problem stringlengths 59 5.02k | reasoning stringlengths 1.51k 124k | deepseek_solution stringlengths 0 5.06k | ground_truth_solution stringlengths 97 9.16k | correct bool 2
classes | judge_reasoning stringlengths 31 1.55k | source stringclasses 1
value | solution stringlengths 97 9.16k | messages stringlengths 325 10.4k | 4o_response stringlengths 286 16.7k | __original_row_idx int64 0 17.3k | conversations listlengths 2 2 |
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A $3\times 6$ grid is filled with the numbers in the list $\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}$ according to the following rules: (1) Both the first three columns and the last three columns contain the integers 1 through 9. (2) No numbers appear more than once in a given row. Let $N$ be the number of ways... | Okay, let's try to figure out this problem. It's about filling a 3x6 grid with numbers from the set {1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9} following two rules. Rule 1 says that both the first three columns and the last three columns must contain the integers 1 through 9. So each set of three columns (columns 1-3 and col... | To solve the problem of determining the largest positive integer \( k \) such that \( 2^k \) divides the number of ways \( N \) to fill a \( 3 \times 6 \) grid according to the given rules, we follow these steps:
1. **Understanding the Grid Constraints**:
- The grid must be filled with numbers from the set \(\{1,1,... | 1. **Understanding the problem**: We need to fill a $3 \times 6$ grid with the numbers from the list $\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}$ such that:
- Both the first three columns and the last three columns contain the integers 1 through 9.
- No numbers appear more than once in a given row.
- We need to f... | false | The provided solution and the ground truth solution differ in how they calculate the total number of ways to fill the grid and the resulting power of 2 that divides it. The correct total of the ways calculated in the response was based on a miscalculation of valid configurations for the second block. The ground truth s... | aops_forum | 1. **Understanding the problem**: We need to fill a $3 \times 6$ grid with the numbers from the list $\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}$ such that:
- Both the first three columns and the last three columns contain the integers 1 through 9.
- No numbers appear more than once in a given row.
- We need to f... | [{'content': 'A $3\\times 6$ grid is filled with the numbers in the list $\\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\\}$ according to the following rules: (1) Both the first three columns and the last three columns contain the integers 1 through 9. (2) No numbers appear more than once in a given row. Let $N$ be t... | To fill the \(3 \times 6\) grid with numbers from the list
\[
\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}
\]
we need to arrange the numbers across three columns on the left and three columns on the right, satisfying two main conditions:
1. Each column must contain the integers \(1\) through \(9\).
2. No row can have du... | 10,000 | [
{
"from": "human",
"value": "A $3\\times 6$ grid is filled with the numbers in the list $\\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\\}$ according to the following rules: (1) Both the first three columns and the last three columns contain the integers 1 through 9. (2) No numbers appear more than once in a... |
Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\ldots,S_n$ such that:
$ 1^\circ$ $ |S_i\cup S_j|\leq 2004$ for each two $ 1\leq i,j\le n$ and
$ 2^\circ$ $ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\}$ for each three integers $ 1\le i<j<k\le n$ . | Okay, let's tackle this problem. It's asking for the largest natural number n such that there exist different sets Sโ, Sโ, ..., Sโ satisfying two conditions:
1. For any two sets S_i and S_j, the union |S_i โช S_j| โค 2004.
2. For any three sets S_i, S_j, S_k, their union is the entire set {1, 2, ..., 2008}.
Hmm, so eac... | The largest natural number \( n \) is \( \boxed{32} \).
**Explanation:**
1. **Key Constraints:**
- Each pair of sets \( S_i \) and \( S_j \) must have a union of at most 2004 elements.
- Any three sets must cover the entire universe \( \{1, 2, \ldots, 2008\} \).
2. **Element Missing Analysis:**
- Each eleme... | 1. **Define the universal set and complementary sets:**
Let \( U = \{1, 2, \ldots, 2008\} \) and define \( A_k = U \setminus S_k \) for each \( k \). This means \( S_k = U \setminus A_k \).
2. **Reformulate the conditions:**
- The first condition \( |S_i \cup S_j| \leq 2004 \) translates to \( |U \setminus (A_i ... | true | The solution correctly followed logical reasoning to derive that the largest natural number n that meets the problem's conditions is indeed 32. Both the given solution and the ground truth solution arrive at the same conclusion, using complementary approaches based on set theory and combinatorial arguments. | aops_forum | 1. **Define the universal set and complementary sets:**
Let \( U = \{1, 2, \ldots, 2008\} \) and define \( A_k = U \setminus S_k \) for each \( k \). This means \( S_k = U \setminus A_k \).
2. **Reformulate the conditions:**
- The first condition \( |S_i \cup S_j| \leq 2004 \) translates to \( |U \setminus (A_i ... | [{'content': 'Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\\ldots,S_n$ such that:\r\n\r $ 1^\\circ$ $ |S_i\\cup S_j|\\leq 2004$ for each two $ 1\\leq i,j\\le n$ and\r $ 2^\\circ$ $ S_i\\cup S_j\\cup S_k\\equal{}\\{1,2,\\ldots,2008\\}$ for each three integers $ 1\\le ... | To start solving the problem, we analyze the two conditions given:
1. The first condition states that the union of any two sets in our collection of sets \( S_i \) must have at most 2004 elements. This means that for any two distinct sets \( S_i \) and \( S_j \), the number of distinct elements in their union is limit... | 10,001 | [
{
"from": "human",
"value": "Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\\ldots,S_n$ such that:\r\n\r $ 1^\\circ$ $ |S_i\\cup S_j|\\leq 2004$ for each two $ 1\\leq i,j\\le n$ and\r $ 2^\\circ$ $ S_i\\cup S_j\\cup S_k\\equal{}\\{1,2,\\ldots,2008\\}$ for ea... |
Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called *discrete coloring* if there exists a circle which encloses all red points and excludes all white points. Determine the number o... | Okay, so I have this problem here about discrete colorings of a set of points S in the plane. The conditions are that any three points aren't colinear and any four aren't concyclic. I need to determine the number of discrete colorings for each such set S. Let me try to break this down step by step.
First, let's make s... | Given a set \( S \) of finite points in the plane where no three points are colinear and no four points are concyclic, we need to determine the number of discrete colorings. A discrete coloring is defined as a coloring of the points such that there exists a circle enclosing all red points and excluding all white points... | 1. **Understanding the Problem:**
We are given a set \( S \) of \( n \) points in the plane, where any 3 points are not collinear and any 4 points are not concyclic. We need to determine the number of ways to color these points red and white such that there exists a circle enclosing all red points and excluding all ... | false | The provided solution claims that the number of discrete colorings is \( 2^n \), which assumes all subsets of points can be colored without restriction. However, the ground truth solution correctly identifies that the formula for calculating the number of discrete colorings based on the geometric constraints involves c... | aops_forum | 1. **Understanding the Problem:**
We are given a set \( S \) of \( n \) points in the plane, where any 3 points are not collinear and any 4 points are not concyclic. We need to determine the number of ways to color these points red and white such that there exists a circle enclosing all red points and excluding all ... | [{'content': 'Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called *discrete coloring* if there exists a circle which encloses all red points and excludes all white points. Determin... | To determine the number of *discrete colorings* for a set \( S \) of finite points in the plane with the properties that any three points are not collinear and any four points are not concyclic, we start by analyzing the implications of these properties.
### Properties of the Set \( S \)
1. **No three points are coll... | 10,002 | [
{
"from": "human",
"value": "Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called *discrete coloring* if there exists a circle which encloses all red points and excludes... |
"Pedrito's lucky number is $34117$ . His friend Ramanujan points out that $34117 = 166^2 + 81^2 = (...TRUNCATED) | "Okay, so Pedrito's lucky number is 34117, which can be written in two different ways as the sum of (...TRUNCATED) | "Pedrito's lucky number is 34117, which can be written in two different ways as the sum of squares: (...TRUNCATED) | "To solve the problem, we need to find the smallest number \\( N \\) that can be expressed as the su(...TRUNCATED) | true | "Both the solution and the ground truth arrive at the same result for the least lucky number, which (...TRUNCATED) | aops_forum | "To solve the problem, we need to find the smallest number \\( N \\) that can be expressed as the su(...TRUNCATED) | "[{'content': \"Pedrito's lucky number is $34117$ . His friend Ramanujan points out that $34117 = (...TRUNCATED) | "To solve the problem, we are looking for positive integers \\( a, b, c, d \\) such that:\n\n\\[\na^(...TRUNCATED) | 10,003 | [{"from":"human","value":"Pedrito's lucky number is $34117$ . His friend Ramanujan points out that (...TRUNCATED) |
"A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_(...TRUNCATED) | "Okay, so I need to find all positive integers n that are called \"good.\" A good number n can be un(...TRUNCATED) | "To determine all good natural numbers, we need to find positive integers \\( n \\) that can be uniq(...TRUNCATED) | "1. **Understanding the Problem:**\n We need to find all positive integers \\( n \\) that can be u(...TRUNCATED) | true | "Both solutions correctly identify that good natural numbers can be expressed as either the square o(...TRUNCATED) | aops_forum | "1. **Understanding the Problem:**\n We need to find all positive integers \\( n \\) that can be u(...TRUNCATED) | "[{'content': 'A positive integer $ n$ is called $ good$ if it can be uniquely written simultane(...TRUNCATED) | "To identify all positive integers \\( n \\) that are classified as good, we begin by analyzing the (...TRUNCATED) | 10,004 | [{"from":"human","value":"A positive integer $ n$ is called $ good$ if it can be uniquely writte(...TRUNCATED) |
"A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop h(...TRUNCATED) | "Okay, so I have this problem where a grasshopper starts at the origin (0,0) on a coordinate plane. (...TRUNCATED) | "The grasshopper starts at the origin (0,0) and needs to reach the point (2021, 2021) using hops of (...TRUNCATED) | "1. **Understanding the Problem:**\n The grasshopper starts at the origin \\((0,0)\\) and makes ho(...TRUNCATED) | true | "The solution provided correctly analyzes the hops required for the grasshopper to reach the point ((...TRUNCATED) | aops_forum | "1. **Understanding the Problem:**\n The grasshopper starts at the origin \\((0,0)\\) and makes ho(...TRUNCATED) | "[{'content': 'A grasshopper starts at the origin in the coordinate plane and makes a sequence of ho(...TRUNCATED) | "To find the smallest number of hops needed for the grasshopper to reach the point \\((2021, 2021)\\(...TRUNCATED) | 10,005 | [{"from":"human","value":"A grasshopper starts at the origin in the coordinate plane and makes a seq(...TRUNCATED) |
"You know that the binary function $\\diamond$ takes in two non-negative integers and has the foll(...TRUNCATED) | "Alright, let's try to figure out this binary function $\\diamond$. So, the problem states that for (...TRUNCATED) | "To find a general formula for \\( x \\diamond y \\) where \\( y \\geq x > 0 \\), we start by analyz(...TRUNCATED) | "To find a general formula for \\( x \\diamond y \\) given the properties of the binary function \\((...TRUNCATED) | true | "The provided solution correctly deduces the general formula for the binary operation \\( x \\diamon(...TRUNCATED) | aops_forum | "To find a general formula for \\( x \\diamond y \\) given the properties of the binary function \\((...TRUNCATED) | "[{'content': 'You know that the binary function $\\\\diamond$ takes in two non-negative integers (...TRUNCATED) | "To derive a general formula for the binary function \\( \\diamond \\) defined by the properties giv(...TRUNCATED) | 10,006 | [{"from":"human","value":"You know that the binary function $\\diamond$ takes in two non-negative (...TRUNCATED) |
"Find all natural number which can be expressed in\r $ \\frac{a\\plus{}b}{c}\\plus{}\\frac{b\\plus{}(...TRUNCATED) | "Okay, let's see. I need to find all natural numbers that can be expressed in the form (a + b)/c + ((...TRUNCATED) | "The natural numbers that can be expressed in the form \\( \\frac{a+b}{c} + \\frac{b+c}{a} + \\frac{(...TRUNCATED) | "1. We start with the given expression:\n \\[\n \\frac{a+b}{c} + \\frac{b+c}{a} + \\frac{c+a}{b}(...TRUNCATED) | false | "The solution suggests the numbers 6, 7, and 8 are the only achievable results, yet analysis in the (...TRUNCATED) | aops_forum | "1. We start with the given expression:\n \\[\n \\frac{a+b}{c} + \\frac{b+c}{a} + \\frac{c+a}{b}(...TRUNCATED) | "[{'content': 'Find all natural number which can be expressed in\\r $ \\\\frac{a\\\\plus{}b}{c}\\\\p(...TRUNCATED) | "To find all natural numbers that can be expressed in the form\n\n\\[\n\\frac{a+b}{c} + \\frac{b+c}{(...TRUNCATED) | 10,007 | [{"from":"human","value":"Find all natural number which can be expressed in\r $ \\frac{a\\plus{}b}{c(...TRUNCATED) |
"The difference of fractions $\\frac{2024}{2023} - \\frac{2023}{2024}$ was represented as an irred(...TRUNCATED) | "Okay, so I need to find the difference between the two fractions 2024/2023 and 2023/2024, and then (...TRUNCATED) | "To find the difference between the fractions \\(\\frac{2024}{2023}\\) and \\(\\frac{2023}{2024}\\) (...TRUNCATED) | "1. Start with the given expression:\n \\[\n \\frac{2024}{2023} - \\frac{2023}{2024}\n \\]\n\n(...TRUNCATED) | true | "The provided solution accurately follows the steps needed to find the difference between the fracti(...TRUNCATED) | aops_forum | "1. Start with the given expression:\n \\[\n \\frac{2024}{2023} - \\frac{2023}{2024}\n \\]\n\n(...TRUNCATED) | "[{'content': 'The difference of fractions $\\\\frac{2024}{2023} - \\\\frac{2023}{2024}$ was repre(...TRUNCATED) | "To find the difference of the fractions \\(\\frac{2024}{2023} - \\frac{2023}{2024}\\), we first nee(...TRUNCATED) | 10,008 | [{"from":"human","value":"The difference of fractions $\\frac{2024}{2023} - \\frac{2023}{2024}$ wa(...TRUNCATED) |
"Sets $A_0, A_1, \\dots, A_{2023}$ satisfies the following conditions:\n\n\n- $A_0 = \\{ 3 \\}$\n-(...TRUNCATED) | "Alright, let's tackle this problem step by step. So, we have these sets Aโ, Aโ, ..., Aโโโ(...TRUNCATED) | "The sets \\( A_0, A_1, \\dots, A_{2023} \\) are defined recursively, starting with \\( A_0 = \\{3\\(...TRUNCATED) | "1. **Initialization**: We start with the set \\( A_0 = \\{3\\} \\).\n\n2. **Recursive Definition**:(...TRUNCATED) | true | "The provided solution correctly follows the recursive definition of the sets, calculates the sizes (...TRUNCATED) | aops_forum | "1. **Initialization**: We start with the set \\( A_0 = \\{3\\} \\).\n\n2. **Recursive Definition**:(...TRUNCATED) | "[{'content': 'Sets $A_0, A_1, \\\\dots, A_{2023}$ satisfies the following conditions:\\n\\n\\n- $(...TRUNCATED) | "Let's analyze the sets \\( A_n \\) and determine their contents step by step.\n\n### Base Case: \\((...TRUNCATED) | 10,009 | [{"from":"human","value":"Sets $A_0, A_1, \\dots, A_{2023}$ satisfies the following conditions:\n\(...TRUNCATED) |
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