input stringlengths 130 3.2k | output stringlengths 332 22.9k | score float64 0 1 | step int64 1 64 |
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user
In one urn, there are two white balls, in another—two black balls, in the third—one white ball and one black ball. Each urn has a label indicating its contents: $W W$, $B B$, $W B$. However, a prankster has swapped all the labels so that now each label incorrectly describes the contents of the urn. You are allowed... | ### Understanding the Problem
First, let's break down the problem to understand what's being asked:
- **Urn Contents**: There are three urns:
1. Urn 1: Two white balls ($WW$)
2. Urn 2: Two black balls ($BB$)
3. Urn 3: One white and one black ball ($WB$)
- **Label Swapping**: The labels on the urns have been in... | 1 | 1 |
user
In one urn, there are two white balls, in another—two black balls, in the third—one white ball and one black ball. Each urn has a label indicating its contents: $W W$, $B B$, $W B$. However, a prankster has swapped all the labels so that now each label incorrectly describes the contents of the urn. You are allowed... | ### Problem Understanding
First, let's break down the problem:
- **Urn Contents (Original Labels):**
- Urn 1: WW (Two White Balls)
- Urn 2: BB (Two Black Balls)
- Urn 3: WB (One White and One Black Ball)
- **Current Labels (After Label Swap):**
- All labels are incorrect. So:
- Urn 1 doesn't have WW.
... | 1 | 1 |
user
In one urn, there are two white balls, in another—two black balls, in the third—one white ball and one black ball. Each urn has a label indicating its contents: $W W$, $B B$, $W B$. However, a prankster has swapped all the labels so that now each label incorrectly describes the contents of the urn. You are allowed... | ### Problem Analysis
We have three urns with the following incorrect labels:
1. Urn labeled \( WW \): Not both white balls.
2. Urn labeled \( BB \): Not both black balls.
3. Urn labeled \( WB \): Not one white and one black ball.
Our goal is to determine the contents of each urn by drawing the minimum number of ball... | 1 | 1 |
user
In one urn, there are two white balls, in another—two black balls, in the third—one white ball and one black ball. Each urn has a label indicating its contents: $W W$, $B B$, $W B$. However, a prankster has swapped all the labels so that now each label incorrectly describes the contents of the urn. You are allowed... | ### Problem Analysis
We need to determine the contents of three urns by drawing balls from them, with the goal of figuring out the correct contents based on the incorrect labels. Here's a step-by-step reasoning process to find the minimum number of draws required.
### Understanding the Problem
1. **Initial Setup:**
... | 1 | 1 |
user
In one urn, there are two white balls, in another—two black balls, in the third—one white ball and one black ball. Each urn has a label indicating its contents: $W W$, $B B$, $W B$. However, a prankster has swapped all the labels so that now each label incorrectly describes the contents of the urn. You are allowed... | ### Understanding the Problem
First, let's break down the problem to understand what is being asked:
- **Urn Contents**:
- Urn 1: Originally labeled $WW$ (two white balls), but after label swapping, it has an incorrect label.
- Urn 2: Originally labeled $BB$ (two black balls), but after label swapping, it has an... | 1 | 1 |
user
In one urn, there are two white balls, in another—two black balls, in the third—one white ball and one black ball. Each urn has a label indicating its contents: $W W$, $B B$, $W B$. However, a prankster has swapped all the labels so that now each label incorrectly describes the contents of the urn. You are allowed... | Let's approach this problem step by step.
### Understanding the Problem
1. **Initial Setup**:
- There are three urns with the following labels (which are incorrect):
- Urn 1: Label \( WW \) (but it doesn't contain two white balls)
- Urn 2: Label \( BB \) (but it doesn't contain two black balls)
- Ur... | 1 | 1 |
user
In one urn, there are two white balls, in another—two black balls, in the third—one white ball and one black ball. Each urn has a label indicating its contents: $W W$, $B B$, $W B$. However, a prankster has swapped all the labels so that now each label incorrectly describes the contents of the urn. You are allowed... | To solve this problem, let's approach it step by step.
### Understanding the Problem
We have three urns with the following incorrect labels:
1. **Urn 1**: Labelled $WB$ (but does not contain one white and one black ball)
2. **Urn 2**: Labelled $BB$ (but does not contain two black balls)
3. **Urn 3**: Labelled $WW$ (... | 1 | 1 |
user
In one urn, there are two white balls, in another—two black balls, in the third—one white ball and one black ball. Each urn has a label indicating its contents: $W W$, $B B$, $W B$. However, a prankster has swapped all the labels so that now each label incorrectly describes the contents of the urn. You are allowed... | Let's approach this problem step by step to determine the minimum number of draws required to correctly identify the contents of all three urns.
### Understanding the Problem
1. **Initial Setup:**
- **Urn 1:** Label claims it contains \( WW \) (two white balls).
- **Urn 2:** Label claims it contains \( BB \) (t... | 1 | 1 |
user
A school is arranging for 5 trainee teachers, including Xiao Li, to be placed in Class 1, Class 2, and Class 3 for teaching practice. If at least one teacher must be assigned to each class and Xiao Li is to be placed in Class 1, the number of different arrangement schemes is ________ (answer with a number only).
P... | To solve this problem, we need to determine the number of ways to assign 5 trainee teachers, including Xiao Li, to 3 classes (Class 1, Class 2, and Class 3) with the following constraints:
1. At least one teacher must be assigned to each class.
2. Xiao Li is to be placed in Class 1.
### Step 1: Understand the Constra... | 1 | 1 |
user
A school is arranging for 5 trainee teachers, including Xiao Li, to be placed in Class 1, Class 2, and Class 3 for teaching practice. If at least one teacher must be assigned to each class and Xiao Li is to be placed in Class 1, the number of different arrangement schemes is ________ (answer with a number only).
P... | ### Problem Analysis
We need to arrange 5 trainee teachers, including Xiao Li, into 3 classes (Class 1, Class 2, and Class 3) with the following constraints:
1. **At least one teacher must be assigned to each class.**
2. **Xiao Li is to be placed in Class 1.**
### Step 1: Understand the Constraints
- **Total Teache... | 0 | 1 |
user
A school is arranging for 5 trainee teachers, including Xiao Li, to be placed in Class 1, Class 2, and Class 3 for teaching practice. If at least one teacher must be assigned to each class and Xiao Li is to be placed in Class 1, the number of different arrangement schemes is ________ (answer with a number only).
P... | ### Problem Analysis
We need to arrange 5 trainee teachers, including Xiao Li, into 3 classes (Class 1, Class 2, and Class 3) with the following constraints:
1. **At least one teacher must be assigned to each class.**
2. **Xiao Li is to be placed in Class 1.**
### Step 1: Understand the Constraints
- **Total Teache... | 0 | 1 |
user
A school is arranging for 5 trainee teachers, including Xiao Li, to be placed in Class 1, Class 2, and Class 3 for teaching practice. If at least one teacher must be assigned to each class and Xiao Li is to be placed in Class 1, the number of different arrangement schemes is ________ (answer with a number only).
P... | ### Problem Analysis
We need to determine the number of ways to assign 5 trainee teachers, including Xiao Li, to 3 classes (Class 1, Class 2, and Class 3) with the following constraints:
1. **At least one teacher must be assigned to each class.**
2. **Xiao Li is to be placed in Class 1.**
### Step-by-Step Reasoning
... | 1 | 1 |
user
A school is arranging for 5 trainee teachers, including Xiao Li, to be placed in Class 1, Class 2, and Class 3 for teaching practice. If at least one teacher must be assigned to each class and Xiao Li is to be placed in Class 1, the number of different arrangement schemes is ________ (answer with a number only).
P... | ### Problem Analysis
We need to arrange 5 trainee teachers, including Xiao Li, into 3 classes (Class 1, Class 2, and Class 3) with the following constraints:
1. **At least one teacher must be assigned to each class.**
2. **Xiao Li is to be placed in Class 1.**
### Step 1: Understand the Constraints
- **Xiao Li is i... | 0 | 1 |
user
A school is arranging for 5 trainee teachers, including Xiao Li, to be placed in Class 1, Class 2, and Class 3 for teaching practice. If at least one teacher must be assigned to each class and Xiao Li is to be placed in Class 1, the number of different arrangement schemes is ________ (answer with a number only).
P... | To solve this problem, we need to determine the number of ways to arrange 5 trainee teachers, including Xiao Li, into 3 classes (Class 1, Class 2, and Class 3) with the following constraints:
1. At least one teacher must be assigned to each class.
2. Xiao Li is to be placed in Class 1.
### Step 1: Understand the Cons... | 1 | 1 |
user
If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning.
Please reas... | ### Problem Analysis
We are given the concept of a "good number" \( n \): a number \( n \) is called "good" if there exists a permutation \( a_1, a_2, \cdots, a_n \) of \( 1, 2, \cdots, n \) such that \( k + a_k \) (for \( k = 1, 2, \cdots, n \)) are all perfect squares.
We are asked to determine which numbers in the... | 1 | 1 |
user
If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning.
Please reas... | ### Problem Analysis
We need to determine which numbers in the set \(\{11, 13, 15, 17, 19\}\) are "good numbers". A number \(n\) is called a "good number" if there exists a permutation \(a_1, a_2, \cdots, a_n\) of \(1, 2, \cdots, n\) such that \(k + a_k\) is a perfect square for all \(k = 1, 2, \cdots, n\).
### Step-... | 0 | 1 |
user
If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning.
Please reas... | ### Problem Analysis
We are given the definition of a "good number" \( n \): there exists a permutation \( a_1, a_2, \cdots, a_n \) of \( 1, 2, \cdots, n \) such that \( k + a_k \) (for \( k = 1, 2, \cdots, n \)) are all perfect squares. We are asked to determine which numbers in the set \( \{11, 13, 15, 17, 19\} \) a... | 0 | 1 |
user
If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning.
Please reas... | ### Problem Analysis
We are given the following:
- A permutation \( a_1, a_2, \cdots, a_n \) of \( 1, 2, \cdots, n \) such that \( k + a_k \) (for \( k = 1, 2, \cdots, n \)) are all perfect squares.
- The set \( \{11, 13, 15, 17, 19\} \).
- We need to determine which numbers in this set are "good numbers" and which a... | 0 | 1 |
user
If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning.
Please reas... | To determine which numbers in the set \(\{11, 13, 15, 17, 19\}\) are "good numbers," we need to check if there exists a permutation \(a_1, a_2, \cdots, a_n\) of \(1, 2, \cdots, n\) such that \(k + a_k\) is a perfect square for all \(k = 1, 2, \cdots, n\).
### Step-by-Step Reasoning:
#### 1. Understanding the Problem:... | 0 | 1 |
user
If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning.
Please reas... | ### Problem Analysis
We need to determine which numbers in the set \(\{11, 13, 15, 17, 19\}\) are "good numbers". A number \(n\) is called a "good number" if there exists a permutation \(a_1, a_2, \cdots, a_n\) of \(1, 2, \cdots, n\) such that \(k + a_k\) is a perfect square for all \(k = 1, 2, \cdots, n\).
### Step-... | 0 | 1 |
user
If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning.
Please reas... | ### Problem Analysis
We are given the definition of a "good number" \( n \): there exists a permutation \( a_1, a_2, \cdots, a_n \) of \( 1, 2, \cdots, n \) such that \( k + a_k \) (for \( k = 1, 2, \cdots, n \)) are all perfect squares. We are asked to determine which numbers in the set \( \{11, 13, 15, 17, 19\} \) a... | 0 | 1 |
user
If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning.
Please reas... | To determine which numbers in the set \(\{11, 13, 15, 17, 19\}\) are "good numbers," we need to check if there exists a permutation \(a_1, a_2, \cdots, a_n\) of \(1, 2, \cdots, n\) such that \(k + a_k\) is a perfect square for all \(k = 1, 2, \cdots, n\).
### Step-by-Step Reasoning:
#### 1. Understand the Problem:
- ... | 0 | 1 |
user
Compute the number of ways to assemble 2 red unit cubes and 25 white unit cubes into a \(3 \times 3 \times 3\) cube such that the red cubes are visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.)
Please reason step by step, and put your final answer within \boxed{{}}... | ### Problem Analysis
We need to assemble a \(3 \times 3 \times 3\) cube using 2 red unit cubes and 25 white unit cubes. The red cubes must be visible on exactly 4 faces of the larger cube. We need to compute the number of ways to arrange the cubes under these constraints, considering that rotations and reflections are... | 0 | 1 |
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