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If $y$ is a number between 1 and 2, which of the following represents the largest value?
A). $y$
B). $y^2$
C). $3y$
D). $\sqrt{y}$
E). $\frac{1}{y}$
Express your answer as A, B, C, D, or E. | If $y$ is a number between 1 and 2, which of the following represents the largest value?
A). $y$
B). $y^2$
C). $3y$
D). $\sqrt{y}$
E). $\frac{1}{y}$
Express your answer as A, B, C, D, or E. | Okay, let's see. The problem is asking which of the given options represents the largest value when y is a number between 1 and 2. The options are A) y, B) y², C) 3y, D) sqrt(y), and E) 1/y. Hmm, I need to figure out which of these functions gives the biggest result for y in (1,2).
First, maybe I should consider the ... | To determine which of the given options represents the largest value when \( y \) is a number between 1 and 2, we analyze each function's behavior in the interval (1, 2):
- **Option A (\( y \))**: Values range from 1 to 2.
- **Option B (\( y^2 \))**: Values range from 1 to 4. However, solving \( y^2 > 3y \) shows that... | 24,841 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is asking which of the given options represents the largest value when y is a number between 1 and 2. The options are A) y, B) y², C) 3y, D) sqrt(y), and E) 1/y. Hmm, I need to figure out which of these functions gives the biggest result for y in (1,2).
First, maybe I should consi... | [
{
"from": "human",
"value": "If $y$ is a number between 1 and 2, which of the following represents the largest value?\n\nA). $y$\nB). $y^2$\nC). $3y$\nD). $\\sqrt{y}$\nE). $\\frac{1}{y}$\n\nExpress your answer as A, B, C, D, or E."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The prob... |
A student walks from home (H) to school (S) via a corner shop (C). The student needs to make X north moves and 1 east move to reach C, then from C to S, 1 north move and 2 east moves. What is the probability they go through C? If we know the answer to the above question is 9/20, what is the value of unknown variable X? | To find the probability that the student goes through point C, we need to determine the total number of ways the student can go from home (H) to school (S) directly and the number of ways to reach school (S) through the corner shop (C). We are given that the student needs to make X north moves and 1 east move to reach ... | Okay, let's try to figure out this problem step by step. So, the student is walking from home (H) to school (S) via a corner shop (C). The path involves some north and east moves. Let me break down the information given.
First, from H to C, the student needs to make X north moves and 1 east move. Then, from C to S, it... | To determine the value of \( X \), we need to find the probability that the student goes through the corner shop (C) on their way from home (H) to school (S). The given probability is \( \frac{9}{20} \).
1. **Total moves from H to S**:
- From H to C: \( X \) north moves and 1 east move.
- From C to S: 1 north mo... | 8,761 | MetaMath-AIME | <think>
Okay, let's try to figure out this problem step by step. So, the student is walking from home (H) to school (S) via a corner shop (C). The path involves some north and east moves. Let me break down the information given.
First, from H to C, the student needs to make X north moves and 1 east move. Then, from C ... | [
{
"from": "human",
"value": "A student walks from home (H) to school (S) via a corner shop (C). The student needs to make X north moves and 1 east move to reach C, then from C to S, 1 north move and 2 east moves. What is the probability they go through C? If we know the answer to the above question is 9/20,... |
[Divisibility of numbers. General properties] $[$ Prime numbers and their properties ]
Find all natural numbers $n$ such that the number $(n-1)!$ does not divide $n^2$.
# | null | Okay, so I need to find all natural numbers n where (n-1)! doesn't divide n². Hmm, let me think. I remember that factorials and divisibility are closely related, especially with primes. Maybe Wilson's theorem comes into play here? Wilson's theorem states that if p is a prime, then (p-1)! ≡ -1 mod p. So, for a prime p, ... | To find all natural numbers \( n \) such that \((n-1)!\) does not divide \( n^2 \), we analyze the divisibility properties and factorial growth:
1. **Wilson's Theorem**: For a prime \( p \), \((p-1)! \equiv -1 \pmod{p}\). This implies \((p-1)! + 1\) is divisible by \( p \), but \((p-1)!\) does not divide \( p^2 \) for... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find all natural numbers n where (n-1)! doesn't divide n². Hmm, let me think. I remember that factorials and divisibility are closely related, especially with primes. Maybe Wilson's theorem comes into play here? Wilson's theorem states that if p is a prime, then (p-1)! ≡ -1 mod p. So, for a p... | [
{
"from": "human",
"value": "[Divisibility of numbers. General properties] $[$ Prime numbers and their properties ]\n\nFind all natural numbers $n$ such that the number $(n-1)!$ does not divide $n^2$.\n\n#"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find all natural numbers n whe... |
A soccer league consists of teams with X players each. Each player must earn at least $12,500, and the maximum total salary for each team cannot exceed $550,000. The highest possible salary for one player is 337500. What is the value of the unknown variable X? | To determine the value of the unknown variable X, which represents the number of players in each soccer team, we need to analyze the information given in the question step by step.
1. **Minimum Salary for Players**: Each player must earn at least $12,500.
2. **Maximum Total Salary for a Team**: The total salary for e... | Alright, let's tackle this problem step by step. So, we have a soccer league where each team has X players. Each player must earn at least $12,500, and the maximum total salary for each team can't exceed $550,000. Additionally, the highest possible salary for one player is $337,500. We need to find the value of X.
Hmm... | To determine the value of \( X \), we need to consider the constraints given: the maximum total salary for each team is $550,000, each player must earn at least $12,500, and the highest possible salary for one player is $337,500.
1. The total salary for a team is the sum of the highest possible salary for one player a... | 17,841 | MetaMath-AIME | <think>
Alright, let's tackle this problem step by step. So, we have a soccer league where each team has X players. Each player must earn at least $12,500, and the maximum total salary for each team can't exceed $550,000. Additionally, the highest possible salary for one player is $337,500. We need to find the value of... | [
{
"from": "human",
"value": "A soccer league consists of teams with X players each. Each player must earn at least $12,500, and the maximum total salary for each team cannot exceed $550,000. The highest possible salary for one player is 337500. What is the value of the unknown variable X?"
},
{
"fro... |
Given the context of Calabi-Yau differential equations of order \(3\) and \(5\), can you provide a detailed explanation of how the level \(\ell\) of a differential equation defined as \( \ell = \frac{4}{\tau_c^2} \) relates to its modular properties, including the connection between the values of \(\tau_c\) and the cor... | Okay, so I need to understand how the level ℓ of a Calabi-Yau differential equation is related to its modular properties, especially given that ℓ is defined as 4 divided by τ_c squared. The user also wants to know how to compute ℓ if τ_c is known and what significance this holds in number theory. Let me start breaking ... | The level \(\ell\) of a Calabi-Yau differential equation, defined as \(\ell = \frac{4}{\tau_c^2}\), plays a pivotal role in connecting the analytic properties of the equation to modular forms in number theory. Here's a detailed breakdown:
### **1. Definition and Computation of \(\ell\)**
- **Relation to \(\tau_c\)**: ... | 39,587 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to understand how the level ℓ of a Calabi-Yau differential equation is related to its modular properties, especially given that ℓ is defined as 4 divided by τ_c squared. The user also wants to know how to compute ℓ if τ_c is known and what significance this holds in number theory. Let me start b... | [
{
"from": "human",
"value": "Given the context of Calabi-Yau differential equations of order \\(3\\) and \\(5\\), can you provide a detailed explanation of how the level \\(\\ell\\) of a differential equation defined as \\( \\ell = \\frac{4}{\\tau_c^2} \\) relates to its modular properties, including the co... | |
"Solve the system:\n\n$$\n\\left\\{\\begin{array}{l}\ny=x(4-x) \\\\\nz=y(4-y) \\\\\nx=z(4-z)\n\\end{(...TRUNCATED) | null | "Okay, so I need to solve this system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\ny = x(4 - x) (...TRUNCATED) | "To solve the system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\ny = x(4 - x) \\\\\nz = y(4 - y(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, so I need to solve this system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\ny = (...TRUNCATED) | [{"from":"human","value":"Solve the system:\n\n$$\n\\left\\{\\begin{array}{l}\ny=x(4-x) \\\\\nz=y(4-(...TRUNCATED) |
"Simplify: $$\\dfrac{\\sqrt{338}}{\\sqrt{288}}+\\dfrac{\\sqrt{150}}{\\sqrt{96}}.$$Express your answe(...TRUNCATED) | "Simplify: $$\\dfrac{\\sqrt{338}}{\\sqrt{288}}+\\dfrac{\\sqrt{150}}{\\sqrt{96}}.$$Express your answe(...TRUNCATED) | "Alright, let's try to solve this problem step by step. Okay, the problem is about a student who was(...TRUNCATED) | "Let the two-digit number be \\(10a + b\\), where \\(a\\) is the tens digit and \\(b\\) is the units(...TRUNCATED) | 16,323 | ai2-adapt-dev/openmath-2-math | "<think>\nAlright, let's try to solve this problem step by step. Okay, the problem is about a studen(...TRUNCATED) | [{"from":"human","value":"Simplify: $$\\dfrac{\\sqrt{338}}{\\sqrt{288}}+\\dfrac{\\sqrt{150}}{\\sqrt{(...TRUNCATED) |
"On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and al(...TRUNCATED) | "On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and al(...TRUNCATED) | "Okay, let me try to figure out this problem. So, we have a single-lane highway where all cars are m(...TRUNCATED) | "To solve the problem, we need to determine the maximum number of cars, \\( M \\), that can pass a p(...TRUNCATED) | 16,138 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, let me try to figure out this problem. So, we have a single-lane highway where all c(...TRUNCATED) | [{"from":"human","value":"On a long straight stretch of one-way single-lane highway, cars all travel(...TRUNCATED) |
"Let $P(x) = (x-1)(x-2)(x-4)$. For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ (...TRUNCATED) | "Let $P(x) = (x-1)(x-2)(x-4)$. For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ (...TRUNCATED) | "Okay, so I need to find how many polynomials Q(x) satisfy the equation P(Q(x)) = P(x) * R(x), where(...TRUNCATED) | "Given the polynomial \\( P(x) = (x-1)(x-2)(x-4) \\), we need to find the number of polynomials \\( (...TRUNCATED) | 3,163 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find how many polynomials Q(x) satisfy the equation P(Q(x)) = P(x) * R((...TRUNCATED) | [{"from":"human","value":"Let $P(x) = (x-1)(x-2)(x-4)$. For how many polynomials $Q(x)$ does there e(...TRUNCATED) |
"Given the set $M=\\{0,1,2\\}$, and $N=\\{x|x^{2}-3x+2 > 0\\}$, determine $M∩(∁\\_RN)=(\\ \\ \\ (...TRUNCATED) | null | "Okay, let me try to figure out this problem step by step. So, we have two sets here: M is given as (...TRUNCATED) | "Given the set \\( M = \\{0, 1, 2\\} \\) and \\( N = \\{x \\mid x^2 - 3x + 2 > 0\\} \\), we need to (...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, let me try to figure out this problem step by step. So, we have two sets here: M is (...TRUNCATED) | [{"from":"human","value":"Given the set $M=\\{0,1,2\\}$, and $N=\\{x|x^{2}-3x+2 > 0\\}$, determine $(...TRUNCATED) |
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