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128 | Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product? | math | qq8933/AIME_1983_2024 | {'ID': '1991-5', 'Year': 1991, 'Problem Number': 5, 'Part': None} |
The problem asks for the number of rational numbers between 0 and 1 such that when the rational number is written as a fraction in lowest terms, the product of the numerator and the denominator is $20!$.
Let the rational number be $r$, where $0 < r < 1$.
We can write $r$ as a fraction $\frac{a}{b}$, where $a$ and $b$... | Solution:
Let the rational number between 0 and 1 be $r$. When written as a fraction in lowest terms, let it be $\frac{a}{b}$, where $a$ and $b$ are positive integers, $\gcd(a, b) = 1$, and $0 < a < b$.
We are given that the product of the numerator and denominator is $20!$, so $a \times b = 20!$.
We need to find the ... | Alright, so I need to figure out how many rational numbers between 0 and 1 have a product of the numerator and denominator (when written in lowest terms) equal to 20 factorial. Let me start by understanding the problem.
First, a rational number between 0 and 1 can be written as a fraction \(\frac{a}{b}\) where \(0 < a... | Given a rational number between 0 and 1, we need to write it as a fraction in lowest terms \(\frac{a}{b}\) and calculate the product of the resulting numerator and denominator. We need to determine how many such rational numbers exist where the product \(a \times b = 20!\).
1. **Prime Factorization of 20!**:
- The ... | [
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"content": "Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?",
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"content": "<think>\nAlright, so I need t... | 7,145 |
109 | Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$ is divided by $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2010-I-2', 'Year': 2010, 'Problem Number': 2, 'Part': 'I'} |
The problem asks for the remainder when the product $P = 9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$ is divided by $1000$.
The terms in the product are of the form $10^k - 1$, where $k$ ranges from $1$ to $999$.
The terms are $10^1 - 1, 10^2 - 1, 10^3 - 1, \ldots, 10^{999} - 1... | Solution:
Let the product be $P = 9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$.
The terms in the product can be written as $10^k - 1$, where $k$ ranges from $1$ to $999$.
The product is $P = (10^1 - 1)(10^2 - 1)(10^3 - 1) \cdots (10^{999} - 1)$.
We need to find the remainder when... | Alright, so I have this problem here: I need to find the remainder when the product of numbers like 9, 99, 999, and so on, up to a number with 999 nines, is divided by 1000. Hmm, okay, let's unpack this step by step.
First, the problem is asking for the remainder of a very large product divided by 1000. When dealing w... | To find the remainder when \(9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\) is divided by 1000, we consider the product modulo 8 and modulo 125 separately, then combine the results using the Chinese Remainder Theorem.
1. **Modulo 8 Calculation:**
- Each term in the product is ... | [
{
"content": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$ .",
"role": "user"
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"content": "<think>\nAlright, so I have this problem here: I need to find the remainder when the product of numbers like 9,... | 7,422 |
167.0 | Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s | math | TIGER-Lab/TheoremQA/float | {'Answer_type': 'float', 'Picture': None} | Here's a thinking process to arrive at the solution:
1. **Understand the Goal:** The request is to calculate the mean molecular speed of radon. This immediately brings to mind concepts from kinetic theory of gases.
2. **Identify the Relevant Formula:** The mean molecular speed is related to the temperature and molar ... | To compute the mean molecular speed (v) of radon (Rn), we can use the following formula derived from the kinetic theory of gases:
**v = √(8RT / (πM))**
Where:
* **v** is the mean molecular speed (in m/s)
* **R** is the ideal gas constant (8.314 J/mol·K)
* **T** is the absolute temperature (in Kelvin)
* **M** is the ... | Okay, I need to calculate the mean molecular speed of radon gas. Hmm, first, I remember there's a formula for the root mean square speed, which is related to temperature and molar mass. Wait, but the question specifically says "mean molecular speed"—is that the same as the root mean square speed, or is it a different k... | To compute the mean molecular speed \( v \) of radon (Rn) gas, we use the formula for the mean (average) speed derived from the Maxwell-Boltzmann distribution:
\[
v = \sqrt{\frac{8RT}{\pi M}}
\]
where:
- \( R \) is the gas constant (\(8.314 \, \text{J/(mol·K)}\)),
- \( T \) is the temperature in Kelvin,
- \( M \) is ... | [
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"content": "Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s",
"role": "user"
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"content": "<think>\nOkay, I need to calculate the mean molecular speed of radon gas. Hmm, first, I remember there's a formula for the root mean square speed, which is related to temperature and... | 11,095 |
1.3 | "Two capacitors with capacitance values $C_{1}=2000 \\pm 10 \\mathrm{pF}$ and $C_{2}=3000 \\pm 15 \\(...TRUNCATED) | math | daman1209arora/jeebench/phy | {'subject': 'phy', 'description': 'JEE Adv 2020 Paper 2', 'index': 15, 'type': 'Numeric'} | "Let $C_1$ and $C_2$ be the capacitance values of the two capacitors, and $\\Delta C_1$ and $\\Delta(...TRUNCATED) | "Given the capacitance values $C_{1}=2000 \\pm 10 \\mathrm{pF}$ and $C_{2}=3000 \\pm 15 \\mathrm{pF}(...TRUNCATED) | "Okay, so I need to find the percentage error in the calculation of the energy stored in two capacit(...TRUNCATED) | "To find the percentage error in the calculation of the energy stored in two capacitors connected in(...TRUNCATED) | [{"content":"Two capacitors with capacitance values $C_{1}=2000 \\pm 10 \\mathrm{pF}$ and $C_{2}=300(...TRUNCATED) | 9,488 |
181 | "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpo(...TRUNCATED) | math | qq8933/AIME_1983_2024 | {'ID': '2000-II-6', 'Year': 2000, 'Problem Number': 6, 'Part': 'II'} | "\nThe problem asks for the value of $\\lfloor x^2/100 \\rfloor$, where $x$ is the length of a segme(...TRUNCATED) | "Solution:\nLet the lengths of the two bases of the trapezoid be $b_1$ and $b_2$, with $b_1 > b_2$. (...TRUNCATED) | "Okay, let's see. I need to solve this problem about a trapezoid with bases differing by 100 units. (...TRUNCATED) | "Let the shorter base be \\( b \\) and the longer base be \\( b + 100 \\). The midline of the trapez(...TRUNCATED) | [{"content":"One base of a trapezoid is $100$ units longer than the other base. The segment that joi(...TRUNCATED) | 3,635 |
"1. **Defining the Sparse Set \\( S \\) and the Turing Machine \\( M \\)**\n\n We need to construc(...TRUNCATED) | "Let's say a language $L \\subseteq \\{0,1\\}^*$ is in $\\textbf{P}_{angel}$ if there exists a p(...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "{'source': 'aops_forum', 'messages': [{'content': \"Let's say a language $L \\\\subseteq \\\\{0,1\(...TRUNCATED) | "Let's break down each part of the problem and build the solutions.\n\n**Part 1: Combining Sparse Se(...TRUNCATED) | "Let's solve each part step by step.\n\n**Part 1: Combining Sparse Sets**\n\nWe are given $k$ sparse(...TRUNCATED) | "Okay, let me try to work through these problems one by one. They all seem related to sparse sets, o(...TRUNCATED) | "**Problem 1:**\n\nTo construct the sparse set \\( S \\), we can combine the sparse sets \\( S_1, S_(...TRUNCATED) | [{"content":"Let's say a language $L \\subseteq \\{0,1\\}^*$ is in $\\textbf{P}_{angel}$ if ther(...TRUNCATED) | 8,675 |
"(a) Let the number of promoted participants be \\( P \\) and the number of repeaters be \\( R \\). (...TRUNCATED) | "In a mathematics test number of participants is $N < 40$ . The passmark is fixed at $65$ . The te(...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "{'source': 'aops_forum', 'messages': [{'content': 'In a mathematics test number of participants is (...TRUNCATED) | "Let $N$ be the total number of participants, $N < 40$.\nLet $P$ be the number of promoted participa(...TRUNCATED) | "Solution:\nLet $N$ be the total number of participants. The passmark is $65$.\nInitial state:\nAver(...TRUNCATED) | "Alright, so I'm trying to solve this math problem about participants in a test. Let me first try to(...TRUNCATED) | "Given a mathematics test with \\( N \\) participants (\\( N < 40 \\)), the pass mark is fixed at 65(...TRUNCATED) | [{"content":"In a mathematics test number of participants is $N < 40$ . The passmark is fixed at $(...TRUNCATED) | 11,498 |
157 | "Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the ca(...TRUNCATED) | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-3', 'Year': 2018, 'Problem Number': 3, 'Part': 'I'} | "\nThe problem asks for the probability that in a random layout of $5$ cards chosen from $5$ red and(...TRUNCATED) | "Solution:\nLet $R$ denote a red card and $G$ denote a green card. There are $5$ red cards and $5$ g(...TRUNCATED) | "Okay, let's see. We need to find the probability that when Kathy lays out 5 cards from a shuffled d(...TRUNCATED) | "Kathy has 5 red cards and 5 green cards. She shuffles the 10 cards and lays out 5 of them in a row.(...TRUNCATED) | [{"content":"Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $(...TRUNCATED) | 9,379 |
25 | "Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ , and the are(...TRUNCATED) | math | qq8933/AIME_1983_2024 | {'ID': '2008-I-2', 'Year': 2008, 'Problem Number': 2, 'Part': 'I'} | "\nThe problem asks for the length of the altitude to the base $EM$ of an isosceles triangle $GEM$, (...TRUNCATED) | "Solution:\nLet the square $AIME$ have vertices $A=(0,10)$, $I=(10,10)$, $M=(10,0)$, $E=(0,0)$. The (...TRUNCATED) | "Okay, so I need to solve this geometry problem where there's a square named AIME with sides of 10 u(...TRUNCATED) | "Square \\(AIME\\) has sides of length 10 units. Isosceles triangle \\(GEM\\) has base \\(EM\\), and(...TRUNCATED) | [{"content":"Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ ,(...TRUNCATED) | 8,407 |
"1. **Define the Game and Initial Lists:**\n - We are given a game where two players alternately t(...TRUNCATED) | "Consider the following two person game. A number of pebbles are situated on the table. Two players (...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "{'source': 'aops_forum', 'messages': [{'content': 'Consider the following two person game. A number(...TRUNCATED) | "Let $N$ be the initial number of pebbles.\nThe possible moves are to subtract $1^2, 2^2, 3^2, \\dot(...TRUNCATED) | "Solution:\nLet a position be the number of pebbles on the table. A position is a losing position if(...TRUNCATED) | "Okay, so I need to prove that in this pebble game, there are infinitely many starting numbers of pe(...TRUNCATED) | "To prove that there are infinitely many initial situations in which the second player can win no ma(...TRUNCATED) | [{"content":"Consider the following two person game. A number of pebbles are situated on the table. (...TRUNCATED) | 13,017 |
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