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conversations list | problem string | solution string | answer string | problem_type string | question_type string | problem_is_valid string | solution_is_valid string | source string | synthetic bool | generations_count int64 | correctness_math_verify sequence | correct_count int64 | generation sequence | correct sequence | judge_reasoning string | __original_row_idx int64 | _majority_responses sequence | verified_generation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
[
{
"from": "human",
"value": "The ratio of the three interior angles of $\\triangle ABC$ is 1:1:1, then $\\triangle ABC$ is ( ) \nA: Isosceles triangle \nB: Right triangle \nC: Equilateral triangle \nD: Obtuse triangle"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to solve this pr... | The ratio of the three interior angles of $\triangle ABC$ is 1:1:1, then $\triangle ABC$ is ( )
A: Isosceles triangle
B: Right triangle
C: Equilateral triangle
D: Obtuse triangle | Let $\angle A = x$. Given that the ratio of the three interior angles of $\triangle ABC$ is 1:1:1, we know $\angle B = \angle C = x$,
Therefore, $\angle A + \angle B + \angle C = 3x = 180^\circ$,
Therefore, $x = 60^\circ$.
Therefore, $\angle A = \angle B = \angle C = 60^\circ$,
Therefore, $\triangle ABC$ is an equi... | C | Geometry | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, so I have this problem here: The ratio of the three interior angles of triangle ABC is 1:1:1, then triangle ABC is which type of triangle? The options are A: Isosceles, B: Right triangle, C: Equilateral triangle, D: Obtuse triangle. Hmm, let me think through this step by step.\n\nFirst, I remember t... | [
true,
true
] | The model correctly reasoned that since all angles in the triangle measure 60 degrees, it defines the triangle as equilateral. The answer choice C matches the ground truth solution. | 495,161 | [
"<think>\nOkay, so I have this problem here: The ratio of the three interior angles of triangle ABC is 1:1:1, then triangle ABC is which type of triangle? The options are A: Isosceles, B: Right triangle, C: Equilateral triangle, D: Obtuse triangle. Hmm, let me think through this step by step.\n\nFirst, I remember t... | <think>
Okay, let me try to solve this problem step by step. So, the question is about a triangle ABC where the ratio of its three interior angles is 1:1:1. We need to determine what type of triangle this is, with the options being isosceles, right, equilateral, or obtuse. Let me recall what each of these terms means.
... |
[
{
"from": "human",
"value": "18. As shown in Figure 5, in $\\triangle ABC$, $BC=6$, $AC=4\\sqrt{2}$, $\\angle C=45^{\\circ}$, $P$ is a moving point on side $BC$. A line through $P$ parallel to $AB$ intersects $AC$ at point $D$. Connect $AP$. The areas of $\\triangle ABP$, $\\triangle APD$, and $\\triangle C... | 18. As shown in Figure 5, in $\triangle ABC$, $BC=6$, $AC=4\sqrt{2}$, $\angle C=45^{\circ}$, $P$ is a moving point on side $BC$. A line through $P$ parallel to $AB$ intersects $AC$ at point $D$. Connect $AP$. The areas of $\triangle ABP$, $\triangle APD$, and $\triangle CDP$ are denoted as $S_{1}$, $S_{2}$, and $S_{3}$... | 18. (1) From the problem, we know that $B P=x, 0<x<6$, and $\frac{A D}{B P}=\frac{A C}{B C}$,
so $A D=\frac{A C}{B C} \cdot B P=\frac{4 \sqrt{2}}{6} x=\frac{2 \sqrt{2}}{3} x$.
$S_{2}=S_{\triangle A P D}=\frac{1}{2} A D \cdot P M$
$=\frac{1}{2} \times \frac{2 \sqrt{2}}{3} x \times \frac{\sqrt{2}}{2}(6-x)=2 x-\frac{1}{3... | S_{1}=S_{2}+S_{3}, S_{2}=S_{3}, S_{1}=2 S_{2}, S_{1}=2 S_{3} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 2 | [
false,
false
] | 0 | [
"<think>\nOkay, let's tackle this geometry problem step by step. First, I need to understand what's given and what needs to be found. \n\nWe have triangle ABC with BC = 6, AC = 4√2, and angle C = 45 degrees. Point P is moving along side BC, and there's a line through P parallel to AB intersecting AC at point D. The... | [
true,
true
] | The model accurately solved the geometry problem by correctly expressing the areas S1, S2, and S3 in terms of x and also provided the correct relationships between the areas when S2 is maximized. All calculations are consistent with geometric principles. | 1,008,963 | [
"<think>\nOkay, let's tackle this geometry problem step by step. First, I need to understand what's given and what needs to be found. \n\nWe have triangle ABC with BC = 6, AC = 4√2, and angle C = 45 degrees. Point P is moving along side BC, and there's a line through P parallel to AB intersecting AC at point D. The... | <think>
Okay, let's tackle this geometry problem step by step. First, I need to understand what's given and what needs to be found.
We have triangle ABC with BC = 6, AC = 4√2, and angle C = 45 degrees. Point P is moving along side BC, and there's a line through P parallel to AB intersecting AC at point D. Then we con... |
[
{
"from": "human",
"value": "Example 5 Let $x_{i}(i=0,1, \\cdots, n)$ be positive numbers, $n \\geqslant 2, n \\in \\mathbf{N}$, then\n$$\n\\left(\\frac{x_{0}}{x_{1}}\\right)^{n}+\\left(\\frac{x_{1}}{x_{2}}\\right)^{n}+\\cdots+\\left(\\frac{x_{n-1}}{x_{n}}\\right)^{n}+\\left(\\frac{x_{n}}{x_{0}}\\right)^{n}... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
[{"from":"human","value":"89. The blackboard is written with a number $\\underbrace{222 \\cdots 2}_{(...TRUNCATED) | "89. The blackboard is written with a number $\\underbrace{222 \\cdots 2}_{9 \\uparrow 2}$, and the (...TRUNCATED) | answer: 6 | 6 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 4 | [
true,
true,
true,
true
] | 4 | ["<think>\nOkay, let's see. The problem starts with a number that's made up of nine 2s. So that's 22(...TRUNCATED) | [
true,
true,
true,
true
] | "The final answer calculated for the sequence of operations based on the specified rules resulted in(...TRUNCATED) | 186,178 | ["<think>\nOkay, let's see. The problem starts with a number that's made up of nine 2s. So that's 22(...TRUNCATED) | "<think>\nOkay, let me try to figure out this problem. So, we start with a number that's made up of (...TRUNCATED) |
[{"from":"human","value":"1. Given that the real number $x$ satisfies $20 \\sin x=22 \\cos x$, then (...TRUNCATED) | "1. Given that the real number $x$ satisfies $20 \\sin x=22 \\cos x$, then the greatest integer not (...TRUNCATED) | "$$\n\\tan x=\\frac{11}{10} \\Rightarrow \\sin 2 x=\\frac{2 \\tan x}{1+\\tan ^{2} x}=\\frac{\\frac{1(...TRUNCATED) | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 8 | [
true,
true,
true,
true,
true,
true,
true,
true
] | 8 | ["<think>\nOkay, let's see. The problem says that for a real number x, 20 sin x equals 22 cos x. The(...TRUNCATED) | [
true,
true,
true,
true,
true,
true,
true,
true
] | "The solution correctly derives the values needed and computes the final answer as 1, which matches (...TRUNCATED) | 198,805 | ["<think>\nOkay, let's see. The problem says that for a real number x, 20 sin x equals 22 cos x. The(...TRUNCATED) | "<think>\nOkay, let's see. I need to find the greatest integer not exceeding the value of \\(\\left((...TRUNCATED) |
[{"from":"human","value":"Given vector $\\overrightarrow{a}=(\\sin α, \\cos α - 2\\sin α)$ and ve(...TRUNCATED) | "Given vector $\\overrightarrow{a}=(\\sin α, \\cos α - 2\\sin α)$ and vector $\\overrightarrow{b}(...TRUNCATED) | "Since $\\overrightarrow{a} \\parallel \\overrightarrow{b}$, their corresponding components are prop(...TRUNCATED) | \frac{1}{4} | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, so I need to find the value of tan α given that the vector a is parallel to vector(...TRUNCATED) | [
true,
true
] | The solution accurately deduces that an α = 1/4, which matches the given ground truth solution. | 473,980 | ["<think>\nOkay, so I need to find the value of tan α given that the vector a is parallel to vector(...TRUNCATED) | "<think>\nOkay, so I need to find the value of tan α given that the vector a is parallel to vector (...TRUNCATED) |
[{"from":"human","value":"6. As shown in Figure 1, the radius of $\\odot O$ is $6, M$ is a point out(...TRUNCATED) | "6. As shown in Figure 1, the radius of $\\odot O$ is $6, M$ is a point outside $\\odot O$, and $O M(...TRUNCATED) | "6. C.\n\nAs shown in Figure 4, extend $M O$ to intersect $\\odot O$ at point $N$, and connect $O B$(...TRUNCATED) | C | Geometry | MCQ | Yes | Yes | cn_contest | false | 8 | [
true,
true,
true,
true,
true,
true,
true,
true
] | 8 | ["<think>\nOkay, so I need to find the length of OP in this geometry problem. Let me try to visualiz(...TRUNCATED) | [
true,
true,
true,
true,
true,
true,
true,
true
] | "The provided solution correctly illustrates the logic and calculations for determining the length o(...TRUNCATED) | 1,034,275 | ["<think>\nOkay, so I need to find the length of OP in this geometry problem. Let me try to visualiz(...TRUNCATED) | "<think>\nOkay, so I need to find the length of OP in this geometry problem. Let me try to visualize(...TRUNCATED) |
[{"from":"human","value":"Given $\\tan \\alpha=2$, find the value of $$\\frac{\\sin (\\pi+\\alpha)-\(...TRUNCATED) | "Given $\\tan \\alpha=2$, find the value of $$\\frac{\\sin (\\pi+\\alpha)-\\cos (\\pi-\\alpha)}{\\si(...TRUNCATED) | "This problem requires us to use cofunction identities to simplify and find the value. Proficiency i(...TRUNCATED) | -\frac{1}{3} | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nAlright, so I have this trigonometry problem here. Let me see. The problem says that give(...TRUNCATED) | [
true,
true
] | "The solution accurately derives the expression from the trigonometric identities and correctly simp(...TRUNCATED) | 568,329 | ["<think>\nAlright, so I have this trigonometry problem here. Let me see. The problem says that give(...TRUNCATED) | "<think>\nAlright, so I have this trigonometry problem here. Let me see. The problem says that given(...TRUNCATED) |
[{"from":"human","value":"19.2 .11 ** Write all natural numbers in sequence after the decimal point (...TRUNCATED) | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
[{"from":"human","value":"Given points A(2,5) and B(1,6), what is the slope angle of line AB?\n\nA: (...TRUNCATED) | "Given points A(2,5) and B(1,6), what is the slope angle of line AB?\n\nA: $\\frac{3π}{4}$\n\nB: $\(...TRUNCATED) | "To find the slope angle of line AB, we first calculate the slope of the line using the coordinates (...TRUNCATED) | \text{A} | Geometry | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, so I need to find the slope angle of the line AB given points A(2,5) and B(1,6). Hm(...TRUNCATED) | [
true,
true
] | "The solution correctly identifies the slope of the line and computes the angle with respect to the (...TRUNCATED) | 459,674 | ["<think>\nOkay, so I need to find the slope angle of the line AB given points A(2,5) and B(1,6). Hm(...TRUNCATED) | "<think>\nOkay, so I need to find the slope angle of the line AB given points A(2,5) and B(1,6). Hmm(...TRUNCATED) |
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