problem stringlengths 11 2.51k | reasoning stringlengths 708 107k | deepseek_solution stringlengths 3 8.97k | ground_truth_solution stringlengths 190 7.13k | correct bool 2
classes | judge_reasoning stringlengths 88 112k | source stringclasses 1
value | solution stringlengths 194 7.22k | messages stringlengths 309 8.58k | 4o_response stringlengths 3 119k | __original_row_idx int64 0 34.1k | conversations listlengths 2 2 |
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On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could hear. Knights... | Alright, let's try to tackle this problem step by step. So, we have an island with knights, liars, and followers. Knights always tell the truth, liars always lie, and followers answer the same as the majority of the preceding respondents. If there's a tie, they can choose either. The total residents are 2018, each answ... | To determine the maximum number of followers among the 2018 island residents, we need to consider the behavior of knights, liars, and followers. Knights always tell the truth, liars always lie, and followers answer according to the majority of preceding answers. The goal is to maximize the number of followers while ens... |
#### Step 1: Define the Participants
We categorize the inhabitants of the island into three types:
- Knights, who always tell the truth.
- Liars, who always lie.
- Copycats, who respond based on the majority response heard so far.
#### Step 2: Analyze Initial and Final Balances
When dealing with the responses of inha... | false | The solution provided inaccurately concludes the maximum number of followers (copycats) as 2016, whereas the ground truth solution deduces that the correct answer is 1009. The misunderstanding arises in the essential mechanics of how knights and liars influence the responses and ultimately the balance needed to maintai... | olympiads |
#### Step 1: Define the Participants
We categorize the inhabitants of the island into three types:
- Knights, who always tell the truth.
- Liars, who always lie.
- Copycats, who respond based on the majority response heard so far.
#### Step 2: Analyze Initial and Final Balances
When dealing with the responses of inha... | [{'content': 'On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could... | To solve the problem, let's first define the types of residents on the island:
- **Knights** always tell the truth.
- **Liars** always lie.
- **Followers** answer based on the majority response heard so far, or randomly if there is a tie.
Let:
- \( k \) be the number of knights,
- \( l \) be the number of liars,
- \(... | 10,000 | [
{
"from": "human",
"value": "On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer \"Yes\" or \"No\" to the question: \"Are there more knights than liars on the island?\" The residents responded one by one in... |
There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth... | Okay, let's try to figure out this problem. So, we have an island with knights, liars, and followers. Knights always tell the truth, liars always lie, and followers answer the same as the majority of previous answers, or either if there's a tie. There are 2018 islanders arranged in a row, each answered "Yes" or "No" to... | To determine the maximum number of followers, we analyze the roles and their responses. Knights and liars answer truthfully or deceitfully about whether there are more knights than liars. Followers mimic the majority of previous answers or choose arbitrarily when tied. Given 1009 "Yes" answers (exactly half), we need t... |
**Initial Setup:**
We have two types of principled inhabitants on the island: knights (who always tell the truth) and liars (who always lie). In addition, there are parrots (подпевавалы), who mimic the majority's answer up to that point. If the answers are tied, a parrot can choose either.
Given:
- The total number ... | true | The analysis presented in the solution and the ground truth solution reaches the same conclusion regarding the maximum number of parrots (followers) possible. Both solutions correctly reason through the roles of knights, liars, and parrots, leading to the conclusion that with an even split of 'Yes' and 'No' responses, ... | olympiads |
**Initial Setup:**
We have two types of principled inhabitants on the island: knights (who always tell the truth) and liars (who always lie). In addition, there are parrots (подпевавалы), who mimic the majority's answer up to that point. If the answers are tied, a parrot can choose either.
Given:
- The total number ... | [{'content': 'There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights ... | To solve the problem, we need to analyze the implications of the answers given by the knights, liars, and followers regarding the statement "Are there more knights than liars?"
1. **Definitions:**
- Knights ALWAYS tell the truth.
- Liars ALWAYS lie.
- Followers copy the majority of responses given before them... | 10,001 | [
{
"from": "human",
"value": "There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer \"Yes\" or \"No\" to the question: \"Are there more knights on the island than liars?\". They answered in turn such that e... |
"Laila writes a list of numbers. Her first number is 4. Each number after the first is 7 more than t(...TRUNCATED) | "Okay, let's tackle this problem step by step. Hmm, so Laila writes a list of numbers where the firs(...TRUNCATED) | "Laila's sequence starts at 4 and increases by 7 each time. The nth term is given by the arithmetic (...TRUNCATED) | ": \n\n1. First, we observe that Laila's list starts with the number `4` and each subsequent number (...TRUNCATED) | true | "The provided solution accurately analyzes the arithmetic sequence, successfully identifying which o(...TRUNCATED) | olympiads | ": \n\n1. First, we observe that Laila's list starts with the number `4` and each subsequent number (...TRUNCATED) | "[{'content': \"Laila writes a list of numbers. Her first number is 4. Each number after the first i(...TRUNCATED) | "Laila's list of numbers starts at 4 and each subsequent number is 7 more than the previous one. \n\(...TRUNCATED) | 10,002 | [{"from":"human","value":"Laila writes a list of numbers. Her first number is 4. Each number after t(...TRUNCATED) |
"Find the points on the surface of a cube that are equidistant from the endpoints of one of its spac(...TRUNCATED) | "Okay, so I need to find the points on the surface of a cube that are equidistant from the endpoints(...TRUNCATED) | "The points on the surface of a cube that are equidistant from the endpoints of one of its space dia(...TRUNCATED) | "\n1. **Identify the Points**: We need to find the points on the surface of a cube that are equidist(...TRUNCATED) | true | "Both solutions correctly identify that the points on the surface of a cube that are equidistant fro(...TRUNCATED) | olympiads | "\n1. **Identify the Points**: We need to find the points on the surface of a cube that are equidist(...TRUNCATED) | "[{'content': 'Find the points on the surface of a cube that are equidistant from the endpoints of o(...TRUNCATED) | "To find the points on the surface of a cube that are equidistant from the endpoints of one of its s(...TRUNCATED) | 10,003 | [{"from":"human","value":"Find the points on the surface of a cube that are equidistant from the end(...TRUNCATED) |
"Prove that among all the regular $n$-gons inscribed in a given regular $n$-gon (with $n > 3$), the (...TRUNCATED) | "Okay, so I need to prove that among all regular n-gons inscribed in a given regular n-gon (where n (...TRUNCATED) | "To prove that among all regular \\( n \\)-gons inscribed in a given regular \\( n \\)-gon (with \\((...TRUNCATED) | "\n1. Consider two regular $n$-gons: the original one denoted as $A_1 \\cdots A_n$ with area $S_A$, (...TRUNCATED) | false | "While both solutions arrive at the conclusion that the area of the inscribed regular n-gon is minim(...TRUNCATED) | olympiads | "\n1. Consider two regular $n$-gons: the original one denoted as $A_1 \\\\cdots A_n$ with area $S_A$(...TRUNCATED) | "[{'content': 'Prove that among all the regular $n$-gons inscribed in a given regular $n$-gon (with (...TRUNCATED) | "Let \\( P \\) be a regular \\( n \\)-gon inscribed in a circle of radius \\( R \\), centered at the(...TRUNCATED) | 10,004 | [{"from":"human","value":"Prove that among all the regular $n$-gons inscribed in a given regular $n$(...TRUNCATED) |
"Fill in the empty boxes with numbers so that each box in the middle of the marked lines contains th(...TRUNCATED) | "Alright, let's tackle this problem step by step. The goal is to fill the empty boxes with numbers s(...TRUNCATED) | "To solve the problem, we need to fill the empty boxes such that each middle box on the marked lines(...TRUNCATED) | "\n1. **Identify numbers based on the given conditions:**\n - According to the first condition, t(...TRUNCATED) | false | "The solutions provided do not match; the proposed grid from the SOLUTION contains a different set o(...TRUNCATED) | olympiads | "\n1. **Identify numbers based on the given conditions:**\n - According to the first condition, t(...TRUNCATED) | "[{'content': 'Fill in the empty boxes with numbers so that each box in the middle of the marked lin(...TRUNCATED) | "To solve this, we need to create a structure that follows the guidelines you've set. The layout see(...TRUNCATED) | 10,005 | [{"from":"human","value":"Fill in the empty boxes with numbers so that each box in the middle of the(...TRUNCATED) |
A knight made $n$ moves and returned to its original square. Prove that $n$ is an even number. | "Okay, let's try to figure out why a knight that makes n moves and returns to its original square mu(...TRUNCATED) | "To prove that a knight must make an even number of moves, \\( n \\), to return to its original squa(...TRUNCATED) | "\n1. **Understanding the Movement of the Knight**:\n - A knight in chess moves in an \"L\" shape:(...TRUNCATED) | true | "Both solutions correctly establish that a knight must make an even number of moves to return to its(...TRUNCATED) | olympiads | "\n1. **Understanding the Movement of the Knight**:\n - A knight in chess moves in an \"L\" shape:(...TRUNCATED) | "[{'content': 'A knight made $n$ moves and returned to its original square. Prove that $n$ is an eve(...TRUNCATED) | "To prove that a knight's move \\( n \\) must be even when a knight returns to its original square, (...TRUNCATED) | 10,006 | [{"from":"human","value":"A knight made $n$ moves and returned to its original square. Prove that $n(...TRUNCATED) |
"A cyclist planned to travel from point A to point B in 5 hours, moving at a constant speed. He trav(...TRUNCATED) | "Okay, let's see. So the problem is about a cyclist who planned to go from point A to point B in 5 h(...TRUNCATED) | "The cyclist's total journey time is **4.5 hours** (or 4 hours and 30 minutes). Here's the breakdown(...TRUNCATED) | "\n1. **Define the variables and conditions**: Let \\( D \\) be the total distance from point A to p(...TRUNCATED) | false | "The solution provided by the judge calculates the total journey time as 4.5 hours, whereas the grou(...TRUNCATED) | olympiads | "\n1. **Define the variables and conditions**: Let \\\\( D \\\\) be the total distance from point A (...TRUNCATED) | "[{'content': 'A cyclist planned to travel from point A to point B in 5 hours, moving at a constant (...TRUNCATED) | "To solve the problem, let's first define the necessary variables and establish a plan.\n\nLet the t(...TRUNCATED) | 10,007 | [{"from":"human","value":"A cyclist planned to travel from point A to point B in 5 hours, moving at (...TRUNCATED) |
"14 * 49k is a natural number and 1001 * 1002 * ... * 1985 * 1986 / 11^k is an integer. What is the (...TRUNCATED) | "Okay, so I need to find the maximum value of k such that both 14 * 49k is a natural number and the (...TRUNCATED) | "To find the maximum value of \\( k \\) such that \\( 14 \\times 49k \\) is a natural number and \\((...TRUNCATED) | "\nTo determine the largest value of \\( k \\) such that \\(\\frac{1001 \\cdot 1002 \\cdot \\cdots \(...TRUNCATED) | true | "The solution correctly calculates the exponent of 11 in the factorials of 1986 and 1000 using Legen(...TRUNCATED) | olympiads | "\nTo determine the largest value of \\\\( k \\\\) such that \\\\(\\\\frac{1001 \\\\cdot 1002 \\\\cd(...TRUNCATED) | "[{'content': '14 * 49k is a natural number and 1001 * 1002 * ... * 1985 * 1986 / 11^k is an integer(...TRUNCATED) | "To determine the maximum value of \\( k \\) such that \\( 14 \\times 49^k \\) is a natural number a(...TRUNCATED) | 10,008 | [{"from":"human","value":"14 * 49k is a natural number and 1001 * 1002 * ... * 1985 * 1986 / 11^k is(...TRUNCATED) |
"Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to sch(...TRUNCATED) | "Okay, let me try to figure out this problem. So, Nikita usually leaves home at 8:00 AM, gets into U(...TRUNCATED) | "Nikita usually leaves home at 8:00 AM and arrives at school on time. On Friday, he left home at 7:1(...TRUNCATED) | "\nTo find out how many times the speed of Uncle Vanya's car exceeded Nikita's running speed, we can(...TRUNCATED) | true | "Both the solution and the ground truth solution arrive at the same conclusion that Uncle Vanya's sp(...TRUNCATED) | olympiads | "\nTo find out how many times the speed of Uncle Vanya's car exceeded Nikita's running speed, we can(...TRUNCATED) | "[{'content': \"Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle dr(...TRUNCATED) | "To solve the problem, we first need to define some variables and analyze the situation step-by-step(...TRUNCATED) | 10,009 | [{"from":"human","value":"Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and hi(...TRUNCATED) |
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