problem stringlengths 17 4.42k | reasoning stringlengths 1.2k 97.8k | deepseek_solution stringlengths 0 5.46k | ground_truth_solution stringlengths 100 7.96k | correct bool 1
class | judge_reasoning stringlengths 92 15.5k | system stringclasses 1
value | conversations listlengths 2 2 |
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Given a square \(ABCD\) with point \(P\) inside such that \(PA = 1\), \(PB = 2\), and \(PC = 3\), calculate the angle \(\widehat{APB}\). | Okay, so I have this geometry problem here: There's a square ABCD with a point P inside such that PA = 1, PB = 2, and PC = 3. I need to find the angle APB, which is the angle at point P between points A and B. Hmm, sounds a bit tricky, but let me try to work through it step by step.
First, let me visualize the square.... | Given a square \(ABCD\) with point \(P\) inside such that \(PA = 1\), \(PB = 2\), and \(PC = 3\), we need to calculate the angle \(\widehat{APB}\).
1. **Assigning Coordinates:**
- Place the square on a coordinate system with \(A(0,0)\), \(B(a,0)\), \(C(a,a)\), and \(D(0,a)\).
- Let \(P\) have coordinates \((x,y)... |
1. **Rotation and Initial Observations:**
Let $\rho$ be the rotation centered at $B$ with an angle of $90^\circ$. This rotation sends point $C$ to point $A$, keeps $B$ fixed, and sends $P$ to a point $P'$ such that $\angle PBP' = 90^\circ$. Additionally, since this is a rotation by $90^\circ$, the distance $P'B$ wi... | true | The provided solution correctly derives the angle \\widehat{APB} using appropriate geometric principles involving the coordinates and distance relations. Both the solution and the ground truth arrive at the same conclusion of \\boxed{135^ullet}. Therefore, the solution is accurate. | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
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"value": "Return your final response within \\boxed{}. Given a square \\(ABCD\\) with point \\(P\\) inside such that \\(PA = 1\\), \\(PB = 2\\), and \\(PC = 3\\), calculate the angle \\(\\widehat{APB}\\)."
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"value": "<|begin_of_thought|>\n\nOkay, so I h... |
Petya and Misha start on a circular track from the same point, running in a counterclockwise direction. Both run at constant speeds, with Misha's speed being $2\%$ greater than Petya's speed. Petya always runs counterclockwise, while Misha can change direction at any moment, provided he has run at least half a lap or m... | Okay, let's try to tackle this problem step by step. So, we have Petya and Misha starting at the same point on a circular track, both initially running counterclockwise. Misha's speed is 2% faster than Petya's. The key point here is that Misha can change direction at any moment, but only if he has run at least half a l... | Petya and Misha start running counterclockwise on a circular track from the same point. Misha's speed is 2% greater than Petya's. The goal is to show that Misha can meet Petya three times while Petya completes his first lap, excluding the starting point.
1. **Initial Setup and Speeds**:
- Petya's speed: \( v \)
... |
1. **Define Variables:**
Let:
- $v_P$ be Petya's running speed.
- $v_M$ be Misha's running speed.
- $L$ be the length of the circular track.
2. **Speed Relationship:**
According to the problem, Misha’s speed is $2\%$ more than Petya’s speed:
\[
v_M = 1.02 \cdot v_P
\]
3. **First Encounter:**
... | true | Both the provided solution and the ground truth solution correctly demonstrate that Misha can meet Petya three times as he completes his first lap on the track. The calculations of distances, times, and speeds align, resulting in the same conclusion of three meetings. Thus, the evaluation of the solution's correctness ... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
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"value": "Return your final response within \\boxed{}. Petya and Misha start on a circular track from the same point, running in a counterclockwise direction. Both run at constant speeds, with Misha's speed being $2\\%$ greater than Petya's speed. Petya always runs counterclockwise, while ... |
"In $\\triangle ABC$, given $a + c = 3b$, with incenter $I$, the incircle touches sides $AB$ and $BC(...TRUNCATED) | "Okay, so I need to prove that points A, C, K, and L are concyclic in triangle ABC where a + c = 3b.(...TRUNCATED) | "To prove that points \\(A\\), \\(C\\), \\(K\\), and \\(L\\) are concyclic in \\(\\triangle ABC\\) w(...TRUNCATED) | "1. **Identify Initial Context and Given Information:**\n In $\\triangle ABC$, we have \\(a + c = (...TRUNCATED) | true | "The solution presented correctly follows the reasoning and steps needed to demonstrate that points (...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. In $\\triangle ABC$, given $a (...TRUNCATED) |
How many positive two-digit integers are there in which each of the two digits is prime? | "Okay, so the problem is asking how many positive two-digit integers there are where each of the two(...TRUNCATED) | "To determine how many positive two-digit integers exist where each digit is prime, we start by iden(...TRUNCATED) | "To solve this problem, we first identify all the prime digits. The prime numbers less than 10 are:\(...TRUNCATED) | true | "The solution accurately identifies the prime digits and correctly calculates the number of two-digi(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. How many positive two-digit in(...TRUNCATED) |
"Let \\( n \\) and \\( k \\) be positive integers, and \\( S \\) be a set of \\( n \\) points in a p(...TRUNCATED) | "Okay, so I need to prove that for a set S of n points in a plane, with no three collinear and each (...TRUNCATED) | "Given a set \\( S \\) of \\( n \\) points in a plane where no three points are collinear, and each (...TRUNCATED) | ": \n\n1. **Initialization and Setup**:\n Let $S = \\{P_1, P_2, \\ldots, P_n\\}$ be a set of $n$ p(...TRUNCATED) | true | "Both the proposed solution and the ground truth solution reach the same conclusion that \\( k < \\f(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Let \\( n \\) and \\( k \\) be(...TRUNCATED) |
"Prove that the product of the first \\( n \\) terms of a geometric progression is equal to \\( a_{1(...TRUNCATED) | "Okay, so I need to prove that the product of the first n terms of a geometric progression (GP) is e(...TRUNCATED) | "The product of the first \\( n \\) terms of a geometric progression (GP) with first term \\( a_1 \\(...TRUNCATED) | "\nTo show that the product of \\( n \\) terms of a geometric progression is equal to \\( a_{1}^{n} (...TRUNCATED) | true | "The provided solution correctly outlines the mathematical proof that the product of the first n ter(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Prove that the product of the (...TRUNCATED) |
What is the smallest integer $n$, greater than $1$, such that $n^{-1}\pmod{1050}$ is defined? | "Okay, so I need to find the smallest integer n, greater than 1, such that the inverse of n modulo 1(...TRUNCATED) | "To determine the smallest integer \\( n \\), greater than 1, such that \\( n^{-1} \\pmod{1050} \\) (...TRUNCATED) | "To find the smallest integer $n > 1$ such that $n^{-1} \\pmod{1050}$ is defined, we need to ensure (...TRUNCATED) | true | "The provided solution correctly identifies that to find the smallest integer greater than 1 that is(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. What is the smallest integer $(...TRUNCATED) |
"Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make l(...TRUNCATED) | "Okay, let's try to figure out how many calories are in 200 grams of Francesca's lemonade. Hmm, the (...TRUNCATED) | "Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonad(...TRUNCATED) | "1. **Calculate Total Calories in the Lemonade Mixture**:\n Francesca uses $100$ grams of lemon ju(...TRUNCATED) | true | "Both the solution and the ground truth arrive at the same conclusion regarding the calorie content (...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Francesca uses $100$ grams of (...TRUNCATED) |
"Andrew flips a fair coin $5$ times, and counts the number of heads that appear. Beth flips a fair(...TRUNCATED) | "Okay, let's see. The problem is: Andrew flips a fair coin 5 times, counts the number of heads. Beth(...TRUNCATED) | "Andrew flips a fair coin 5 times, and Beth flips a fair coin 6 times. We need to compute the probab(...TRUNCATED) | "To solve this problem, we need to compute the probability that Andrew counts at least as many heads(...TRUNCATED) | true | "Both the provided solution and the ground truth solution compute the probability that Andrew counts(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Andrew flips a fair coin $5$ (...TRUNCATED) |
"A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle tha(...TRUNCATED) | "Okay, let's see. I need to figure out how much larger the area of the large semicircles is compared(...TRUNCATED) | "A rectangle measures 6 meters by 10 meters. On each side of the rectangle, a semicircle is drawn wi(...TRUNCATED) | "To solve this problem, we start by calculating the areas of the semicircles drawn on each side of t(...TRUNCATED) | true | "Both the provided solution and the ground truth solution correctly calculate the areas of the semic(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. A rectangle measures 6 meters (...TRUNCATED) |
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