI_InfinityMATH | "<think>\nOkay, let's see. I need to solve this math problem: 9 - 3 ÷ 1/3 + 6. The answer choices a(...TRUNCATED) | [{"from":"human","value":"9 - 3 ÷ 1/3 + 6 = ?\nAnswer Choices: (A) 1 (B) 3 (C) 6 (D) 9 (E) 12"},{"f(...TRUNCATED) | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
"# Question\n\nAditi can walk 1.7 miles in half an hour. At the same constant speed, how many hours (...TRUNCATED) | null | "Okay, let's see. Aditi can walk 1.7 miles in half an hour. I need to figure out how long it would t(...TRUNCATED) | "To determine how long it would take Aditi to walk 17 miles at her constant speed, follow these step(...TRUNCATED) | 3,726 | Asap7772/hendrycks-math-mc-llama-sft-intermediate-parallel | "<think>\nOkay, let's see. Aditi can walk 1.7 miles in half an hour. I need to figure out how long i(...TRUNCATED) | [{"from":"human","value":"# Question\n\nAditi can walk 1.7 miles in half an hour. At the same consta(...TRUNCATED) | null | null | null | "# Question\n\nAditi can walk 1.7 miles in half an hour. At the same constant speed, how many hours (...TRUNCATED) | false | ["# Question\n\nAditi can walk 1.7 miles in half an hour. At the same constant speed, how many hours(...TRUNCATED) | [0.09042000226322036,0.1994039641466454,0.07481223235862071,0.0667391023255814,0.042327159292035406,(...TRUNCATED) | [0.288135593220339,0.5718849840255591,0.19310344827586207,0.15503875968992248,0.08849557522123894,0.(...TRUNCATED) | 5 | 8 | 14,840 | null | null | null |
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