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The dataset generation failed
Error code: DatasetGenerationError
Exception: CastError
Message: Couldn't cast
answer: string
subject: string
level: int64
unique_id: string
data_source: string
prompt: list<element: struct<content: string, role: string>>
child 0, element: struct<content: string, role: string>
child 0, content: string
child 1, role: string
ability: string
reward_model: struct<ground_truth: string, style: string>
child 0, ground_truth: string
child 1, style: string
extra_info: struct<index: int64, split: string>
child 0, index: int64
child 1, split: string
-- schema metadata --
huggingface: '{"info": {"features": {"answer": {"dtype": "string", "_type' + 610
to
{'data_source': Value('string'), 'prompt': List({'content': Value('string'), 'role': Value('string')}), 'ability': Value('string'), 'golden_answer': Value('string'), 'reward_model': {'ground_truth': Value('string'), 'style': Value('string')}, 'extra_info': {'index': Value('int64'), 'split': Value('string')}}
because column names don't match
Traceback: Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1405, in compute_config_parquet_and_info_response
fill_builder_info(builder, hf_endpoint=hf_endpoint, hf_token=hf_token, validate=validate)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 578, in fill_builder_info
) = retry_validate_get_features_num_examples_size_and_compression_ratio(
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 497, in retry_validate_get_features_num_examples_size_and_compression_ratio
validate(pf)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 535, in validate
raise TooBigRowGroupsError(
worker.job_runners.config.parquet_and_info.TooBigRowGroupsError: Parquet file has too big row groups. First row group has 513085229 which exceeds the limit of 300000000
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1815, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 692, in wrapped
for item in generator(*args, **kwargs):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/parquet/parquet.py", line 106, in _generate_tables
yield f"{file_idx}_{batch_idx}", self._cast_table(pa_table)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/parquet/parquet.py", line 73, in _cast_table
pa_table = table_cast(pa_table, self.info.features.arrow_schema)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/table.py", line 2272, in table_cast
return cast_table_to_schema(table, schema)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/table.py", line 2218, in cast_table_to_schema
raise CastError(
datasets.table.CastError: Couldn't cast
answer: string
subject: string
level: int64
unique_id: string
data_source: string
prompt: list<element: struct<content: string, role: string>>
child 0, element: struct<content: string, role: string>
child 0, content: string
child 1, role: string
ability: string
reward_model: struct<ground_truth: string, style: string>
child 0, ground_truth: string
child 1, style: string
extra_info: struct<index: int64, split: string>
child 0, index: int64
child 1, split: string
-- schema metadata --
huggingface: '{"info": {"features": {"answer": {"dtype": "string", "_type' + 610
to
{'data_source': Value('string'), 'prompt': List({'content': Value('string'), 'role': Value('string')}), 'ability': Value('string'), 'golden_answer': Value('string'), 'reward_model': {'ground_truth': Value('string'), 'style': Value('string')}, 'extra_info': {'index': Value('int64'), 'split': Value('string')}}
because column names don't match
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1428, in compute_config_parquet_and_info_response
parquet_operations, partial, estimated_dataset_info = stream_convert_to_parquet(
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 994, in stream_convert_to_parquet
builder._prepare_split(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1702, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1858, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
datasets.exceptions.DatasetGenerationError: An error occurred while generating the datasetNeed help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
data_source string | prompt list | ability string | golden_answer string | reward_model dict | extra_info dict |
|---|---|---|---|---|---|
math_verify_numina | [
{
"content": "Tommy is looking at his change collection. He has some dimes and pennies. He has twice as many nickels as dimes. He has 4 quarters. He has 10 times as many pennies as quarters. He has 100 nickels. How many more dimes does he have than pennies? Let's think step by step and output the final answer w... | math | Tommy has 100 nickels, and since he has twice as many nickels as dimes, we can find the number of dimes by dividing the number of nickels by 2.
Number of dimes = 100 nickels / 2 = 50 dimes
He has 4 quarters, and since he has 10 times as many pennies as quarters, we can find the number of pennies by multiplying the nu... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 0,
"split": "train"
} |
math_verify_numina | [
{
"content": "A point $R$ is located outside of circle $\\omega$ with center $O$ and radius $10$ units. Find the locus of the midpoint $M$ of the segment from $R$ to a point $S$ as $S$ varies over circle $\\omega$, where the segment $RS$ is scaled by $\\frac{1}{3}$ towards $S$.\nA) Circle with center at $\\left... | math | 1. **Understanding the Transformation**: A homothety centered at $R$ with scale factor $\frac{1}{3}$ is applied to point $S$ on circle $\omega$. The midpoint $M$ of segment $RS$ is where this transformation maps $S$ to.
2. **New Circle Properties**:
- **Center**: The new center $N$ of the circle is the image of $O$... | {
"ground_truth": "\\text{B}",
"style": "rule"
} | {
"index": 1,
"split": "train"
} |
math_verify_numina | [
{
"content": "Diego can carry a certain amount of fruit home in his bookbag. He buys a pound of watermelon, a pound of grapes, a pound of oranges, and 17 pounds of apples. What is the total weight of fruit Diego can carry in his bookbag? Let's think step by step and output the final answer within \\boxed{}.",
... | math | To find the total weight of the fruit Diego can carry in his bookbag, we need to add the weight of each type of fruit he bought:
Weight of watermelon = 1 pound
Weight of grapes = 1 pound
Weight of oranges = 1 pound
Weight of apples = 17 pounds
Total weight = Weight of watermelon + Weight of grapes + Weight of oranges... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 2,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given the function $$f(x)= \\begin{cases} a^{x}, x<0 \\\\ ( \\frac {1}{4}-a)x+2a, x\\geq0\\end{cases}$$ such that for any $x\\_1 \\neq x\\_2$, the inequality $$\\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}}<0$$ holds true. Determine the range of values for the real number $a$. Let's think step by step an... | math | Since the function satisfies $$\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}}<0$$ for any $x\_1 \neq x\_2$, it implies that the function is strictly decreasing.
For $x < 0$, we have $f(x) = a^x$. Since $a^x$ is strictly decreasing for $0 < a < 1$, we get $0 < a < 1$.
For $x \geq 0$, we have $f(x) = (\frac{1}{4} - a)x + 2a$. ... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 3,
"split": "train"
} |
math_verify_numina | [
{
"content": "For a cubic function $f(x)=ax^3+bx^2+cx+d\\ (a\\neq 0)$, define: if the equation $f''(x)=0$ has a real solution $x_0$, then the point $(x_0, f(x_0))$ is called an \"inflection point\" of the function $y=f(x)$. A student found that \"any cubic function has an 'inflection point'; any cubic function ... | math | 1. First, we need to find the inflection point of the given cubic function $f(x)=x^3- \frac{3}{2}x^2+3x-\frac{1}{4}$ by setting the second derivative to zero.
The first derivative is:
\[
f'(x)=3x^2-3x+3,
\]
and the second derivative is:
\[
f''(x)=6x-3.
\]
Setting $f''(x)=0$ gives us $x=\f... | {
"ground_truth": "2012",
"style": "rule"
} | {
"index": 4,
"split": "train"
} |
math_verify_numina | [
{
"content": "Write an \\( n \\)-digit number using the digits 1 and 2, such that no two consecutive digits are both 1. Denote the number of such \\( n \\)-digit numbers as \\( f(n) \\). Find \\( f(10) \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math |
Let $f(n)$ be the number of $n$-digit numbers formed using digits 1 and 2, where no two consecutive digits are both 1.
1. **Base Cases:**
- For $n=1$, we can have either 1 or 2. So, $f(1)=2$.
- For $n=2$, the possible numbers are 12, 21, and 22. Hence, $f(2)=3$.
2. **Recurrence Relation:**
- Consider the... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": 5,
"split": "train"
} |
math_verify_numina | [
{
"content": "Let $f(x)$ be a function defined on $R$. For any $x_{1}$, $x_{2} \\in R$, it holds that $f(x_{1}+x_{2})=f(x_{1})+f(x_{2})+1$. Also, when $x > 0$, $f(x) > f(0)$ always holds. Which of the following statements is correct?\n\nA: $f(0)=-1$\n\nB: The monotonic increasing interval of the function $f(x)$... | math | To analyze the problem and the given properties of the function $f(x)$, we proceed in a step-by-step manner, closely following the provided solution.
1. **Determining $f(0)$**:
Given that $f(x_{1}+x_{2})=f(x_{1})+f(x_{2})+1$ for any $x_{1}, x_{2} \in R$, we set $x_{1}=x_{2}=0$. This gives us:
\begin{align*}
f... | {
"ground_truth": "ACD",
"style": "rule"
} | {
"index": 6,
"split": "train"
} |
math_verify_numina | [
{
"content": "Points \\( E \\) and \\( F \\) are chosen on the sides \\( BC \\) and \\( CD \\) of the square \\( ABCD \\) in such a way that the angle \\( EAF \\) is \\( 45^\\circ \\). The side length of the square is 1. Find the perimeter of the triangle \\( CEF \\). Justify your solution. Let's think step by ... | math |
1. **Recognize the Given Information**:
We have a square $ABCD$ with side lengths of 1. Points $E$ and $F$ are selected on sides $BC$ and $CD$ respectively, such that the angle $\angle EAF$ is $45^\circ$.
2. **Reflect Point $D$**:
Reflect point $D$ over the line $AF$. Let the image be $D'$. Likewise, reflect ... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 7,
"split": "train"
} |
math_verify_numina | [
{
"content": "Billy and Bobbi each selected a positive integer less than 150. Billy's number is a multiple of 15, and Bobbi's number is a multiple of 20. What is the probability that they selected the same number, given that at least one of their numbers is a square? Express your answer as a common fraction. Le... | math | First, find the least common multiple (LCM) of 15 and 20. The prime factorizations are $15 = 3 \times 5$ and $20 = 2^2 \times 5$, thus the LCM is $2^2 \times 3 \times 5 = 60$. An integer is a common multiple of 15 and 20 if and only if it is a multiple of 60.
Next, calculate how many multiples of 60 there are less tha... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 8,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given \\\\(m,n\\\\) represent two different lines, and \\\\(\\alpha\\\\) represents a plane, which of the following statements is correct? \\\\((\\quad)\\)\nA: If \\\\(m\\parallel\\alpha\\\\) and \\\\(n\\parallel\\alpha\\\\), then \\\\(m\\parallel n\\\\)\nB: If \\\\(m\\perp \\alpha\\\\) and \\\\(n... | math | **Analysis**
This question examines the spatial relationship between lines and planes, the criteria for parallelism between a line and a plane, the criteria for perpendicularity between a line and a plane, and the properties of perpendicularity between a line and a plane.
Using the spatial relationship between lines ... | {
"ground_truth": "\\text{B}",
"style": "rule"
} | {
"index": 9,
"split": "train"
} |
math_verify_numina | [
{
"content": "There are five different books, of which two are Chinese language books, two are math books, and one is a physics book. If they are randomly placed side by side on the same shelf, the probability that the books of the same subject are adjacent to each other is ( ).\n\nA: $$\\frac {1}{5}$$\nB: $$\\... | math | The total number of different books is five, including two Chinese language books, two math books, and one physics book. To determine the probability that books of the same subject are adjacent when randomly placed side by side on the same shelf:
First, we calculate the total number of basic events (n), which represen... | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": 10,
"split": "train"
} |
math_verify_numina | [
{
"content": "Heather is going to sew some aprons for a kiddie crew program. She already was able to sew 13 aprons, and today, she sewed three times as many aprons. Heather needs to sew 49 aprons tomorrow if she wants to sew half of the remaining number of aprons needed. How many aprons does Heather need to sew... | math | Heather sewed 13 aprons initially and then sewed three times as many today. So today, she sewed 3 * 13 = 39 aprons.
The total number of aprons she has sewn so far is 13 (initial) + 39 (today) = 52 aprons.
If Heather needs to sew 49 aprons tomorrow to sew half of the remaining number of aprons needed, this means there... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": 11,
"split": "train"
} |
math_verify_numina | [
{
"content": "A group of men decided to do a work in 50 days. Some of them became absent, and the rest of the group did the work in 60 days. The original number of men was 60. How many men became absent? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Let's assume that the original group of 60 men could do the entire work in 50 days. This means that the work done by one man in one day is \( \frac{1}{60 \times 50} \) of the total work.
Now, if some men became absent and the remaining men completed the work in 60 days, we can say that the work done by the remaining m... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 12,
"split": "train"
} |
math_verify_numina | [
{
"content": "Find the greatest value of $x$ such that \\[\\frac{x^2 - x - 90}{x-9} = \\frac{2}{x+7}.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | The expression when simplified by factoring the numerator transforms into:
\[
\frac{(x-9)(x+10)}{x-9} = \frac{2}{x+7}.
\]
Canceling the $(x-9)$ factor on both sides, provided $x \neq 9$, we get:
\[
x + 10 = \frac{2}{x + 7}.
\]
Multiplying both sides by $(x+7)$ to eliminate the fraction yields:
\[
(x + 10)(x + 7) = 2.
\... | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": 13,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given that the complex number $z$ satisfies $(z-3i)(2+i)=5i$ (where $i$ is the imaginary unit), find the conjugate of $z$.\n\nA: $(-2-2i)$\nB: $(-2+2i)$\nC: $(2-2i)$\nD: $(2+2i)$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Analysis:
This problem tests the multiplication and division operations of complex numbers in algebraic form and examines basic concepts of complex numbers. This is a basic-level question.
Solution Enhancement:
First, isolate $z$ by dividing both sides by $(2+i)$:
$$z-3i=\frac{5i}{2+i}.$$
To eliminate the imaginary n... | {
"ground_truth": "2-5i",
"style": "rule"
} | {
"index": 14,
"split": "train"
} |
math_verify_numina | [
{
"content": "How many even natural-number factors does \\( n = 2^3 \\cdot 5^1 \\cdot 11^2 \\) have? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Every factor of \( n \) is in the form \( 2^a \cdot 5^b \cdot 11^c \) where \( 0 \le a \le 3 \), \( 0 \le b \le 1 \), and \( 0 \le c \le 2 \). To find the number of even factors, the power of 2 must be at least 1 (\( 1 \le a \le 3 \)). The number of choices for \( a \) is 3 (since \( a \) can be 1, 2, or 3), the number... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": 15,
"split": "train"
} |
math_verify_numina | [
{
"content": "When Erick went to the market to sell his fruits, he realized that the price of lemons had risen by $4 for each lemon. The price of grapes had also increased by half the price that the price of lemon increased by per grape. He had planned to sell the lemons at $8 and the grapes at $7, and he had 8... | math | First, let's calculate the new price for each lemon. The price of lemons increased by $4, so the new price per lemon is $8 (original price) + $4 (increase) = $12 per lemon.
Next, let's calculate the new price for each grape. The price of grapes increased by half the price that the price of lemon increased by, which is... | {
"ground_truth": "\\$2220",
"style": "rule"
} | {
"index": 16,
"split": "train"
} |
math_verify_numina | [
{
"content": "You need to subtract 46 from a certain number, but mistakenly subtracted 64, and got 122. Find the result of the correct calculation. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | If you mistakenly subtracted 64 instead of 46 and got 122, you can find the original number by adding 64 back to 122:
122 + 64 = 186
Now, to find the result of the correct calculation, you need to subtract 46 from the original number:
186 - 46 = 140
So, the result of the correct calculation is $\boxed{140}$ . | {
"ground_truth": "140",
"style": "rule"
} | {
"index": 17,
"split": "train"
} |
math_verify_numina | [
{
"content": "Determine the difference between a 7.5% sales tax and a 6.25% sales tax on an item priced at $50 before tax.\nA) $0.50\nB) $0.625\nC) $0.75\nD) $1.00\nE) $1.25 Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | 1. **Convert percentages to decimal form**:
- 7.5% as a decimal is 0.075
- 6.25% as a decimal is 0.0625
2. **Calculate the sales tax for each rate**:
- Sales tax at 7.5% is \(50 \times 0.075 = 3.75\)
- Sales tax at 6.25% is \(50 \times 0.0625 = 3.125\)
3. **Calculate the difference in sales tax**:
- Di... | {
"ground_truth": "B",
"style": "rule"
} | {
"index": 18,
"split": "train"
} |
math_verify_numina | [
{
"content": "A retailer buys a radio for Rs 225. His overhead expenses are Rs 30. He sells the radio for some amount. The profit percent of the retailer is 17.64705882352942%. What is the selling price of the radio? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}... | math | To calculate the selling price of the radio, we first need to determine the total cost for the retailer, which includes the purchase price and the overhead expenses.
Total cost = Purchase price + Overhead expenses
Total cost = Rs 225 + Rs 30
Total cost = Rs 255
The retailer makes a profit of 17.64705882352942% on the... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": 19,
"split": "train"
} |
math_verify_numina | [
{
"content": "In the polar coordinate system, the distance from the center of the circle $\\rho = 4 \\cos \\theta \\left( \\,\\rho \\in R \\, \\right)$ to the line $\\theta = \\frac{\\pi}{3}$ is\nA: $\\sqrt{3}$\nB: $2\\sqrt{3}$\nC: $1$\nD: $2$ Let's think step by step and output the final answer within \\boxed{... | math | **Step 1:** Convert the polar coordinate equation to a standard Cartesian equation.
Given circle equation, $\rho = 4 \cos \theta$, squaring both sides gives us $\rho^2 = 4 \rho \cos \theta$.
Now, converting to Cartesian coordinates, we have $x^2 + y^2 = 4x$.
**Step 2:** Identify the circle's center.
Completing the ... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": 20,
"split": "train"
} |
math_verify_numina | [
{
"content": "What is the base ten equivalent of $45321_{7}$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | We are given the number $45321_{7}$ and need to find its base ten equivalent. The process involves expanding the number with respect to powers of 7 and calculating each term's value.
$$45321_{7} = 1 \cdot 7^0 + 2 \cdot 7^1 + 3 \cdot 7^2 + 5 \cdot 7^3 + 4 \cdot 7^4$$
$$45321_{7} = 1 \cdot 1 + 2 \cdot 7 + 3 \cdot 49 + 5... | {
"ground_truth": "11481",
"style": "rule"
} | {
"index": 21,
"split": "train"
} |
math_verify_numina | [
{
"content": "Try to fill a $4 \\times 4$ grid with 16 non-zero numbers so that the sum of the numbers in each row, each column, and each diagonal (including all 14 diagonals that pass through 1, 2, or 3 cells) equals zero. Let's think step by step and output the final answer within \\boxed{}.",
"role": "us... | math | ### Problem:
Fill a $4 \times 4$ grid with 16 non-zero numbers such that the sum of the numbers in each row, each column, and every diagonal (including those crossing 1, 2, or 3 cells—the total being 14 diagonals) is equal to zero.
1. Consider the $4 \times 4$ grid. Let us denote the entries of the grid as follows:
... | {
"ground_truth": "\\text{A possible solution is:}",
"style": "rule"
} | {
"index": 22,
"split": "train"
} |
math_verify_numina | [
{
"content": "A trader bought a car at a 20% discount on its original price, with an additional 5% depreciation per year for 2 years. He also paid 3% of the purchase price in taxes and 2% of the purchase price in insurance fees for each year. Furthermore, he spent 1% of the original price on maintenance. After ... | math | Let's break down the problem step by step:
1. The trader bought the car at a 20% discount on its original price.
2. The car depreciated by 5% per year for 2 years.
3. The trader paid 3% of the purchase price in taxes.
4. The trader paid 2% of the purchase price in insurance fees for each year (so 4% for 2 years).
5. T... | {
"ground_truth": "33.96\\%",
"style": "rule"
} | {
"index": 23,
"split": "train"
} |
math_verify_numina | [
{
"content": "Greg's mom made some cookies for the family and cut them in half. Greg ate some halves, his brother, Brad, had 6 halves, and there were 18 halves left. Greg's mom made 14 whole cookies. How many halves did Greg eat? Let's think step by step and output the final answer within \\boxed{}.",
"role... | math | If Greg's mom made 14 whole cookies and cut them in half, there would be a total of 14 x 2 = 28 halves.
Brad ate 6 halves, and there were 18 halves left. So, the total number of halves eaten by Greg and Brad together is 6 (Brad's halves) + 18 (left halves) = 24 halves.
Since there were originally 28 halves, and 24 ha... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 24,
"split": "train"
} |
math_verify_numina | [
{
"content": "When positive integer x is divided by positive integer y, the remainder is some number. If x / y = 6.12, and the value of y is 49.99999999999996, what is the remainder when x is divided by y? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | To find the remainder when x is divided by y, we can use the following relationship:
x = y * quotient + remainder
Given that x / y = 6.12, we can say that the quotient is 6 (since the quotient must be an integer, and the decimal part represents the fraction of y that will be the remainder).
Now, let's find the value... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 25,
"split": "train"
} |
math_verify_numina | [
{
"content": "A number, when divided by 125, gives a remainder of 40. When the same number is divided by another value, the remainder is 11. What is the value that the number is divided by to get a remainder of 11? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Let's call the number in question "N" and the unknown divisor "D".
When N is divided by 125, the remainder is 40. This means that N can be expressed as:
N = 125k + 40
where k is some integer.
Now, when N is divided by D, the remainder is 11. This means that N can also be expressed as:
N = Dm + 11
where m is some... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": 26,
"split": "train"
} |
math_verify_numina | [
{
"content": "Rachel strolled to Nicholas's house. It is 10 miles from Rachel's house to Nicholas's house. Rachel went at a speed of 2 miles per hour. How long did it take Rachel to get there? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | To find out how long it took Rachel to get to Nicholas's house, we need to divide the total distance by her speed.
Distance = 10 miles
Speed = 2 miles per hour
Time = Distance / Speed
Time = 10 miles / 2 miles per hour
Time = 5 hours
So, it took Rachel $\boxed{5}$ hours to get to Nicholas's house. | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 27,
"split": "train"
} |
math_verify_numina | [
{
"content": "An order was placed for the supply of a carpet whose breadth was 6 m and length was 1.44 times the breadth. What would be the cost of a carpet whose length and breadth are 42.7% more and 27.5% more respectively than the first carpet? Given that the cost of the carpet is RS. 46.35 per sq. m. Let's ... | math | First, let's calculate the area of the first carpet.
The breadth of the first carpet is 6 m, and the length is 1.44 times the breadth, so the length is:
1.44 * 6 m = 8.64 m
The area of the first carpet is:
Area = length * breadth
Area = 8.64 m * 6 m = 51.84 sq. m
Now, let's calculate the dimensions of the second car... | {
"ground_truth": "4371.78",
"style": "rule"
} | {
"index": 28,
"split": "train"
} |
math_verify_numina | [
{
"content": "In triangle $PQR$, $PQ = 21$, $PR = 28$, $QR = 35$, and $PS$ is an angle bisector. Find the ratio of the area of triangle $PQS$ to the area of triangle $PRS$. (Express your answer as a fraction in lowest terms.) Let's think step by step and output the final answer within \\boxed{}.",
"role": "... | math | - By the properties of an angle bisector in a triangle:
- The ratio of the areas of triangle $PQS$ to triangle $PRS$ is equal to the ratio $QS/RS$.
- By the Angle Bisector Theorem applied to triangle $PQR$:
- $QS/RS = PQ/PR = 21/28 = \frac{3}{4}$.
- Simplify the fraction $21/28$ by dividing both the numerator ... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": 29,
"split": "train"
} |
math_verify_numina | [
{
"content": "Schoolchildren were given 7 sheets of paper and asked to cut some of them into 7 pieces. The resulting pieces of paper were mixed and they were again asked to cut some of them into 7 pieces. This process was repeated several times. How many pieces of paper need to be cut in order to end up with 33... | math |
Let us denote by \(k\) the number of times the pieces of paper were cut. Initially, we start with 7 pieces of paper.
1. **Observe the effect of one cut on the number of pieces:**
Each piece that is cut into 7 parts will result in 6 additional pieces since 1 piece is turned into 7 pieces (6 new pieces are created ... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": 30,
"split": "train"
} |
math_verify_numina | [
{
"content": "John wants to lose weight. He eats 1800 calories a day and burns 2300 a day. If he needs to burn 4000 calories to lose 1 pound how many days will it take to lose 10 pounds? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | To solve this problem, we follow these steps:
1. First, we calculate the net calories John burns each day by subtracting the calories he eats from the calories he burns:
\[2300 \text{ calories burned} - 1800 \text{ calories eaten} = 500 \text{ calories net burned per day}\]
2. Next, we calculate the total number of ... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": 31,
"split": "train"
} |
math_verify_numina | [
{
"content": "One-fourth of a number is greater than one-fifth of the number succeeding it by 1. Find the number. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Let's denote the number we are looking for as \( x \). According to the problem, one-fourth of this number is greater than one-fifth of the number succeeding it by 1. The number succeeding \( x \) is \( x + 1 \).
So, we can write the equation as:
\[ \frac{1}{4}x = \frac{1}{5}(x + 1) + 1 \]
Now, let's solve for \( x ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": 32,
"split": "train"
} |
math_verify_numina | [
{
"content": "In a square with a side length of 1, there are 100 figures placed such that their total area is greater than 99. Prove that there exists a point in the square that belongs to all these figures. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | 1. Let us denote the 100 given figures by $\mathrm{A}_{1}, \mathrm{A}_{2}, \ldots, \mathrm{A}_{100}$, and their areas by $S_{1}, S_{2}, \ldots, S_{100}$ respectively.
According to the problem, we have:
\[
S_{1} + S_{2} + \ldots + S_{100} > 99
\]
2. We introduce $\mathrm{B}_{i}$ as the figure that is th... | {
"ground_truth": "",
"style": "rule"
} | {
"index": 33,
"split": "train"
} |
math_verify_numina | [
{
"content": "A theater has 80 seats. On opening night, they sold out, selling adult tickets for 12 dollars each and child tickets for 5 dollars each. They sold 63 child tickets. How much money did they make in total? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... | math | First, we need to find out how many adult tickets were sold. Since there are 80 seats and they sold out, and we know they sold 63 child tickets, we can subtract the number of child tickets from the total number of seats to find the number of adult tickets sold:
80 seats - 63 child tickets = 17 adult tickets
Now we ca... | {
"ground_truth": "\\$519",
"style": "rule"
} | {
"index": 34,
"split": "train"
} |
math_verify_numina | [
{
"content": "If the random variable $X$ follows a binomial distribution with $P(X=0)=0.8$ and $P(X=1)=0.2$, and let $Y=3X-2$, then $P(Y=-2)=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | To solve this, we set $Y=-2$, which implies $X=0$. Therefore, $P(Y=-2)=P(X=0)=0.8$.
Hence, the answer is $\boxed{0.8}$.
This question tests the understanding of the binomial distribution of the random variable $X$ and examines the student's computational ability. It is quite basic. | {
"ground_truth": "0.8",
"style": "rule"
} | {
"index": 35,
"split": "train"
} |
math_verify_numina | [
{
"content": "Find the smallest positive real number $x$ such that\n\\[\\lfloor x^2 \\rfloor - x \\lfloor x \\rfloor = 4.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Let $n = \lfloor x \rfloor$ and $f = \{x\}$, where $x = n + f$. We can then express the equation as:
\[\lfloor n^2 + 2nf + f^2 \rfloor - (n + f)n = 4.\]
Extracting the $n^2$ term out of the floor function gives:
\[n^2 + \lfloor 2nf + f^2 \rfloor - n^2 - nf = 4.\]
This simplifies to:
\[\lfloor 2nf + f^2 \rfloor - nf = 4... | {
"ground_truth": "\\frac{29}{5}",
"style": "rule"
} | {
"index": 36,
"split": "train"
} |
math_verify_numina | [
{
"content": "For real numbers \\(t \\neq 0,\\) the point\n\\[\n(x, y) = \\left( \\frac{2t + 1}{t}, \\frac{t - 2}{t} \\right)\n\\]\nis plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Let's thi... | math | For \(x = \frac{2t + 1}{t}\) and \(y = \frac{t - 2}{t},\)
\[
x + y = \frac{2t + 1}{t} + \frac{t - 2}{t} = \frac{3t - 1}{t} = 3 - \frac{1}{t}.
\]
Thus, the expression \(3 - \frac{1}{t}\) is a hyperbola, because it represents a rational function where \(y\) depends inversely on \(t\), which does not remain constant but v... | {
"ground_truth": "\\text{(E)}",
"style": "rule"
} | {
"index": 37,
"split": "train"
} |
math_verify_numina | [
{
"content": "Let \\(T_{0}=1, T_{1}=x, T_{k+1}=2 x T_{k}-T_{k-1}\\) for \\(k \\in \\mathbb{N}\\). Express \\(T_{5}\\) and \\(T_{0}+T_{1}+T_{2}+T_{3}\\) in standard form. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math |
1. First, note the initial conditions and recursive relation for the Chebyshev polynomials of the first kind:
\[
T_0 = 1, \quad T_1 = x, \quad \text{and} \quad T_{k+1} = 2x T_k - T_{k-1} \quad \text{for} \quad k \in \mathbb{N}.
\]
2. Calculate \(T_2\) using the given recursion:
\[
T_2 = 2x T_1 - T_0 = ... | {
"ground_truth": "T_5 = 16x^5 - 20x^3 + 5x, \\quad T_0 + T_1 + T_2 + T_3 = 4x^3 + 2x^2 - 2x.",
"style": "rule"
} | {
"index": 38,
"split": "train"
} |
math_verify_numina | [
{
"content": "Prove that for any values of \\(a\\), \\(b\\), and \\(c\\), the graph of the function \\(y = (x - a)(x - b) - c^2\\) has at least one point in common with the \\(x\\)-axis. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math |
We need to prove that for any values of \(a\), \(b\), and \(c\), the graph of the function \(y = (x-a)(x-b) - c^2\) intersects the x-axis at least once.
1. **Rewrite the function in the standard quadratic form**:
\[
y = (x-a)(x-b) - c^2
\]
2. **Expand the expression**:
\[
y = x^2 - (a+b)x + ab - c^2
... | {
"ground_truth": "",
"style": "rule"
} | {
"index": 39,
"split": "train"
} |
math_verify_numina | [
{
"content": "Calculate the number of revolutions of a wheel, with a fixed center and an outside diameter of $10$ feet, required for a point on the rim to travel half a mile.\nA) $\\frac{132}{\\pi}$\nB) $264$\nC) $\\frac{264}{\\pi}$\nD) $132\\pi$\nE) $\\text{none of these}$ Let's think step by step and output t... | math | 1. **Determine the radius of the wheel**: Given the diameter of the wheel is $10$ feet, the radius $r$ is:
\[
r = \frac{10}{2} = 5 \text{ feet}
\]
2. **Calculate the wheel's circumference**: The circumference $C$ of a circle is given by $C = 2\pi r$. Substituting the radius:
\[
C = 2\pi \times 5 = 10\pi... | {
"ground_truth": "C",
"style": "rule"
} | {
"index": 40,
"split": "train"
} |
math_verify_numina | [
{
"content": "Let $f(x)$ be a function defined on $\\mathbb{R}$ that is never zero, and for any $x, y \\in \\mathbb{R}$, it satisfies $f(x) \\bullet f(y) = f(x + y)$. If ${a_1} = \\frac{1}{2}, {a_n} = f(n), (n \\in \\mathbb{N}^+)$, then the range of the sum of the first $n$ terms of the sequence ${{a_n}}$ denot... | math | First, let's find the expression for $f(n)$ in terms of $f(1)$. Since $f(x) \bullet f(y) = f(x + y)$, we can write $f(n)$ as:
$$f(n) = f(\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}}) = f(1) \bullet f(1) \bullet \cdots \bullet f(1) = (f(1))^n$$
Now, recall that ${a_1} = \frac{1}{2}$ and ${a_n} = f(n)$. Therefore,... | {
"ground_truth": "[\\frac{1}{2}, 1)",
"style": "rule"
} | {
"index": 41,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given the function $f(x)=2\\sqrt{3}\\sin{x}\\cos{x}+2\\cos^2{x}-1$ where $x\\in\\mathbb{R}$. \n$(1)$ Find the intervals where the function $f(x)$ is decreasing, and determine its maximum and minimum values on the interval $[0,\\frac{\\pi}{2}]$. \n$(2)$ If $f(x_0)=\\frac{6}{5}$ and $x_0\\in[\\fra... | math | ### Solution:
#### Part (1)
Given the function $f(x)=2\sqrt{3}\sin{x}\cos{x}+2\cos^2{x}-1$, we can simplify it using trigonometric identities:
\[
\begin{align*}
f(x) & = 2\sqrt{3}\sin{x}\cos{x}+2\cos^2{x}-1 \\
& = \sqrt{3}\sin{2x}+\cos{2x} \\
& = 2\left(\frac{\sqrt{3}}{2}\sin{2x}+\frac{1}{2}\cos{2x}\right) \\
& = 2\... | {
"ground_truth": "\\cos{2x_0} = \\frac{3 - 4\\sqrt{3}}{10}",
"style": "rule"
} | {
"index": 42,
"split": "train"
} |
math_verify_numina | [
{
"content": "If four people are selected at random from a group of nine men and six women, what is the probability that at least one woman is selected? Express your answer as a common fraction. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | First, determine the total number of ways to select 4 people out of 15 (9 men + 6 women). This is done using the combination formula:
\[ \binom{15}{4} = \frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1} = 1365. \]
Next, calculate the number of ways to select 4 people such that all are men:
\[ \bino... | {
"ground_truth": "\\frac{137}{151}",
"style": "rule"
} | {
"index": 43,
"split": "train"
} |
math_verify_numina | [
{
"content": "Nicholas starts with 8 bottle caps. He gets some more from Catherine and ends up with 93 bottle caps. How many bottle caps did Catherine give to Nicholas? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | To find out how many bottle caps Catherine gave to Nicholas, we need to subtract the number of bottle caps Nicholas started with from the total number he ended up with.
Nicholas ended up with 93 bottle caps and started with 8 bottle caps. So, the calculation is:
93 (total bottle caps) - 8 (initial bottle caps) = 85 b... | {
"ground_truth": "85",
"style": "rule"
} | {
"index": 44,
"split": "train"
} |
math_verify_numina | [
{
"content": "Let $\\{a_n\\}$ be an arithmetic sequence with a non-zero common difference, satisfying $a_4^2 + a_5^2 = a_6^2 + a_7^2$. Determine the sum of the first 10 terms of this sequence ($S_{10} = \\underline{\\quad}$).\nA: $0$\nB: $-5$\nC: $-10$\nD: $5$ Let's think step by step and output the final answe... | math | This problem tests our understanding of the general term formula and the sum of the first $n$ terms of an arithmetic sequence. We first denote the first term of the sequence as $a_1$ and the common difference as $d (d \neq 0)$.
Given $a_4^2 + a_5^2 = a_6^2 + a_7^2$, we can substitute the terms with their respective ex... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 45,
"split": "train"
} |
math_verify_numina | [
{
"content": "Points $C$ and $D$ are $7$ units apart. How many lines in a given plane containing $C$ and $D$ are $3$ units from $C$ and $4$ units from $D$?\n$\\text{(A) } 0\\quad \\text{(B) } 1\\quad \\text{(C) } 2\\quad \\text{(D) } 3\\quad \\text{(E) more than }3$ Let's think step by step and output the final... | math | 1. **Identify the circles**:
- Circle centered at $C$ (denoted as $C_C$) has radius $3$ units.
- Circle centered at $D$ (denoted as $C_D$) has radius $4$ units.
- The distance between centers $C$ and $D$ is $7$ units.
2. **Analyze the relative positions of the circles**:
- The sum of the radii of $C_C$ and... | {
"ground_truth": "\\text{(D) } 3",
"style": "rule"
} | {
"index": 46,
"split": "train"
} |
math_verify_numina | [
{
"content": "If $0 < x < 1$, then the maximum value of $f(x) = x(1-x)$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Given that $0 < x < 1$ and $f(x) = x(1-x)$, we can rewrite the function as $f(x) = -\left(x - \frac {1}{2}\right)^2 + \frac {1}{4}$. Since the term $-\left(x - \frac{1}{2}\right)^2$ is always non-positive and has its maximum value of $0$ when $x = \frac{1}{2}$, we can deduce that the function $f(x)$ reaches its maximum... | {
"ground_truth": "\\frac {1}{4}",
"style": "rule"
} | {
"index": 47,
"split": "train"
} |
math_verify_numina | [
{
"content": "There are 840 male and female participants in a meeting. Half the female participants and some proportion of the male participants are democrats. One-third of all the participants are democrats. There are 140 female democrats. What is the ratio of male democrats to the total number of male partici... | math | Let's denote the total number of female participants as F and the total number of male participants as M. We know that the total number of participants is 840, so:
F + M = 840
We are told that half the female participants are democrats, and there are 140 female democrats. Therefore, the total number of female partici... | {
"ground_truth": "1:4",
"style": "rule"
} | {
"index": 48,
"split": "train"
} |
math_verify_numina | [
{
"content": "The number of 2007-digit decimal numbers $\\overline{2 a_{1} a_{2} a_{3} \\cdots a_{2006}}$ in which the digits $a_{1}, a_{2}, a_{3}, \\cdots, a_{2006}$ contain an odd number of 9's is\nA. $\\frac{1}{2}\\left(10^{2006}+8^{2006}\\right)$\nB. $\\frac{1}{2}\\left(10^{2006}-8^{2006}\\right)$\nC. $10^{... | math |
1. Consider the number $ \overline{2 a_{1} a_{2} a_{3} \cdots a_{2006}} $, which is a 2007-digit number written in base 10.
2. We are given that this 2007-digit number has an odd number of the digit '9' among the digits $a_1, a_2, \ldots, a_{2006}$.
3. Let $B$ be the number of ways to select an odd number of 9’s fro... | {
"ground_truth": "\\text{B}",
"style": "rule"
} | {
"index": 49,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given events $A$, $B$, $P(B)=\\frac{1}{3}$, $P(\\overline{B}|A)=\\frac{3}{4}$, $P(\\overline{B}|\\overline{A})=\\frac{1}{2}$, then $P\\left(A\\right)=\\left(\\ \\ \\right)$\n\nA: $\\frac{1}{4}$\n\nB: $\\frac{1}{3}$\n\nC: $\\frac{2}{3}$\n\nD: $\\frac{1}{2}$ Let's think step by step and output the f... | math | Given the problem statement, let's break down the solution step by step:
1. We are given that $P(\overline{B}|A)=\frac{3}{4}$. This means that the probability of $B$ not happening given $A$ happens is $\frac{3}{4}$. Therefore, the probability of $B$ happening given $A$ happens, $P(B|A)$, is the complement of $P(\overl... | {
"ground_truth": "C",
"style": "rule"
} | {
"index": 50,
"split": "train"
} |
math_verify_numina | [
{
"content": "For all integers $n$ greater than 1, define $a_n = \\dfrac{1}{\\log_n 3003}$. Let $d = a_3 + a_4 + a_5 + a_6$ and $e = a_{15} + a_{16} + a_{17} + a_{18} + a_{19}$. Find $d - e$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | We start by noting that $a_n = \frac{1}{\log_n 3003} = \log_{3003} n$. Thus, we can write:
$$
d = \log_{3003} 3 + \log_{3003} 4 + \log_{3003} 5 + \log_{3003} 6
$$
$$
e = \log_{3003} 15 + \log_{3003} 16 + \log_{3003} 17 + \log_{3003} 18 + \log_{3003} 19
$$
The difference $d - e$ is:
$$
d - e = \left(\log_{3003} 3 + \log... | {
"ground_truth": "\\log_{3003} \\frac{1}{323}",
"style": "rule"
} | {
"index": 51,
"split": "train"
} |
math_verify_numina | [
{
"content": "In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\\omega$ that has no common points with these lines. For any line $\\ell||b$ , define $A=\\ell\\cap a$ , and $\\{B,C\\}=\\ell\\cap \\omega$ such that $B$ is on segment $AC$ . Construct the line $\\ell$ ... | math | To solve this problem, we need to maximize the ratio \(\frac{|BC|}{|AB|}\) for a line \(\ell\) parallel to \(b\). Let's break down the solution step by step.
1. **Define the Geometry:**
- Let \(D\) be the center of the circle \(\omega\).
- Let \(x\) be a line parallel to \(b\) passing through \(D\).
- Let \(O... | {
"ground_truth": "\\frac{r^2}{ka}",
"style": "rule"
} | {
"index": 52,
"split": "train"
} |
math_verify_numina | [
{
"content": "Following the directions of the arrows, how many different paths are there from $A$ to $D$?\n\n[asy]\npair A,B,C,D;\nA=(0,0);\nB=(3,0);\nC=(6,0);\nD=(9,0);\ndot(A);\ndot(B);\ndot(C);\ndot(D);\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,S);\nlabel(\"$D$\",D,S);\ndraw((0,0)--(1.5,1)--... | math | 1. From $A$ to $B$, there are two paths.
2. From $B$ to $C$, there are two paths: multiplying paths from $A$ to $B$ by paths from $B$ to $C$ gives $2 \times 2 = 4$ paths.
3. From $C$ to $D$, there are two paths, leading to $4 \times 2 = 8$ paths from $A$ to $D$ passing through $B$ and $C$.
4. There is a direct path fro... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 53,
"split": "train"
} |
math_verify_numina | [
{
"content": "**p1.** Point $P$ has coordinate value $(3,2)$ , and point $Q$ has coordinate value $(6, x)$ . For what value of $x$ does Line $PQ$ have a slope of $6$ ?**p2.** I have fewer than one hundred pens. When my pens are grouped into nines, I have one pen left over. When grouped into fives, I ... | math | 1. Let \( X \) be the total number of pens.
2. We are given the following congruences:
\[
\begin{align*}
X &\equiv 1 \pmod{9}, \\
X &\equiv 3 \pmod{5}, \\
X &\equiv 1 \pmod{2}.
\end{align*}
\]
3. First, we solve the congruences \( X \equiv 1 \pmod{9} \) and \( X \equiv 1 \pmod{2} \). Since \( 9 \) ... | {
"ground_truth": "73",
"style": "rule"
} | {
"index": 54,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given the ellipse $\\frac{x^{2}}{9}+ \\frac{y^{2}}{m^{2}}=1$ (where $0<m<3$), the length of its major axis is twice that of its minor axis. Find the value of $m$.\nA: $\\frac{3}{2}$\nB: $\\frac{8}{3}$\nC: 1\nD: 2 Let's think step by step and output the final answer within \\boxed{}.",
"role": ... | math | The given ellipse $\frac{x^{2}}{9}+ \frac{y^{2}}{m^{2}}=1$ has its foci on the x-axis, with a major axis length of $6$ and minor axis length of $2m$ (since $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ has the major axis $a$ and minor axis $b$ when $a>b$).
According to the problem, the major axis is twice the length of the mino... | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": 55,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given the polar equation of curve C is $\\rho=2\\cos\\theta$, with the pole as the origin of the Cartesian coordinate system and the polar axis as the positive half-axis of the x-axis, establish a Cartesian coordinate system. The parametric equation of line $l$ is $$\\begin{cases} x= \\frac { \\sq... | math | Solution:
(Ⅰ) From $\rho=2\cos\theta$, we get: $\rho^2=2\rho\cos\theta$, thus $x^2+y^2=2x$, which leads to $(x-1)^2+y^2=1$.
Therefore, the Cartesian equation of curve C is $(x-1)^2+y^2=1$.
From $$\begin{cases} x= \frac { \sqrt {3}}{2}t+m \\ y= \frac {1}{2}t \end{cases}$$ (t is the parameter), we get $x= \sqrt {3}y+m$, ... | {
"ground_truth": "1, 1+ \\sqrt {2}, 1- \\sqrt {2}",
"style": "rule"
} | {
"index": 56,
"split": "train"
} |
math_verify_numina | [
{
"content": "Mr. Fast eats a pizza slice in 5 minutes, Mr. Slow eats a pizza slice in 10 minutes, and Mr. Steady eats a pizza slice in 8 minutes. How long does it take for them to eat 24 pizza slices together? Express your answer in minutes. Let's think step by step and output the final answer within \\boxed{}... | math | Mr. Fast's rate of eating is $\frac{1}{5}$ slice per minute.
Mr. Slow's rate of eating is $\frac{1}{10}$ slice per minute.
Mr. Steady's rate of eating is $\frac{1}{8}$ slice per minute.
The combined eating rate is:
\[
\frac{1}{5} + \frac{1}{10} + \frac{1}{8} = \frac{8}{40} + \frac{4}{40} + \frac{5}{40} = \frac{17}{40}... | {
"ground_truth": "56",
"style": "rule"
} | {
"index": 57,
"split": "train"
} |
math_verify_numina | [
{
"content": "Let \\( a, b, \\) and \\( c \\) be the roots of the equation \\( x^3 - 18x^2 + 20x - 8 = 0 \\). Find the value of \\( (2+a)(2+b)(2+c) \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Expanding the expression, we have:
\[
(2+a)(2+b)(2+c) = 2^3 + 2^2(a+b+c) + 2(ab+bc+ca) + abc
\]
Using Vieta's formulas:
- \( a+b+c = 18 \)
- \( ab+bc+ca = 20 \)
- \( abc = 8 \)
Substitute these values into the expanded form:
\[
(2+a)(2+b)(2+c) = 8 + 4 \times 18 + 2 \times 20 + 8 = 8 + 72 + 40 + 8
\]
\[
= 128
\]
Concl... | {
"ground_truth": "128",
"style": "rule"
} | {
"index": 58,
"split": "train"
} |
math_verify_numina | [
{
"content": "The function $f(x)=x^{3}+ax^{2}+(a+6)x+1$ has extreme values on $R$. Determine the range of values for the real number $a$.\nA: $-3\\leqslant a\\leqslant 6$\nB: $a\\geqslant 6$ or $a\\leqslant -3$\nC: $-3 < a < 6$\nD: $a > 6$ or $a < -3$ Let's think step by step and output the final answer within ... | math | Given $f(x)=x^{3}+ax^{2}+(a+6)x+1$,
Then $f′(x)=3x^{2}+2ax+a+6$,
Since the function $f(x)=x^{3}+ax^{2}+(a+6)x+1$ has extreme values on $R$,
It follows that the function $f(x)=x^{3}+ax^{2}+(a+6)x+1$ is not monotonic on $R$,
Thus, $f′(x)=3x^{2}+2ax+a+6=0$ has two distinct roots,
That is, $\Delta =4a^{2}-12a-72 > 0$,... | {
"ground_truth": "D",
"style": "rule"
} | {
"index": 59,
"split": "train"
} |
math_verify_numina | [
{
"content": "In \\\\(\\triangle ABC\\\\), the sides opposite to angles \\\\(A\\\\), \\\\(B\\\\), and \\\\(C\\\\) are denoted as \\\\(a\\\\), \\\\(b\\\\), and \\\\(c\\\\) respectively. Given that \\\\(b=3\\\\), \\\\(\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = -6\\\\), and the area of \\\\(\\triangle ABC\... | math | From \\(\overrightarrow{AB} \cdot \overrightarrow{AC} = -6\\), we can derive \\(bc\cos A = -6\\), (1),
Using the formula for the area of a triangle, we get \\(S_{\triangle ABC} = \frac{1}{2}bc\sin A = 3\\), (2)
\\(\therefore \tan A = -1\\),
Since \\(0 < A < 180^{\circ}\\),
\\(\therefore A = 135^{\circ}\\),
\\... | {
"ground_truth": "A = 135^{\\circ}, a = \\sqrt{29}",
"style": "rule"
} | {
"index": 60,
"split": "train"
} |
math_verify_numina | [
{
"content": "a farmer has an apple orchard consisting of fuji and gala apple trees . due to high winds this year 10 % of his trees cross pollinated . the number of his trees that are pure fuji plus the cross - pollinated ones totals 153 , while 3 / 4 of all his trees are pure fuji . how many of his trees are p... | math | Let's denote the total number of trees as T, the number of pure Fuji trees as F, and the number of pure Gala trees as G. According to the problem, 10% of the trees cross-pollinated, so the number of cross-pollinated trees is 0.10T.
We are given two pieces of information:
1. The number of pure Fuji trees plus the cros... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 61,
"split": "train"
} |
math_verify_numina | [
{
"content": "A trader marked the selling price of an article at 50% above the cost price. At the time of selling, he allows a certain discount and suffers a loss of 1%. What is the percentage of the discount he allowed? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"... | math | Let's assume the cost price (CP) of the article is $100 (for ease of calculation).
The trader marks up the selling price (SP) by 50% above the cost price. So the marked price (MP) would be:
MP = CP + (50% of CP)
MP = $100 + ($100 * 0.5)
MP = $100 + $50
MP = $150
Now, the trader suffers a loss of 1% on the cost price ... | {
"ground_truth": "34\\%",
"style": "rule"
} | {
"index": 62,
"split": "train"
} |
math_verify_numina | [
{
"content": "A multinational company has a yearly subscription to three categories of professional magazines: Category A, Category B, and Category C. The subscription costs for these categories in USD are $520, $860, and $620, respectively. The company wants to make different percentage cuts in each category's... | math | First, let's calculate the reduced subscription costs for each category after the budget cuts:
Category A: $520 - (25% of $520) = $520 - ($520 * 0.25) = $520 - $130 = $390
Category B: $860 - (35% of $860) = $860 - ($860 * 0.35) = $860 - $301 = $559
Category C: $620 - (30% of $620) = $620 - ($620 * 0.30) = $620 - $186 ... | {
"ground_truth": "€561.1615",
"style": "rule"
} | {
"index": 63,
"split": "train"
} |
math_verify_numina | [
{
"content": "(5 points) (2015 Zaozhuang School-Level Mock) The negation of the proposition \"There exists an $x \\in \\mathbb{R}$ such that $x^3 - 2x + 1 = 0$\" is ( ).\n\nA. There exists an $x \\in \\mathbb{R}$ such that $x^3 - 2x + 1 \\neq 0$ \nB. There does not exist an $x \\in \\mathbb{R}$ such that $x^3 ... | math | The statement "There exists an $x \in \mathbb{R}$ such that $x^3 - 2x + 1 = 0$" is an existential proposition. Its negation becomes a universal proposition.
In more detail, the original proposition claims that there is at least one real number $x$ for which the polynomial $x^3 - 2x + 1$ equals zero. Negating this stat... | {
"ground_truth": "\\forall x \\in \\mathbb{R}, x^3 - 2x + 1 \\neq 0",
"style": "rule"
} | {
"index": 64,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given the curve $y= \\frac {1}{3}x^{3}+ \\frac {4}{3}$,\n(1) Find the equation of the tangent line at point P(2,4);\n(2) Find the equation of the tangent line passing through point P(2,4). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | (1) The derivative of $y= \frac {1}{3}x^{3}+ \frac {4}{3}$ is $y'=x^2$.
Thus, the slope of the tangent line at point P(2,4) is 4.
Hence, the equation of the tangent line at point P(2,4) is $y-4=4(x-2)$, which simplifies to $\boxed{4x-y-4=0}$.
(2) Let the point of tangency be $(m,n)$. Then, $n= \frac {1}{3}m^{3}+ \frac... | {
"ground_truth": "4x-y-4=0",
"style": "rule"
} | {
"index": 65,
"split": "train"
} |
math_verify_numina | [
{
"content": "Every bedtime, Juwella reads a book. Three nights ago, she read 15 pages. She read some ratio of pages two nights ago compared to three nights ago, while last night she read 5 pages more than the previous night. She promised to read the remaining pages of the book tonight. If the book has 100 page... | math | Let's break down the information given:
1. Three nights ago, Juwella read 15 pages.
2. Two nights ago, she read some ratio of pages compared to three nights ago.
3. Last night, she read 5 pages more than the previous night.
4. The book has 100 pages in total.
5. She will read 20 pages tonight.
We know that three nigh... | {
"ground_truth": "2:1",
"style": "rule"
} | {
"index": 66,
"split": "train"
} |
math_verify_numina | [
{
"content": "Find all angles $\\theta,$ $0 \\le \\theta \\le 2\\pi,$ with the following property: For all real numbers $x,$ $0 \\le x \\le 1,$\n\\[\nx^3 \\cos \\theta + x(1 - x) - (1 - x)^3 \\sin \\theta < 0.\n\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | - **Step 1**: Evaluate boundary conditions $x = 0$ and $x = 1$.
- For $x = 0$, the expression simplifies to $-\sin \theta < 0$, thus $\sin \theta > 0$.
- For $x = 1$, the expression simplifies to $\cos \theta < 0$, so $\cos \theta > 0$ is invalid; instead, it confirms $\cos \theta < 0$.
- **Step 2**: Analyze and s... | {
"ground_truth": "\\left(\\frac{\\pi}{2}, \\pi\\right)",
"style": "rule"
} | {
"index": 67,
"split": "train"
} |
math_verify_numina | [
{
"content": "A triangle has side lengths of 10, 24, and 26 units. What is the area of this triangle, in square units? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | First, verify whether a triangle with sides 10, 24, and 26 is a right triangle using the Pythagorean theorem:
$$ 10^2 + 24^2 = 100 + 576 = 676 $$
$$ 26^2 = 676 $$
Since $10^2 + 24^2 = 26^2$, the triangle is a right triangle.
Now, calculate the area of the right triangle with legs 10 and 24:
$$ \text{Area} = \frac{1}{2... | {
"ground_truth": "120 \\text{ square units}",
"style": "rule"
} | {
"index": 68,
"split": "train"
} |
math_verify_numina | [
{
"content": "Let x be the original number, and y be an additional number such that y = x^2 + 5x - 12. Find the number x which, when multiplied by 15, is increased by 280, while also satisfying the relationship between x and y. Let's think step by step and output the final answer within \\boxed{}.",
"role":... | math | First, let's set up the equation for the condition that when x is multiplied by 15, it is increased by 280:
15x = x + 280
Now, let's solve for x:
15x - x = 280
14x = 280
x = 280 / 14
x = 20
Now we have the value of x, we can check if it satisfies the relationship between x and y:
y = x^2 + 5x - 12
Substitute x = ... | {
"ground_truth": "x^2",
"style": "rule"
} | {
"index": 69,
"split": "train"
} |
math_verify_numina | [
{
"content": "Let $\\theta$ be the angle between the line\n\\[\\frac{x - 2}{4} = \\frac{y + 3}{5} = \\frac{z + 1}{7}\\]\nand the plane $5x - 3y + 4z = 6.$ Find $\\sin \\theta.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | The direction vector of the line is \(\mathbf{d} = \begin{pmatrix} 4 \\ 5 \\ 7 \end{pmatrix}\), and the normal vector to the plane is \(\mathbf{n} = \begin{pmatrix} 5 \\ -3 \\ 4 \end{pmatrix}\).
Note that if \(\theta\) is the angle between \(\mathbf{d}\) and the plane, then the angle between \(\mathbf{d}\) and \(\math... | {
"ground_truth": "\\frac{\\sqrt{3411}}{67}",
"style": "rule"
} | {
"index": 70,
"split": "train"
} |
math_verify_numina | [
{
"content": "How many positive 3-digit numbers are divisible by 13? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | We first find the smallest and largest 3-digit numbers divisible by 13.
The smallest 3-digit number is 100, but it needs to be divisible by 13. We calculate $\lceil 100 / 13 \rceil = \lceil 7.6923 \rceil = 8$. Thus, the smallest 3-digit number divisible by 13 is $13 \times 8 = 104$.
The largest 3-digit number is 999, ... | {
"ground_truth": "69",
"style": "rule"
} | {
"index": 71,
"split": "train"
} |
math_verify_numina | [
{
"content": "Calligraphy is a cultural treasure of the Chinese nation, a precious wealth of human civilization, and an important part of basic education in our country. A certain school is preparing to purchase a batch of brushes and rice paper for students' calligraphy classes. It is known that purchasing 40 ... | math | ### Step-by-Step Solution
#### Part 1: Finding the Unit Prices
Let's denote the unit price of a brush as $x$ yuan and the unit price of rice paper as $y$ yuan.
Given the information, we can set up the following system of equations:
1. For 40 brushes and 100 sheets of rice paper, the cost is 280 yuan, leading to the ... | {
"ground_truth": "\\text{The school can purchase at most 50 brushes.}",
"style": "rule"
} | {
"index": 72,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given $x=\\sqrt{6}-\\sqrt{2}$, the value of the algebraic expression $2x^2+4\\sqrt{2}x$ is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Given $x=\sqrt{6}-\sqrt{2}$, we want to find the value of $2x^2+4\sqrt{2}x$. Let's proceed step by step:
1. First, notice that adding $\sqrt{2}$ to $x$ gives us $x + \sqrt{2} = (\sqrt{6}-\sqrt{2}) + \sqrt{2} = \sqrt{6}$.
2. Squaring both sides of the equation $x + \sqrt{2} = \sqrt{6}$ yields $(x + \sqrt{2})^2 = (\sqr... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 73,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given the line $x+y=1$ and the circle $x^{2}+y^{2}=a$ intersect at points $A$ and $B$. $O$ is the origin, and $C$ is a point on the circle. If $\\overrightarrow{OA} + \\overrightarrow{OB} = \\overrightarrow{OC}$, find the value of $a$. Let's think step by step and output the final answer within \\... | math | Let $A(x_{1},y_{1})$, $B(x_{2},y_{2})$, and $C(x_{0},y_{0})$.
Solve the system of equations:
$
\begin{cases}
x+y=1 \\
x^{2}+y^{2}=a
\end{cases}
$
Rearrange the equations to obtain $2x^{2}-2x+1-a=0$. Since the line $x+y=1$ intersects the circle $x^{2}+y^{2}=a$ at two points, the discriminant $\triangle = 4-8(1-a) > 0$... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 74,
"split": "train"
} |
math_verify_numina | [
{
"content": "Consider a 10-sided polygon $ABCDEFGHIJ$, where each side has alternating lengths of 5 and 3 units, and each two consecutive sides form a right angle. Suppose that $\\overline{AE}$ and $\\overline{CG}$ meet at point $M$. Find the area of quadrilateral $ABCM$.\nA) $36$\nB) $40$\nC) $44$\nD) $48$\nE... | math | 1. **Identify the Key Points and Setup:**
- We have a 10-sided polygon with sides alternating between 5 and 3 units, with right angles at each vertex.
- We need to determine the area of quadrilateral $ABCM$ where $\overline{AE}$ and $\overline{CG}$ intersect at point $M$.
2. **Calculate the Area of Rectangle $AB... | {
"ground_truth": "B",
"style": "rule"
} | {
"index": 75,
"split": "train"
} |
math_verify_numina | [
{
"content": "The general term formula of the sequence $\\{a_n\\}$ is $a_n=2n-49$. When $S_n$ is minimized, $n$ equals \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Given $a_n=2n-49$, we can derive
$a_{n+1}-a_n=2(n+1)-49-(2n-49)=2$, which is a constant,
∴ The sequence $\{a_n\}$ is an arithmetic sequence,
∴ $S_{n}= \frac {(a_{1}+a_{n})n}{2}$, and $a_1=2\times1-49=-47$,
∴ $S_{n}= \frac {-47+2n-49}{2}\times n=n^{2}-48n$ = $(n-24)^2-24^2$
Based on the properties of quadratic... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": 76,
"split": "train"
} |
math_verify_numina | [
{
"content": "Person A and person B play a game to fill in the coefficients of a polynomial of degree at least 4:\n\\[ x^{2n} + \\square x^{2n-1} + \\square x^{2n-2} + \\cdots + \\square x + 1 \\]\n\nwhere they take turns to choose one of the blanks and fill it with a real number as the coefficient, until all b... | math |
1. **Define the problem scope and terms:**
We are dealing with a polynomial of degree $2n$ with coefficients partially filled.
The polynomial is:
\[
P(x) = x^{2n} + \square x^{2n-1} + \square x^{2n-2} + \cdots + \square x + 1
\]
Two players, \(A\) and \(B\), alternately fill in the coefficients. If t... | {
"ground_truth": "B",
"style": "rule"
} | {
"index": 77,
"split": "train"
} |
math_verify_numina | [
{
"content": "Logan recorded the snowfall every day during a snowstorm . He recorded 0.33 centimeter on Wednesday , 0.33 centimeter on Thursday , and 0.22 centimeter on Friday. Logan recorded _____ centimeters of snow . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... | math | To find the total amount of snow Logan recorded, we need to add the snowfall from each day together:
Wednesday: 0.33 cm
Thursday: 0.33 cm
Friday: 0.22 cm
Total snowfall = Wednesday + Thursday + Friday
Total snowfall = 0.33 cm + 0.33 cm + 0.22 cm
Total snowfall = 0.88 cm
Logan recorded $\boxed{0.88}$ centimeters of ... | {
"ground_truth": "0.88",
"style": "rule"
} | {
"index": 78,
"split": "train"
} |
math_verify_numina | [
{
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$. If $a_1=2$ and $S=12$, then $a_6$ equals ( )\nA: $8$\nB: $12$\nC: $10$\nD: $14$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | **Analysis**
This question examines the general formula for an arithmetic sequence and its sum formula. By using the properties of an arithmetic sequence and the given information, we can find $a_2$, and then the common difference, which leads to finding $a_6$.
**Solution**
Given the information, we have $S_3=a_1+a_... | {
"ground_truth": "B",
"style": "rule"
} | {
"index": 79,
"split": "train"
} |
math_verify_numina | [
{
"content": "Find the coordinates of the point in the plane at which the sum of the distances from it to the three points $(0, 0)$ , $(2, 0)$ , $(0, \\sqrt{3})$ is minimal. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | To find the coordinates of the point in the plane at which the sum of the distances from it to the three points \((0, 0)\), \((2, 0)\), and \((0, \sqrt{3})\) is minimal, we need to determine the Fermat point of the triangle formed by these vertices. The Fermat point minimizes the total distance to the vertices of the t... | {
"ground_truth": "\\left(1, \\frac{5.5\\sqrt{3} - 3}{13}\\right)",
"style": "rule"
} | {
"index": 80,
"split": "train"
} |
math_verify_numina | [
{
"content": "On the base \\(AD\\) of trapezoid \\(ABCD\\), a point \\(K\\) is given. Find a point \\(M\\) on the base \\(BC\\) for which the area of the common part of triangles \\(AMD\\) and \\(BKC\\) is maximized. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}... | math |
1. **Identify the geometrical properties and introduce the points:**
Given a trapezoid $ABCD$ with $AD$ as one base and $BC$ as another base. A point $K$ is situated on $AD$, and we need to find a point $M$ on $BC$ such that the area of the intersection of triangles $AMD$ and $BKC$ is maximized.
2. **Determine th... | {
"ground_truth": "\\frac{BM}{MC} = \\frac{AK}{KD}",
"style": "rule"
} | {
"index": 81,
"split": "train"
} |
math_verify_numina | [
{
"content": "Hadassah takes six hours to paint 12 small paintings and eight hours to paint 6 large paintings. If she plans to paint 15 more small paintings, 10 more large paintings, and takes a 30-minute break after every third painting, what would be the total time she's taken to finish all the paintings? Let... | math | First, let's calculate the time Hadassah takes to paint one small painting and one large painting.
For small paintings:
6 hours / 12 small paintings = 0.5 hours per small painting
For large paintings:
8 hours / 6 large paintings = 1.33 hours (or 1 hour and 20 minutes) per large painting
Now, let's calculate the time... | {
"ground_truth": "24.8",
"style": "rule"
} | {
"index": 82,
"split": "train"
} |
math_verify_numina | [
{
"content": "As shown in the figure, a rectangular prism with length, width, and height of $15, 5, 4$ respectively has a smaller rectangular prism with length, width, and height of $y, 5, x$ (where $x$ and $y$ are integers) cut out from it. The volume of the remaining part is 120. Find $x$ and $y$. Let's think... | math |
1. **Understanding the Problem**:
We start with a rectangular solid with dimensions \(15 \, \text{units} \times 5 \, \text{units} \times 4 \, \text{units}\). A smaller rectangular solid is cut from this original solid with dimensions \(y \times 5 \times x\), where \(x\) and \(y\) are integers. The remaining volume ... | {
"ground_truth": "3, 12",
"style": "rule"
} | {
"index": 83,
"split": "train"
} |
math_verify_numina | [
{
"content": "The numbers \\(1, 2, \\ldots, 20\\) are put into a hat. Claire draws two numbers from the hat uniformly at random, \\(a < b\\), and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, \\(c < d\\). Let \\(N\\) denote the number of integers \\(n\\) tha... | math |
1. **Understanding the problem:**
We need to determine the probability that \(N\), the number of integers \(n\) that satisfy exactly one of \(a \leq n \leq b\) and \(c \leq n \leq d\), is even. Here, \(a\) and \(b\) are two numbers drawn by Claire, and \(c\) and \(d\) are two numbers drawn by William.
2. **Calcula... | {
"ground_truth": "\\frac{181}{361}",
"style": "rule"
} | {
"index": 84,
"split": "train"
} |
math_verify_numina | [
{
"content": "A circle with a radius of 6 cm is tangent to the two shorter sides and one of the longer sides of a rectangle. The area of the rectangle is three times the area of the circle. Calculate the length of the shorter side of the rectangle in centimeters, expressing your answer in terms of $\\pi$. Let's... | math | 1. The area of the circle, given the radius \(r = 6\) cm, is:
\[
A_{\text{circle}} = \pi r^2 = 36\pi \ \text{cm}^2
\]
2. The area of the rectangle is three times the area of the circle:
\[
A_{\text{rectangle}} = 3 \times A_{\text{circle}} = 108\pi \ \text{cm}^2
\]
3. The shorter side of the rectangl... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 85,
"split": "train"
} |
math_verify_numina | [
{
"content": "In the acute triangle \\( \\triangle ABC \\), \\(\\angle A = 45^\\circ\\). Points \\( O \\) and \\( H \\) are the circumcenter and orthocenter, respectively. \\(BD\\) is the altitude from \\(B\\) to side \\(AC\\) with foot \\(D\\). In the circumcircle of \\( \\triangle ADH \\), let \\( X \\) be th... | math |
To prove: \( DX = DO \)
**Step 1:** Identify and understand the given geometric entities:
- \( \triangle ABC \) is a right-angled triangle with \(\angle A = 45^\circ\).
- \( O \) is the circumcenter and \( H \) is the orthocenter of \( \triangle ABC \).
- \( BD \) is the altitude from vertex \( B \) to the hypotenuse... | {
"ground_truth": "DX = DO",
"style": "rule"
} | {
"index": 86,
"split": "train"
} |
math_verify_numina | [
{
"content": "Compute the indefinite integral:\n\n$$\n\\int \\frac{2 x^{3}+6 x^{2}+5 x+4}{(x-2)(x+1)^{3}} \\, dx\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math |
1. We start by expressing the given integrand as a sum of partial fractions. The given integrand is:
\[
\int \frac{2 x^{3}+6 x^{2}+5 x+4}{(x-2)(x+1)^{3}} \, dx
\]
2. Let's decompose the rational function into partial fractions. We assume the decomposition has the form:
\[
\frac{2 x^{3}+6 x^{2}+5 x+4}{(x-2)(x+1)^{3}... | {
"ground_truth": "2 \\ln |x-2| + \\frac{1}{2(x+1)^2} + C",
"style": "rule"
} | {
"index": 87,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given: The lengths of the two sides $AB$ and $AD$ of rectangle $ABCD$ are the two real roots of the equation $x^2-mx+\\frac{m}{2}-\\frac{1}{4}=0$ in terms of $x$. <br/>$(1)$ For what value of $m$ is the quadrilateral $ABCD$ a square? Provide a reason for your answer; <br/>$(2)$ If the length of $A... | math | ### Step-by-Step Solution
#### Part 1: Finding the value of $m$ for which $ABCD$ is a square
Given the quadratic equation for the sides of the rectangle $ABCD$ is $x^2-mx+\frac{m}{2}-\frac{1}{4}=0$.
For $ABCD$ to be a square, both sides $AB$ and $AD$ must be equal, implying that the quadratic equation has a single r... | {
"ground_truth": "\\frac{\\sqrt{17}}{2}",
"style": "rule"
} | {
"index": 88,
"split": "train"
} |
math_verify_numina | [
{
"content": "In a 3-dimensional space, there are 2016 spheres, none of which coincide. Some of the spheres are red, and the others are green. Each contact point between a red sphere and a green sphere is painted blue. Find the maximum possible number of blue points. Let's think step by step and output the fina... | math |
1. Let us define \(r\) as the number of red spheres and \(2016 - r\) as the number of green spheres.
2. Since any two spheres can intersect at most at one point, the number of blue points is at most the number of red-green sphere pairs. This can be given by the product of the number of red spheres and the number of ... | {
"ground_truth": "1016064",
"style": "rule"
} | {
"index": 89,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given that \\(f(x)=\\begin{cases} \\ln x, & x > 0 \\\\ a+\\int_{0}^{x}{\\left( 1-\\cos t \\right)dt}, & x\\leqslant 0 \\end{cases}\\) and \\(f(f(1))=2\\), find the value of \\(a\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | This problem primarily tests the computation of function values and the application of piecewise functions. By finding the expression for \(f(x)\) using the piecewise function and substituting it into the equation, we can obtain the conclusion.
When \(x \leqslant 0\), \(f(x) = a + (t - \sin t)\big|_0^x = a + x - \sin ... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 90,
"split": "train"
} |
math_verify_numina | [
{
"content": "Oliver bought eight boxes of candy in the morning. In the afternoon, he bought three times the number of boxes he bought in the morning. Later, he gave away 25% of the total boxes he had to his friends. How many boxes of candy does Oliver have now? Let's think step by step and output the final ans... | math | In the morning, Oliver bought 8 boxes of candy.
In the afternoon, he bought 3 times the number of boxes he bought in the morning, which is 3 * 8 = 24 boxes.
The total number of boxes he had before giving any away is 8 (morning) + 24 (afternoon) = 32 boxes.
He gave away 25% of his total boxes to his friends. To find ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": 91,
"split": "train"
} |
math_verify_numina | [
{
"content": "Let \\(a\\), \\(b\\), and \\(c\\) be distinct real numbers. Simplify the expression\n\\[\n\\frac{(x + a)^2}{(a - b)(a - c)} + \\frac{(x + b)^2}{(b - a)(b - c)} + \\frac{(x + c)^2}{(c - a)(c - b)}.\n\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
... | math | Let
\[
p(x) = \frac{(x + a)^2}{(a - b)(a - c)} + \frac{(x + b)^2}{(b - a)(b - c)} + \frac{(x + c)^2}{(c - a)(c - b)}.
\]
Evaluate \(p(x)\) at \(x = -a\):
\[
p(-a) = \frac{(-a + a)^2}{(a - b)(a - c)} + \frac{(-a + b)^2}{(b - a)(b - c)} + \frac{(-a + c)^2}{(c - a)(c - b)} = 0 + \frac{(b - a)^2}{(b - a)(b - c)} + \frac{(c... | {
"ground_truth": "ax + bx + cx - a - b - c",
"style": "rule"
} | {
"index": 92,
"split": "train"
} |
math_verify_numina | [
{
"content": "A convenience store sold 100 bags of chips in a month. In the first week, some bags of chips were sold. In the second week, thrice as many bags of chips were sold. There was an equal number of chips sold in the third and fourth week, which was 20 bags each. How many bags of chips were sold in the ... | math | Let's denote the number of bags of chips sold in the first week as X.
According to the information given, in the second week, thrice as many bags of chips were sold as in the first week. So, the number of bags sold in the second week is 3X.
It is also given that in the third and fourth week, an equal number of chips ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 93,
"split": "train"
} |
math_verify_numina | [
{
"content": "Solve the congruence $15x + 3 \\equiv 9 \\pmod{21}$ for $x$, and express the solution as $x \\equiv a \\pmod{m}$, where $a < m$ and find $a + m$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | 1. **Simplify the Congruence**: Begin by subtracting 3 from both sides:
\[
15x + 3 \equiv 9 \pmod{21} \implies 15x \equiv 6 \pmod{21}
\]
2. **Reduce the Equation**: Divide both sides by the greatest common divisor of 15 and 21, which is 3, to further simplify:
\[
\frac{15x}{3} \equiv \frac{6}{3} \pmod{\f... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 94,
"split": "train"
} |
math_verify_numina | [
{
"content": "Every data point in a set of data is decreased by 80, resulting in a new set of data with an average value of 1.2 and a variance of 4.4. What are the average value and variance of the original data set? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}... | math | Since every data point in the original data set is decreased by 80, the average value will also decrease by 80, but the variance remains unchanged. Given that the new data set has an average value of 1.2 and a variance of 4.4, the average value and variance of the original data set are respectively 81.2 and 4.4.
There... | {
"ground_truth": "81.2, 4.4",
"style": "rule"
} | {
"index": 95,
"split": "train"
} |
math_verify_numina | [
{
"content": "Jean and her three friends are playing a game of dominoes. There are 28 dominoes in the set, and Jean wants each player to receive the same number of dominoes. How many dominoes will Jean and her friends each receive? Let's think step by step and output the final answer within \\boxed{}.",
"ro... | math | To solve this problem, we start by determining the total number of players. Jean is playing with her three friends, which means:
- Jean = 1
- Her three friends = 3
Adding these together gives us the total number of players:
\[1 + 3 = 4\]
There are 4 players in total.
Next, we need to divide the total number of dom... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 96,
"split": "train"
} |
math_verify_numina | [
{
"content": "What will be the cost of painting a building with a certain area, if the price per foot of painting is Rs. 15 and the total cost of painting the building is Rs. 840? How large is the area to be painted? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}... | math | To find the area to be painted, we can use the given information about the cost of painting per foot and the total cost of painting the building.
Let's denote the area to be painted as A (in square feet).
Given:
- Cost of painting per square foot = Rs. 15
- Total cost of painting the building = Rs. 840
We can set up... | {
"ground_truth": "56",
"style": "rule"
} | {
"index": 97,
"split": "train"
} |
math_verify_numina | [
{
"content": "Given positive numbers $a$ and $b$ satisfying $\\dfrac{1}{a} + \\dfrac{1}{b} = 1$, find the minimum value of $\\dfrac{1}{a-1} + \\dfrac{4}{b-1}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | Since positive numbers $a$ and $b$ satisfy $\dfrac{1}{a} + \dfrac{1}{b} = 1$, we have $b = \dfrac{a}{a-1} > 0$, which leads to $a > 1$. Similarly, we can deduce $b > 1$.
Therefore, $\dfrac{1}{a-1} + \dfrac{4}{b-1} = \dfrac{1}{a-1} + \dfrac{4}{\dfrac{a}{a-1}-1} = \dfrac{1}{a-1} + 4(a-1) \geqslant 2\sqrt{\dfrac{1}{a-1} ... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 98,
"split": "train"
} |
math_verify_numina | [
{
"content": "James took a bike ride for some hours. In the second hour, he traveled 12 miles, which was 20 percent farther than he traveled the first hour. If he traveled 25 percent farther in the third hour than he did in the second hour, James traveled 37 miles during the entire ride. How many hours did Jame... | math | Let's denote the distance James traveled in the first hour as \( x \) miles.
According to the information given, in the second hour, he traveled 20 percent farther than the first hour. This means he traveled \( x + 0.20x = 1.20x \) miles in the second hour.
In the third hour, he traveled 25 percent farther than he di... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 99,
"split": "train"
} |
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