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class | generations_count int64 1 8 ⌀ | correctness_math_verify sequencelengths 1 8 ⌀ | correct_count int64 0 8 ⌀ | generation sequencelengths 1 8 ⌀ | generation_extracted_answer sequencelengths 1 8 ⌀ | verified_generation stringlengths 74 74.3k ⌀ |
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[
{
"from": "human",
"value": "A convex polygon with $n$ sides has its interior angles forming an arithmetic sequence, with a common difference $d=10^\\circ$, and the smallest interior angle is $100^\\circ$. Then $n=$ （ ）\nA: 5 or 6\nB: 9\nC: 8\nD: 8 or 9"
},
{
"from": "gpt",
"value": "<think>Okay... | A convex polygon with $n$ sides has its interior angles forming an arithmetic sequence, with a common difference $d=10^\circ$, and the smallest interior angle is $100^\circ$. Then $n=$ （ ）
A: 5 or 6
B: 9
C: 8
D: 8 or 9 | Let the sequence of the degrees of the interior angles be $\{a_n\}$, then $a_1=100^\circ$, $d=10^\circ$
Thus, $a_n=a_1+(n-1)d=100^\circ+(n-1)\cdot10^\circ<180^\circ$
Therefore, $n<9$
Hence, the number of sides is 8
Therefore, the correct choice is $\boxed{\text{C}}$. | \text{C} | Geometry | MCQ | Yes | Incomplete | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, so I need to solve this problem about a convex polygon where the interior angles form an arithmetic sequence with a common difference of 10 degrees, and the smallest angle is 100 degrees. The question is asking for the number of sides n, and the options are given as A: 5 or 6, B: 9, C: 8, D: 8 or 9.... | [
"C",
"C"
] | <think>Okay, so I need to solve this problem about a convex polygon with n sides where the interior angles form an arithmetic sequence. The common difference is 10 degrees, and the smallest interior angle is 100 degrees. The question is asking for n, and the options are A: 5 or 6, B: 9, C: 8, D: 8 or 9.
First, let me ... |
[
{
"from": "human",
"value": "A certain school wants to investigate whether students' preferences for a certain sport are related to their gender. By randomly surveying 110 students, the school obtained the following $2\\times 2$ contingency table:\n\n| Gender | Attitude towards the Sport | Total |\n|-------... | A certain school wants to investigate whether students' preferences for a certain sport are related to their gender. By randomly surveying 110 students, the school obtained the following $2\times 2$ contingency table:
| Gender | Attitude towards the Sport | Total |
|--------|-----------------------------|-------|
| Ma... | To solve this problem, we will use the given contingency table and the formula for ${K^2}$. The contingency table provided is as follows:
| Gender | Likes the Sport | Does not like the Sport | Total |
|--------|-----------------|-------------------------|-------|
| Male | 40 | 20 | ... | D | Other | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, let's see. So the problem is about determining whether there's a relationship between gender and preference for a sport based on a survey of 110 students. They've given a contingency table and calculated the K² statistic as approximately 7.82. Then there's a table of critical values for different si... | [
"D",
"D"
] | <think>
Okay, let me try to figure out this statistics problem. So, the school did a survey of 110 students about their preference for a sport and gender. They presented the data in a 2x2 contingency table. Then they calculated K² ≈ 7.82 using that formula given. Now, we need to use the provided table of critical value... |
[
{
"from": "human",
"value": "Two fair octahedral dice, each with the numbers $1$ through $8$ on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by $8$. Find the expected value of $N$."
},
{
"from": "gpt",
"value": "<think>\nOka... | Two fair octahedral dice, each with the numbers $1$ through $8$ on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by $8$. Find the expected value of $N$. |
To find the expected value of \( N \), where \( N \) is the remainder when the product of the numbers showing on two fair octahedral dice is divided by 8, we need to consider all possible outcomes of rolling two dice and calculate the remainders of their products modulo 8. Each die has faces numbered from 1 to 8, so t... | \frac{75}{16} | NaN | NaN | NaN | NaN | aops_forum | false | 2 | [
false,
false
] | 0 | [
"<think>\nOkay, so I need to find the expected value of N, where N is the remainder when the product of two numbers rolled on two fair octahedral dice (each with numbers 1 through 8) is divided by 8. Hmm, let's see. \n\nFirst, let me recall that the expected value is calculated by summing each possible outcome mult... | [
"\\frac{11}{4}",
"\\frac{11}{4}"
] | <think>
Okay, so I need to find the expected value of N, where N is the remainder when the product of two numbers rolled on two fair octahedral dice (each with numbers 1 through 8) is divided by 8. Hmm, let's see.
First, let me recall that the expected value is calculated by summing each possible outcome multiplied b... |
[{"from":"human","value":"Given vectors $\\overrightarrow{m}$ and $\\overrightarrow{n}$, it is known(...TRUNCATED) | "Given vectors $\\overrightarrow{m}$ and $\\overrightarrow{n}$, it is known that for any $\\lambda \(...TRUNCATED) | "Given the consistent inequality $|\\overrightarrow{m}-\\lambda (\\overrightarrow{m}-\\overrightarro(...TRUNCATED) | B | Inequalities | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, let's try to tackle this problem step by step. So, we have two vectors, m and n, an(...TRUNCATED) | [
"B",
"B"
] | "<think>\nOkay, let's try to tackle this problem step by step. So, we have two vectors, m and n, and(...TRUNCATED) |
[{"from":"human","value":"11. Real numbers $r, s, t$ satisfy $1 \\leqslant r \\leqslant s \\leqslant(...TRUNCATED) | "11. Real numbers $r, s, t$ satisfy $1 \\leqslant r \\leqslant s \\leqslant t \\leqslant 4$, then $((...TRUNCATED) | "11. $12-8 \\sqrt{2}$ Since $(x-1)^{2}-((2 \\sqrt{2}-2) x-1)=(x-\\sqrt{2})^{2} \\geqslant 0$, we hav(...TRUNCATED) | 12-8\sqrt{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 4 | [
true,
true,
true,
true
] | 4 | ["<think>\nOkay, let's see. I have this problem where I need to find the minimum value of this expre(...TRUNCATED) | [
"12 - 8\\sqrt{2}",
"12 - 8\\sqrt{2}",
"12 - 8\\sqrt{2}",
"12 - 8\\sqrt{2}"
] | "<think>\nOkay, so I need to find the minimum value of this expression: \\((r-1)^2 + \\left(\\frac{s(...TRUNCATED) |
[{"from":"human","value":"Let $x$ and $y$ be positive real numbers that satisfy $x \\leqslant 2$, $y(...TRUNCATED) | "Let $x$ and $y$ be positive real numbers that satisfy $x \\leqslant 2$, $y \\leqslant 3$, and $x + (...TRUNCATED) | "Given $x + y = 3$, we have $y = 3 - x$.\n\nSubstitute $y$ into the expression $4x^3 + y^3$:\n\n$4x^(...TRUNCATED) | 33 | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
false,
false
] | 0 | ["<think>\nOkay, so I need to find the maximum value of the expression \\(4x^3 + y^3\\) given the co(...TRUNCATED) | [
"C",
"C"
] | "<think>\nOkay, so I need to find the maximum value of the expression \\(4x^3 + y^3\\) given the con(...TRUNCATED) |
[{"from":"human","value":"Find the minimum value of the function $f(x)=|3-x|+|x-2|$.\n\nA: $-1$\nB: (...TRUNCATED) | Find the minimum value of the function $f(x)=|3-x|+|x-2|$.
A: $-1$
B: $2$
C: $1$
D: $3$ | "By the triangle inequality for absolute values, we have\n$$f(x)=|3-x|+|x-2|\\geqslant |(3-x)+(x-2)|(...TRUNCATED) | C | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, let's see. I need to find the minimum value of the function f(x) = |3 - x| + |x - 2(...TRUNCATED) | [
"C",
"C"
] | "<think>\nOkay, let's see. I need to find the minimum value of the function f(x) = |3 - x| + |x - 2|(...TRUNCATED) |
[{"from":"human","value":"Given vectors $$\\overrightarrow {a}$$\\=(1, -1) and $$\\overrightarrow {b(...TRUNCATED) | "Given vectors $$\\overrightarrow {a}$$\\=(1, -1) and $$\\overrightarrow {b}$$\\=(-2, 3), and $$\\ov(...TRUNCATED) | "We have $$\\overrightarrow {a}+m \\overrightarrow {b}=(1-2m, 3m-1)$$.\n\nSince $$\\overrightarrow {(...TRUNCATED) | A | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, so I have this problem here where I need to find the value of m such that vector a (...TRUNCATED) | [
"A",
"A"
] | "<think>\nOkay, so I have this problem here where I need to find the value of m such that vector a i(...TRUNCATED) |
[{"from":"human","value":"(Stage Valbonne 2018)\n\nLet $ABCD$ be a parallelogram. Let $H$ be the ort(...TRUNCATED) | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
[{"from":"human","value":"If $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightar(...TRUNCATED) | "If $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=5$, $|\\overright(...TRUNCATED) | "To find the maximum and minimum values of $|\\overrightarrow{a}+\\overrightarrow{b}|$, we start by (...TRUNCATED) | 3 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, so I have this problem here about vectors. It says that vectors a and b have magnit(...TRUNCATED) | [
"3",
"3"
] | "<think>\nOkay, so I have this problem here about vectors. It says that vectors a and b have magnitu(...TRUNCATED) |
End of preview. Expand in Data Studio
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