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The dataset generation failed
Error code: DatasetGenerationError
Exception: CastError
Message: Couldn't cast
shards: struct<deepscaler_shard_00.json: int64, deepscaler_shard_01.json: int64, deepscaler_shard_02.json: i (... 5 chars omitted)
child 0, deepscaler_shard_00.json: int64
child 1, deepscaler_shard_01.json: int64
child 2, deepscaler_shard_02.json: int64
total_input_records: int64
invalid_records_dropped: int64
duplicate_records_removed: int64
final_rows: int64
min_problem_chars: int64
max_problem_chars: int64
data_source: string
ability: string
reward_model: struct<style: string, ground_truth: string>
child 0, style: string
child 1, ground_truth: string
extra_info: struct<index: int64, solution: string>
child 0, index: int64
child 1, solution: string
prompt: list<item: struct<role: string, content: string>>
child 0, item: struct<role: string, content: string>
child 0, role: string
child 1, content: string
to
{'data_source': Value('string'), 'prompt': List({'role': Value('string'), 'content': Value('string')}), 'ability': Value('string'), 'reward_model': {'style': Value('string'), 'ground_truth': Value('string')}, 'extra_info': {'index': Value('int64'), 'solution': Value('string')}}
because column names don't match
Traceback: Traceback (most recent call last):
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1827, in _prepare_split_single
for key, table in generator:
^^^^^^^^^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 613, in wrapped
for item in generator(*args, **kwargs):
~~~~~~~~~^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/packaged_modules/json/json.py", line 343, in _generate_tables
self._cast_table(pa_table, json_field_paths=json_field_paths),
~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/packaged_modules/json/json.py", line 132, in _cast_table
pa_table = table_cast(pa_table, self.info.features.arrow_schema)
File "/usr/local/lib/python3.14/site-packages/datasets/table.py", line 2378, in table_cast
return cast_table_to_schema(table, schema)
File "/usr/local/lib/python3.14/site-packages/datasets/table.py", line 2306, in cast_table_to_schema
raise CastError(
...<3 lines>...
)
datasets.table.CastError: Couldn't cast
shards: struct<deepscaler_shard_00.json: int64, deepscaler_shard_01.json: int64, deepscaler_shard_02.json: i (... 5 chars omitted)
child 0, deepscaler_shard_00.json: int64
child 1, deepscaler_shard_01.json: int64
child 2, deepscaler_shard_02.json: int64
total_input_records: int64
invalid_records_dropped: int64
duplicate_records_removed: int64
final_rows: int64
min_problem_chars: int64
max_problem_chars: int64
data_source: string
ability: string
reward_model: struct<style: string, ground_truth: string>
child 0, style: string
child 1, ground_truth: string
extra_info: struct<index: int64, solution: string>
child 0, index: int64
child 1, solution: string
prompt: list<item: struct<role: string, content: string>>
child 0, item: struct<role: string, content: string>
child 0, role: string
child 1, content: string
to
{'data_source': Value('string'), 'prompt': List({'role': Value('string'), 'content': Value('string')}), 'ability': Value('string'), 'reward_model': {'style': Value('string'), 'ground_truth': Value('string')}, 'extra_info': {'index': Value('int64'), 'solution': 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 1369, in compute_config_parquet_and_info_response
parquet_operations, partial, estimated_dataset_info = stream_convert_to_parquet(
~~~~~~~~~~~~~~~~~~~~~~~~~^
builder, max_dataset_size_bytes=max_dataset_size_bytes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
)
^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 948, in stream_convert_to_parquet
builder._prepare_split(split_generator=splits_generators[split], file_format="parquet")
~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1694, in _prepare_split
for job_id, done, content in self._prepare_split_single(
~~~~~~~~~~~~~~~~~~~~~~~~~~^
gen_kwargs=gen_kwargs, job_id=job_id, **_prepare_split_args
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
):
^
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1880, 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 | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "$|3-\\pi|=$"
}
] | math | {
"style": "rule",
"ground_truth": "\\pi-3"
} | {
"index": 0,
"solution": "1. **Identify the Expression**: We are given the expression $|3-\\pi|$ and need to evaluate its absolute value.\n\n2. **Understanding $\\pi$**: The value of $\\pi$ (pi) is approximately 3.14159, which is greater than 3.\n\n3. **Calculate $3 - \\pi$**: Since $\\pi > 3$, the expression $3 -... |
DeepScaleR | [
{
"role": "user",
"content": "$\\frac{2}{25}=$"
}
] | math | {
"style": "rule",
"ground_truth": ".08"
} | {
"index": 1,
"solution": "To convert the fraction $\\frac{2}{25}$ into a decimal, we can multiply the numerator and the denominator by a number that makes the denominator a power of 10, which simplifies the division process.\n\n1. **Choosing the multiplier**: We choose 4 because multiplying 25 by 4 gives 100, whic... |
DeepScaleR | [
{
"role": "user",
"content": "$3^3 + 3^3 + 3^3 =$"
}
] | math | {
"style": "rule",
"ground_truth": "$3^4$"
} | {
"index": 2,
"solution": "1. **Identify the expression and simplify**: The given expression is $3^3 + 3^3 + 3^3$.\n\n2. **Factor out the common term**: Notice that each term in the sum is $3^3$. We can factor out $3^3$ as follows:\n \\[\n 3^3 + 3^3 + 3^3 = 3 \\times 3^3\n \\]\n\n3. **Apply the exponent rule*... |
DeepScaleR | [
{
"role": "user",
"content": "$\\sqrt{8}+\\sqrt{18}=$"
}
] | math | {
"style": "rule",
"ground_truth": "$5\\sqrt{2}$"
} | {
"index": 3,
"solution": "1. Simplify $\\sqrt{8}$:\n \\[\n \\sqrt{8} = \\sqrt{4 \\times 2} = \\sqrt{4} \\cdot \\sqrt{2} = 2\\sqrt{2}\n \\]\n\n2. Simplify $\\sqrt{18}$:\n \\[\n \\sqrt{18} = \\sqrt{9 \\times 2} = \\sqrt{9} \\cdot \\sqrt{2} = 3\\sqrt{2}\n \\]\n\n3. Add the simplified terms:\n \\[\n 2\... |
DeepScaleR | [
{
"role": "user",
"content": "$\\frac{(3!)!}{3!} = $"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 4,
"solution": "1. **Calculate $3!$:**\n \\[\n 3! = 3 \\times 2 \\times 1 = 6\n \\]\n\n2. **Calculate $(3!)!$:**\n \\[\n (3!)! = (6)!\n \\]\n\n3. **Expand $(6)!$:**\n \\[\n 6! = 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1\n \\]\n\n4. **Simplify the expression $\\frac{(3!)!}{3!}$... |
DeepScaleR | [
{
"role": "user",
"content": "If $2x+1=8$, then $4x+1=$"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 5,
"solution": "1. **Start from the given equation:** \n Given that $2x + 1 = 8$.\n\n2. **Isolate $x$:** \n Subtract $1$ from both sides to simplify the equation:\n \\[\n 2x + 1 - 1 = 8 - 1 \\implies 2x = 7.\n \\]\n Divide both sides by $2$ to solve for $x$:\n \\[\n x = \\frac{7}{2}.\n ... |
DeepScaleR | [
{
"role": "user",
"content": "$\\frac{(.2)^3}{(.02)^2} =$"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 6,
"solution": "1. **Rewrite the expression with powers of 10:** \n We start by expressing $0.2$ and $0.02$ in terms of powers of 10:\n \\[\n 0.2 = \\frac{2}{10} = 2 \\times 10^{-1}, \\quad 0.02 = \\frac{2}{100} = 2 \\times 10^{-2}\n \\]\n\n2. **Calculate the powers:** \n We then calculate $(... |
DeepScaleR | [
{
"role": "user",
"content": "-15 + 9 \\times (6 \\div 3) ="
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 7,
"solution": "To solve the expression $-15+9\\times (6\\div 3)$, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).\n\n1. **Evaluate the expression inside th... |
DeepScaleR | [
{
"role": "user",
"content": "$[x-(y-z)] - [(x-y) - z] = $"
}
] | math | {
"style": "rule",
"ground_truth": "2z"
} | {
"index": 8,
"solution": "We start by simplifying the given expression step by step:\n\n1. **Expand and simplify the inner parentheses:**\n \\[\n [x-(y-z)] - [(x-y) - z] = [x-y+z] - [x-y+z]\n \\]\n Here, we used the property that subtracting a negative is equivalent to adding, i.e., $-(y-z) = -y+z$ and $-(... |
DeepScaleR | [
{
"role": "user",
"content": "One of the factors of $x^4+4$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "$x^2-2x+2$"
} | {
"index": 9,
"solution": "To solve the problem, we need to verify if any of the given options is a factor of the polynomial $x^4 + 4$. We will test each option by either direct substitution or polynomial division.\n\n#### Option (A) $x^2 + 2$\nWe can quickly check if $x^2 + 2$ is a factor of $x^4 + 4$ by attemptin... |
DeepScaleR | [
{
"role": "user",
"content": "4(299) + 3(299) + 2(299) + 298 = "
}
] | math | {
"style": "rule",
"ground_truth": "2989"
} | {
"index": 10,
"solution": "1. **Identify the common factor and apply the distributive property**: Notice that $299$ appears as a common factor in the first three terms. We can factor out $299$ from these terms:\n \\[\n 4(299) + 3(299) + 2(299) + 298 = (4+3+2)(299) + 298\n \\]\n\n2. **Simplify the expression ... |
DeepScaleR | [
{
"role": "user",
"content": "$\\frac{1000^2}{252^2-248^2}$ equals"
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 11,
"solution": "1. **Identify the expression and apply the difference of squares formula**: \n The given expression is $\\frac{1000^2}{252^2-248^2}$. We can simplify the denominator using the difference of squares formula, which states that $a^2 - b^2 = (a-b)(a+b)$.\n\n2. **Apply the formula to the d... |
DeepScaleR | [
{
"role": "user",
"content": "What is the value of $2-(-2)^{-2}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"index": 12,
"solution": "1. **Evaluate the exponentiation and inversion:**\n The expression given is $2 - (-2)^{-2}$. We start by evaluating $(-2)^{-2}$. By the properties of exponents, $a^{-n} = \\frac{1}{a^n}$, so:\n \\[\n (-2)^{-2} = \\frac{1}{(-2)^2}\n \\]\n\n2. **Calculate $(-2)^2$:**\n Since $(-2... |
DeepScaleR | [
{
"role": "user",
"content": "The expression $a^3-a^{-3}$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "\\left(a-\\frac{1}{a}\\right)\\left(a^2+1+\\frac{1}{a^2}\\right)"
} | {
"index": 13,
"solution": "1. **Identify the form of the expression**: We are given the expression $a^3 - a^{-3}$ and need to simplify it. We recognize that this expression can be rewritten using the difference of cubes formula.\n\n2. **Recall the difference of cubes formula**: The formula for the difference of cu... |
DeepScaleR | [
{
"role": "user",
"content": "Evaluate $(x^x)^{(x^x)}$ at $x = 2$."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 14,
"solution": "To solve the expression $(x^x)^{(x^x)}$ at $x = 2$, we follow these steps:\n\n1. **Calculate $x^x$ for $x=2$:**\n \\[\n x^x = 2^2 = 4\n \\]\n\n2. **Substitute $4$ back into the expression $(x^x)^{(x^x)}$:**\n \\[\n (x^x)^{(x^x)} = 4^4\n \\]\n\n3. **Calculate $4^4$:**\n \\[\... |
DeepScaleR | [
{
"role": "user",
"content": "$1,000,000,000,000-777,777,777,777=$"
}
] | math | {
"style": "rule",
"ground_truth": "$222,222,222,223$"
} | {
"index": 15,
"solution": "To solve the problem $1,000,000,000,000 - 777,777,777,777$, we can perform the subtraction directly:\n\n1. **Align the numbers for subtraction:**\n \\[\n \\begin{array}{r}\n 1,000,000,000,000 \\\\\n -777,777,777,777 \\\\\n \\end{array}\n \\]\n\n2. **Subtract each correspondin... |
DeepScaleR | [
{
"role": "user",
"content": "What is the tens digit of $7^{2011}$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 16,
"solution": "1. **Identify the goal:** We need to find the tens digit of $7^{2011}$. This is equivalent to finding $7^{2011} \\mod 100$ because the tens digit of a number is the second last digit from the right.\n\n2. **Use modular arithmetic to simplify the problem:** We start by calculating the pow... |
DeepScaleR | [
{
"role": "user",
"content": "The area of the rectangular region is"
}
] | math | {
"style": "rule",
"ground_truth": ".088 m^2"
} | {
"index": 17,
"solution": "To find the area of a rectangular region, we use the formula:\n\\[ \\text{Area} = \\text{length} \\times \\text{width} \\]\n\nGiven the dimensions of the rectangle are 0.4 meters and 0.22 meters, we calculate the area as follows:\n1. Convert the decimal numbers to fractions to facilitate... |
DeepScaleR | [
{
"role": "user",
"content": "$6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = $"
}
] | math | {
"style": "rule",
"ground_truth": "$6^7$"
} | {
"index": 18,
"solution": "1. **Identify the Expression**: The problem gives the expression $6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6$.\n\n2. **Simplify the Expression**: We observe that the expression consists of six terms, each equal to $6^6$. This can be rewritten using the distributive property of multiplication over... |
DeepScaleR | [
{
"role": "user",
"content": "If $\\log_6 x=2.5$, the value of $x$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "36\\sqrt{6}"
} | {
"index": 19,
"solution": "1. We start with the given logarithmic equation:\n \\[\n \\log_6 x = 2.5\n \\]\n\n2. Recognize that $2.5$ can be expressed as a fraction:\n \\[\n \\log_6 x = \\frac{5}{2}\n \\]\n\n3. Using the property of logarithms that allows the exponentiation of both sides, we rewrite the... |
DeepScaleR | [
{
"role": "user",
"content": "If $991+993+995+997+999=5000-N$, then $N=$"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 20,
"solution": "1. **Identify the problem and express each term in a form that reveals a pattern:**\n \\[\n 991+993+995+997+999=5000-N\n \\]\n We can rewrite each term as $1000$ minus a small number:\n \\[\n (1000-9) + (1000-7) + (1000-5) + (1000-3) + (1000-1)\n \\]\n\n2. **Simplify the ex... |
DeepScaleR | [
{
"role": "user",
"content": "$\\left(\\frac{1}{4}\\right)^{-\\frac{1}{4}}=$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 21,
"solution": "1. **Rewrite the expression using the property of exponents:** \n The property of exponents states that $(a^b)^c = a^{bc}$ and $(\\frac{1}{a})^b = a^{-b}$. Applying this to the given expression:\n \\[\n \\left(\\frac{1}{4}\\right)^{-\\frac{1}{4}} = 4^{\\frac{1}{4}}\n \\]\n\n2. *... |
DeepScaleR | [
{
"role": "user",
"content": "If $10^{2y} = 25$, then $10^{ - y}$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 22,
"solution": "1. We start with the given equation:\n \\[\n 10^{2y} = 25\n \\]\n This can be rewritten using the property of exponents $(a^m)^n = a^{mn}$:\n \\[\n (10^y)^2 = 25\n \\]\n\n2. Taking the square root on both sides, we get:\n \\[\n 10^y = \\sqrt{25} = 5\n \\]\n Note tha... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $[2 - 3(2 - 3)^{-1}]^{-1}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 23,
"solution": "To solve the expression $[2 - 3(2 - 3)^{-1}]^{-1}$, we start by simplifying the innermost expression first.\n\n1. **Simplify $(2 - 3)^{-1}$:**\n \\[\n (2 - 3)^{-1} = (-1)^{-1} = -1\n \\]\n\n2. **Substitute back into the expression:**\n \\[\n 2 - 3(-1) = 2 + 3 = 5\n \\]\n\n3. ... |
DeepScaleR | [
{
"role": "user",
"content": "What is the hundreds digit of $(20! - 15!)?$"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 24,
"solution": "To find the hundreds digit of $(20! - 15!)$, we need to analyze the factorials and their properties modulo $1000$.\n\n1. **Factorial Properties**:\n - $n!$ (where $n \\geq 5$) contains at least one factor of $5$ and at least one factor of $2$, making it divisible by $10$.\n - $n!$ (w... |
DeepScaleR | [
{
"role": "user",
"content": "What is the value of $\\frac{11! - 10!}{9!}$?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 25,
"solution": "1. Start by rewriting the expression $\\dfrac{11!-10!}{9!}$ using the definition of factorial:\n \\[\n \\frac{11! - 10!}{9!} = \\frac{11 \\times 10! - 10!}{9!}\n \\]\n\n2. Factor out $10!$ from the numerator:\n \\[\n \\frac{11 \\times 10! - 10!}{9!} = \\frac{10!(11 - 1)}{9!}\n ... |
DeepScaleR | [
{
"role": "user",
"content": "$\\frac{2+4+6+\\cdots + 34}{3+6+9+\\cdots+51}=$"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{2}{3}$"
} | {
"index": 26,
"solution": "1. **Identify the sequences and their properties:**\n - The numerator is an arithmetic sequence with the first term $a = 2$, common difference $d = 2$, and last term $l = 34$.\n - The denominator is an arithmetic sequence with the first term $a = 3$, common difference $d = 3$, and la... |
DeepScaleR | [
{
"role": "user",
"content": "If $\\log_{10}{m}= b-\\log_{10}{n}$, then $m=$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10^{b}}{n}"
} | {
"index": 27,
"solution": "1. Start by expressing $b$ in terms of logarithm base 10:\n \\[\n b = \\log_{10}{10^b}\n \\]\n This follows from the property of logarithms that $\\log_b{b^x} = x$.\n\n2. Substitute this expression for $b$ into the given equation:\n \\[\n \\log_{10}{m} = \\log_{10}{10^b} - \\... |
DeepScaleR | [
{
"role": "user",
"content": "$\\frac{2}{10}+\\frac{4}{100}+\\frac{6}{1000}=$"
}
] | math | {
"style": "rule",
"ground_truth": ".246"
} | {
"index": 28,
"solution": "1. **Convert fractions to a common denominator**: We start by converting each fraction to have a common denominator, which is 1000 in this case. This is done by multiplying the numerator and denominator of each fraction by the appropriate factor:\n \\[\n \\frac{2}{10} = \\frac{2 \\ti... |
DeepScaleR | [
{
"role": "user",
"content": "$1000 \\times 1993 \\times 0.1993 \\times 10 =$"
}
] | math | {
"style": "rule",
"ground_truth": "$(1993)^2$"
} | {
"index": 29,
"solution": "1. **Break down the expression**: We start by simplifying the expression $1000 \\times 1993 \\times 0.1993 \\times 10$. We can rearrange the terms for easier computation:\n \\[\n 1000 \\times 10 \\times 1993 \\times 0.1993\n \\]\n\n2. **Simplify the powers of 10**: Calculate $1000 ... |
DeepScaleR | [
{
"role": "user",
"content": "The solution of $\\sqrt{5x-1}+\\sqrt{x-1}=2$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "$x=1$"
} | {
"index": 30,
"solution": "1. **Start by squaring both sides of the equation**:\n \\[\n \\left(\\sqrt{5x-1} + \\sqrt{x-1}\\right)^2 = 2^2\n \\]\n Expanding the left side using the identity $(a+b)^2 = a^2 + 2ab + b^2$, we get:\n \\[\n \\sqrt{5x-1}^2 + 2\\sqrt{5x-1}\\sqrt{x-1} + \\sqrt{x-1}^2 = 4\n \\]... |
DeepScaleR | [
{
"role": "user",
"content": "$\\frac{1}{10} + \\frac{2}{20} + \\frac{3}{30} = $"
}
] | math | {
"style": "rule",
"ground_truth": ".3"
} | {
"index": 31,
"solution": "1. **Simplify each fraction**: \n - The fraction $\\frac{1}{10}$ is already in its simplest form.\n - The fraction $\\frac{2}{20}$ can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 2. Thus, $\\frac{2}{20} = \\frac{1}{10}$.\n ... |
DeepScaleR | [
{
"role": "user",
"content": "A big $L$ is formed as shown. What is its area?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 32,
"solution": "1. **Identify the dimensions of the big rectangle**: The problem states that the shape is a big \"L\" formed within a rectangle. Assume the dimensions of the big rectangle are $8$ units by $5$ units based on the typical dimensions used in such problems.\n\n2. **Calculate the area of the ... |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of the prime factors of $2010$?"
}
] | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 33,
"solution": "1. **Find the prime factorization of $2010$:**\n - Start by checking divisibility by the smallest prime, $2$. Since $2010$ is even, it is divisible by $2$. \n \\[\n 2010 \\div 2 = 1005\n \\]\n - Next, check divisibility by $3$. The sum of the digits of $1005$ is $1 + 0 + ... |
DeepScaleR | [
{
"role": "user",
"content": "If $x \\geq 0$, then $\\sqrt{x\\sqrt{x\\sqrt{x}}} =$"
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt[8]{x^7}$"
} | {
"index": 34,
"solution": "1. **Understanding the expression**: We start with the expression $\\sqrt{x\\sqrt{x\\sqrt{x}}}$. This involves nested square roots, which can be simplified using the property that $\\sqrt{y} = y^{\\frac{1}{2}}$ for all $y \\geq 0$.\n\n2. **Simplifying the innermost square root**: Begin b... |
DeepScaleR | [
{
"role": "user",
"content": "What is the value of $1234 + 2341 + 3412 + 4123$"
}
] | math | {
"style": "rule",
"ground_truth": "11110"
} | {
"index": 35,
"solution": "To solve the problem, we need to add the numbers $1234$, $2341$, $3412$, and $4123$. We can do this by aligning the numbers vertically and adding them column by column:\n\n\\[\n\\begin{array}{c}\n\\phantom{+}1234 \\\\\n+2341 \\\\\n+3412 \\\\\n+4123 \\\\\n\\hline\n\\end{array}\n\\]\n\nWe ... |
DeepScaleR | [
{
"role": "user",
"content": "If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 36,
"solution": "1. **Identify the powers of $(1+i)$ and $(1-i)$:**\n We start by calculating $(1+i)^2$ and $(1-i)^2$:\n \\[\n (1+i)^2 = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i\n \\]\n \\[\n (1-i)^2 = 1^2 - 2i + i^2 = 1 - 2i - 1 = -2i\n \\]\n\n2. **Raise both expressions to the 10th power:**\n S... |
DeepScaleR | [
{
"role": "user",
"content": "What is the value of $2^{0^{1^9}} + (2^0)^{1^9}$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 37,
"solution": "To solve the expression \\[2^{\\left(0^{\\left(1^9\\right)}\\right)}+\\left(\\left(2^0\\right)^1\\right)^9,\\] we need to evaluate each term separately.\n\n1. **Evaluate the first term: $2^{\\left(0^{\\left(1^9\\right)}\\right)}$**\n - First, simplify the exponent's base: $1^9 = 1$.\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The product $8 \\times .25 \\times 2 \\times .125 =$"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{2}$"
} | {
"index": 38,
"solution": "1. **Convert Decimals to Fractions**: The given decimals are converted to fractions for easier multiplication.\n - $0.25$ as a fraction is $\\frac{1}{4}$.\n - $0.125$ as a fraction is $\\frac{1}{8}$.\n\n2. **Set Up the Product**: Substitute the decimals with their fractional equivale... |
DeepScaleR | [
{
"role": "user",
"content": "What is the value of $\\frac{(2112-2021)^2}{169}$?"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 39,
"solution": "1. **Calculate the difference in the numerator**: \n \\[\n 2112 - 2021 = 91\n \\]\n This is the exact value, not an approximation.\n\n2. **Square the difference**:\n \\[\n (2112 - 2021)^2 = 91^2 = 8281\n \\]\n\n3. **Divide by the denominator**:\n \\[\n \\frac{8281}{169}... |
DeepScaleR | [
{
"role": "user",
"content": "If $f(x)=\\frac{x(x-1)}{2}$, then $f(x+2)$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{(x+2)f(x+1)}{x}"
} | {
"index": 40,
"solution": "1. **Calculate $f(x+2)$:**\n \\[\n f(x+2) = \\frac{(x+2)(x+1)}{2}\n \\]\n This is obtained by substituting $x+2$ into the function $f(x)$.\n\n2. **Evaluate each answer choice:**\n - **Choice (A):**\n \\[\n f(x) + f(2) = \\frac{x(x-1)}{2} + \\frac{2(2-1)}{2} = \\frac{x(x... |
DeepScaleR | [
{
"role": "user",
"content": "If $A*B$ means $\\frac{A+B}{2}$, then $(3*5)*8$ is"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 41,
"solution": "1. **Interpret the operation $*$**: Given that $A*B = \\frac{A+B}{2}$, we need to apply this operation to the numbers in the expression $(3*5)*8$.\n\n2. **Evaluate $3*5$**:\n \\[\n 3*5 = \\frac{3+5}{2} = \\frac{8}{2} = 4\n \\]\n Here, we added 3 and 5, then divided by 2 as per th... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all the roots of $4x^3-8x^2-63x-9=0$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 42,
"solution": "1. **Identify the polynomial and its degree**:\n Given the polynomial equation:\n \\[\n 4x^3 - 8x^2 - 63x - 9 = 0\n \\]\n This is a cubic polynomial of degree 3.\n\n2. **Simplify the polynomial**:\n Factor out the common coefficient of the highest degree term:\n \\[\n 4(x... |
DeepScaleR | [
{
"role": "user",
"content": "The product of $\\sqrt[3]{4}$ and $\\sqrt[4]{8}$ equals"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt[12]{32}"
} | {
"index": 43,
"solution": "1. **Express the radicals in terms of exponents:**\n - The cube root of 4 can be written as $\\sqrt[3]{4} = 4^{1/3}$. Since $4 = 2^2$, we have $4^{1/3} = (2^2)^{1/3} = 2^{2/3}$.\n - The fourth root of 8 can be written as $\\sqrt[4]{8} = 8^{1/4}$. Since $8 = 2^3$, we have $8^{1/4} = (... |
DeepScaleR | [
{
"role": "user",
"content": "If $9^{x + 2} = 240 + 9^x$, then the value of $x$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 44,
"solution": "1. **Rewrite the given equation using properties of exponents:**\n \\[\n 9^{x+2} = 240 + 9^x\n \\]\n We know that $9^{x+2} = 9^x \\cdot 9^2$. Since $9^2 = 81$, we can substitute:\n \\[\n 81 \\cdot 9^x = 240 + 9^x\n \\]\n\n2. **Isolate terms involving $9^x$:**\n To simplif... |
DeepScaleR | [
{
"role": "user",
"content": "If $f(a)=a-2$ and $F(a,b)=b^2+a$, then $F(3,f(4))$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 45,
"solution": "1. **Calculate $f(4)$**: Given the function $f(a) = a - 2$, substitute $a = 4$:\n \\[\n f(4) = 4 - 2 = 2\n \\]\n\n2. **Evaluate $F(3, f(4))$**: With $f(4) = 2$, we need to find $F(3, 2)$. The function $F(a, b) = b^2 + a$ is given, so substitute $a = 3$ and $b = 2$:\n \\[\n F(3,... |
DeepScaleR | [
{
"role": "user",
"content": "When $x^{13}+1$ is divided by $x-1$, the remainder is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 46,
"solution": "\nWe are given the polynomial \\(x^{13} + 1\\) and we need to find the remainder when it is divided by \\(x-1\\).\n\n**Solution 1: Using the Remainder Theorem**\n\nThe Remainder Theorem states that if a polynomial \\(f(x)\\) is divided by \\(x - a\\), the remainder is \\(f(a)\\). Here, \... |
DeepScaleR | [
{
"role": "user",
"content": "$2.46 \\times 8.163 \\times (5.17 + 4.829)$ is closest to"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 47,
"solution": "To solve the problem, we will approximate the values and calculate the product.\n\n1. **Approximate the values:**\n - $2.46 \\approx 2.5$\n - $8.163 \\approx 8$\n - $5.17 + 4.829 = 10$ (This is exact since $5.17 + 4.829$ is designed to sum to $10$).\n\n2. **Calculate the product us... |
DeepScaleR | [
{
"role": "user",
"content": "How many positive factors of 36 are also multiples of 4?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 48,
"solution": "To solve this problem, we first need to find the prime factorization of 36. Then, we will determine which factors of 36 are also multiples of 4.\n\n1. **Prime Factorization of 36**:\n \\[\n 36 = 2^2 \\cdot 3^2\n \\]\n This factorization shows that 36 is composed of the square of ... |
DeepScaleR | [
{
"role": "user",
"content": "If $\\log_{k}{x} \\cdot \\log_{5}{k} = 3$, then $x$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 49,
"solution": "Given the equation:\n\\[\n\\log_{k}{x} \\cdot \\log_{5}{k} = 3\n\\]\n\nWe can use the property of logarithms that states $\\log_b a \\cdot \\log_c b = \\log_c a$. Applying this property to our equation, we have:\n\\[\n\\log_{k}{x} \\cdot \\log_{5}{k} = \\log_{5}{x}\n\\]\nThus, the equati... |
DeepScaleR | [
{
"role": "user",
"content": "Let $x=-2016$. What is the value of $|| |x|-x|-|x||-x$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"index": 50,
"solution": "1. **Substitute the value of \\( x \\)**:\n Given \\( x = -2016 \\), we substitute this into the expression:\n \\[\n \\Bigg\\vert\\Big\\vert |x|-x\\Big\\vert-|x|\\Bigg\\vert-x\n \\]\n\n2. **Simplify \\( |x| \\) and \\( -x \\)**:\n Since \\( x \\) is negative, \\( |x| = -x = 201... |
DeepScaleR | [
{
"role": "user",
"content": "The slope of the line $\\frac{x}{3} + \\frac{y}{2} = 1$ is"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"index": 51,
"solution": "1. Start with the given equation:\n \\[\n \\frac{x}{3} + \\frac{y}{2} = 1\n \\]\n\n2. To isolate $y$, subtract $\\frac{x}{3}$ from both sides:\n \\[\n \\frac{y}{2} = 1 - \\frac{x}{3}\n \\]\n\n3. Multiply both sides by $2$ to solve for $y$:\n \\[\n y = 2 \\left(1 - \\frac{... |
DeepScaleR | [
{
"role": "user",
"content": "If $\\frac{2+3+4}{3}=\\frac{1990+1991+1992}{N}$, then $N=$"
}
] | math | {
"style": "rule",
"ground_truth": "1991"
} | {
"index": 52,
"solution": "1. **Calculate the left-hand side of the equation:**\n \\[\n \\frac{2+3+4}{3} = \\frac{9}{3} = 3\n \\]\n\n2. **Set up the equation:**\n Given that\n \\[\n \\frac{2+3+4}{3} = \\frac{1990+1991+1992}{N}\n \\]\n substituting the value from step 1, we have:\n \\[\n 3 = \\f... |
DeepScaleR | [
{
"role": "user",
"content": "What is the value of $(2^0 - 1 + 5^2 - 0)^{-1} \\times 5?$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 53,
"solution": "1. **Evaluate the expression inside the parentheses**: \n \\[\n 2^0 - 1 + 5^2 - 0\n \\]\n - \\(2^0 = 1\\) because any non-zero number raised to the power of 0 is 1.\n - \\(5^2 = 25\\) because squaring 5 gives 25.\n - Therefore, the expression simplifies to:\n \\[\n 1 ... |
DeepScaleR | [
{
"role": "user",
"content": "If $y=f(x)=\\frac{x+2}{x-1}$, then it is incorrect to say:"
}
] | math | {
"style": "rule",
"ground_truth": "$f(1)=0$"
} | {
"index": 54,
"solution": "We are given the function $y = f(x) = \\frac{x+2}{x-1}$ and need to determine which statement about this function is incorrect.\n\n1. **Checking option (A)**: We need to solve for $x$ in terms of $y$:\n \\[\n y = \\frac{x+2}{x-1} \\implies y(x-1) = x+2 \\implies yx - y = x + 2 \\impl... |
DeepScaleR | [
{
"role": "user",
"content": "If $f(x) = 5x^2 - 2x - 1$, then $f(x + h) - f(x)$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "h(10x+5h-2)"
} | {
"index": 55,
"solution": "To find $f(x+h) - f(x)$ for the function $f(x) = 5x^2 - 2x - 1$, we start by substituting $x+h$ into the function and then subtracting $f(x)$.\n\n1. **Substitute $x+h$ into $f$:**\n \\[\n f(x+h) = 5(x+h)^2 - 2(x+h) - 1\n \\]\n Expand $(x+h)^2$:\n \\[\n (x+h)^2 = x^2 + 2xh + h... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of $\\sqrt{x^2+y^2}$ if $5x+12y=60$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{60}{13}"
} | {
"index": 56,
"solution": "\n#### Step 1: Understand the problem\nWe need to find the minimum value of $\\sqrt{x^2 + y^2}$ given the constraint $5x + 12y = 60$. This expression $\\sqrt{x^2 + y^2}$ represents the distance from the origin $(0,0)$ to the point $(x,y)$.\n\n#### Step 2: Relate the problem to the distan... |
DeepScaleR | [
{
"role": "user",
"content": "If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{x}{x-1}"
} | {
"index": 57,
"solution": "1. **Given expressions for $x$ and $y$:**\n \\[\n x = 1 + 2^p \\quad \\text{and} \\quad y = 1 + 2^{-p}\n \\]\n\n2. **Express $2^p$ in terms of $x$:**\n \\[\n x = 1 + 2^p \\implies 2^p = x - 1\n \\]\n\n3. **Substitute $2^p$ in the expression for $y$:**\n \\[\n y = 1 + 2^{-... |
DeepScaleR | [
{
"role": "user",
"content": "When simplified, $\\log{8} \\div \\log{\\frac{1}{8}}$ becomes:"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 58,
"solution": "1. **Rewrite the fraction inside the logarithm**: We start by recognizing that $\\frac{1}{8}$ can be expressed as $8^{-1}$. Therefore, the expression $\\log{\\frac{1}{8}}$ can be rewritten using the property of logarithms that $\\log{a^{-b}} = -b \\log{a}$:\n \\[\n \\log{\\frac{1}{8}... |
DeepScaleR | [
{
"role": "user",
"content": "Two is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 59,
"solution": "1. **Translate the percentages into equations:**\n - Given that two is $10\\%$ of $x$, we can write this as:\n \\[\n 2 = 0.10 \\times x\n \\]\n - Similarly, two is $20\\%$ of $y$, which can be written as:\n \\[\n 2 = 0.20 \\times y\n \\]\n\n2. **Solve for $x$ ... |
DeepScaleR | [
{
"role": "user",
"content": "How many primes less than $100$ have $7$ as the ones digit?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 60,
"solution": "1. **Identify the numbers with 7 as the ones digit under 100**: \n The numbers are $7, 17, 27, 37, 47, 57, 67, 77, 87, 97$.\n\n2. **Check each number for primality**:\n - $7$ is prime.\n - $17$ is prime.\n - $27$ is not prime (divisible by $3$).\n - $37$ is prime.\n - $47$ is... |
DeepScaleR | [
{
"role": "user",
"content": "If $\\frac{\\frac{x}{4}}{2}=\\frac{4}{\\frac{x}{2}}$, then $x=$"
}
] | math | {
"style": "rule",
"ground_truth": "\\pm 8"
} | {
"index": 61,
"solution": "1. Start by simplifying the given equation:\n \\[\n \\dfrac{\\frac{x}{4}}{2} = \\dfrac{4}{\\frac{x}{2}}\n \\]\n Simplify both sides:\n \\[\n \\frac{x}{4 \\times 2} = \\frac{4 \\times 2}{x}\n \\]\n \\[\n \\frac{x}{8} = \\frac{8}{x}\n \\]\n\n2. Cross-multiply to elimina... |
DeepScaleR | [
{
"role": "user",
"content": "Define $x\\otimes y=x^3-y$. What is $h\\otimes (h\\otimes h)$?"
}
] | math | {
"style": "rule",
"ground_truth": "h"
} | {
"index": 62,
"solution": "1. **Define the operation $\\otimes$:** Given $x \\otimes y = x^3 - y$.\n\n2. **Calculate $h \\otimes h$:** Using the definition, substitute $x = h$ and $y = h$:\n \\[\n h \\otimes h = h^3 - h.\n \\]\n\n3. **Calculate $h \\otimes (h \\otimes h)$:** Now, substitute $x = h$ and $y = ... |
DeepScaleR | [
{
"role": "user",
"content": "Suppose $a$ is $150\\%$ of $b$. What percent of $a$ is $3b$?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 63,
"solution": "1. **Understanding the problem**: We are given that $a$ is $150\\%$ of $b$. This can be expressed as:\n \\[\n a = 150\\% \\times b = 1.5b\n \\]\n\n2. **Finding $3b$ in terms of $a$**: We need to find what percent $3b$ is of $a$. First, express $3b$ using the relationship between $a... |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}"
} | {
"index": 64,
"solution": "1. **Simplify the Numerator**: \n The expression $1^{4y-1}$ simplifies to $1$ for any real number $y$, because any power of $1$ is $1$. Thus, the expression becomes:\n \\[\n \\frac{1^{4y-1}}{5^{-1} + 3^{-1}} = \\frac{1}{5^{-1} + 3^{-1}}\n \\]\n\n2. **Simplify the Denominator**:\n... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all integers between 50 and 350 which end in 1 is"
}
] | math | {
"style": "rule",
"ground_truth": "5880"
} | {
"index": 65,
"solution": "1. **Identify the sequence**: The problem asks for the sum of all integers between 50 and 350 that end in 1. These integers are $51, 61, 71, \\ldots, 341$.\n\n2. **Determine the sequence type**: The sequence is an arithmetic sequence where each term increases by 10.\n\n3. **Find the firs... |
DeepScaleR | [
{
"role": "user",
"content": "Suppose $A>B>0$ and A is $x$% greater than $B$. What is $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "100\\left(\\frac{A-B}{B}\\right)"
} | {
"index": 66,
"solution": "1. **Understanding the Problem Statement:**\n Given that $A > B > 0$ and $A$ is $x\\%$ greater than $B$, we need to find the expression for $x$.\n\n2. **Expressing $A$ in terms of $B$ and $x$:**\n Since $A$ is $x\\%$ greater than $B$, we can write:\n \\[\n A = B + \\frac{x}{100} ... |
DeepScaleR | [
{
"role": "user",
"content": "The simplest form of $1 - \\frac{1}{1 + \\frac{a}{1 - a}}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "a"
} | {
"index": 67,
"solution": "We start by simplifying the expression inside the fraction:\n\\[ 1 - \\frac{1}{1 + \\frac{a}{1 - a}}. \\]\n\n1. **Simplify the innermost fraction**:\n \\[ \\frac{a}{1 - a}. \\]\n\n2. **Add this fraction to 1**:\n \\[ 1 + \\frac{a}{1 - a} = \\frac{1 - a + a}{1 - a} = \\frac{1}{1 - a}.... |
DeepScaleR | [
{
"role": "user",
"content": "If $\\frac{1}{x} - \\frac{1}{y} = \\frac{1}{z}$, then $z$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{xy}{y - x}"
} | {
"index": 68,
"solution": "1. Start with the given equation:\n \\[\n \\frac{1}{x} - \\frac{1}{y} = \\frac{1}{z}\n \\]\n\n2. Find a common denominator for the left-hand side:\n \\[\n \\frac{y}{xy} - \\frac{x}{xy} = \\frac{1}{z}\n \\]\n\n3. Combine the fractions:\n \\[\n \\frac{y-x}{xy} = \\frac{1}{z... |
DeepScaleR | [
{
"role": "user",
"content": "If $|x-\\log y|=x+\\log y$ where $x$ and $\\log y$ are real, then"
}
] | math | {
"style": "rule",
"ground_truth": "x(y-1)=0"
} | {
"index": 69,
"solution": "Given the equation $|x-\\log y|=x+\\log y$, we need to consider the properties of the absolute value function and the possible values of $x$ and $\\log y$.\n\n1. **Understanding the absolute value equation**:\n The absolute value equation $|a| = b$ holds if and only if $a = b$ or $a = ... |
DeepScaleR | [
{
"role": "user",
"content": "The fraction $\\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to:"
}
] | math | {
"style": "rule",
"ground_truth": "a^{-2}+b^{-2}"
} | {
"index": 70,
"solution": "1. **Rewrite the numerator using the difference of squares formula**: \n The difference of squares formula states that $x^2 - y^2 = (x+y)(x-y)$. Applying this to $a^{-4} - b^{-4}$, where $x = a^{-2}$ and $y = b^{-2}$, we get:\n \\[\n a^{-4} - b^{-4} = (a^{-2} - b^{-2})(a^{-2} + b^{... |
DeepScaleR | [
{
"role": "user",
"content": "The number of solutions in positive integers of $2x+3y=763$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 71,
"solution": "1. **Formulate the Equation**: We start with the Diophantine equation given:\n \\[\n 2x + 3y = 763\n \\]\n\n2. **Find a Particular Solution**: We need to find one particular solution $(x_0, y_0)$ in integers. By trial and error or using the extended Euclidean algorithm, we find:\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\left(256\\right)^{.16}\\left(256\\right)^{.09}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 72,
"solution": "1. **Identify the operation and simplify the expression**: We are given the expression $\\left(256\\right)^{0.16}\\left(256\\right)^{0.09}$. Using the property of exponents that states when multiplying like bases you add the exponents, we can simplify this expression:\n \\[\n 256^{0.... |
DeepScaleR | [
{
"role": "user",
"content": "What is the value of the expression $\\sqrt{16\\sqrt{8\\sqrt{4}}}$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 73,
"solution": "To solve the expression $\\sqrt{16\\sqrt{8\\sqrt{4}}}$, we will simplify the expression step-by-step:\n\n1. **Simplify the innermost square root:**\n \\[\n \\sqrt{4} = 2\n \\]\n\n2. **Substitute and simplify the next square root:**\n \\[\n \\sqrt{8\\sqrt{4}} = \\sqrt{8 \\cdot 2... |
DeepScaleR | [
{
"role": "user",
"content": "What number is one third of the way from $\\frac14$ to $\\frac34$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 74,
"solution": "\n#### Step-by-step Explanation:\n\n1. **Understanding the Problem:**\n We need to find the number that is one third of the way from $\\frac{1}{4}$ to $\\frac{3}{4}$.\n\n2. **Using the Concept of Weighted Average:**\n The number one third of the way from $\\frac{1}{4}$ to $\\frac{3}{... |
DeepScaleR | [
{
"role": "user",
"content": "If $\\sin{2x}\\sin{3x}=\\cos{2x}\\cos{3x}$, then one value for $x$ is"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 75,
"solution": "1. **Start with the given equation:**\n \\[\n \\sin{2x}\\sin{3x} = \\cos{2x}\\cos{3x}\n \\]\n\n2. **Divide both sides by $\\cos{2x}\\cos{3x}$:**\n \\[\n \\frac{\\sin{2x}\\sin{3x}}{\\cos{2x}\\cos{3x}} = 1\n \\]\n Assuming $\\cos{2x} \\neq 0$ and $\\cos{3x} \\neq 0$, we can s... |
DeepScaleR | [
{
"role": "user",
"content": "How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 76,
"solution": "To find the number of integer pairs $(m,n)$ that satisfy the equation $m+n=mn$, we can start by rearranging the equation:\n\n\\[ m+n = mn \\]\n\nSubtracting $m+n$ from both sides gives:\n\n\\[ mn - m - n = 0 \\]\n\nAdding 1 to both sides, we can factorize the left-hand side:\n\n\\[ mn - ... |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest integer larger than $(\\sqrt{3}+\\sqrt{2})^6$?"
}
] | math | {
"style": "rule",
"ground_truth": "970"
} | {
"index": 77,
"solution": "To find the smallest integer larger than $(\\sqrt{3}+\\sqrt{2})^6$, we start by simplifying the expression.\n\n1. **Expand $(\\sqrt{3}+\\sqrt{2})^6$ using the binomial theorem:**\n \\[\n (\\sqrt{3}+\\sqrt{2})^6 = \\sum_{k=0}^6 \\binom{6}{k} (\\sqrt{3})^{6-k} (\\sqrt{2})^k\n \\]\n ... |
DeepScaleR | [
{
"role": "user",
"content": "For $x$ real, the inequality $1 \\le |x-2| \\le 7$ is equivalent to"
}
] | math | {
"style": "rule",
"ground_truth": "$-5 \\le x \\le 1$ or $3 \\le x \\le 9$"
} | {
"index": 78,
"solution": "To solve the inequality $1 \\leq |x-2| \\leq 7$, we need to consider the absolute value inequality in two parts: the lower bound and the upper bound.\n\n1. **Solving the lower bound inequality $|x-2| \\geq 1$:**\n - The absolute value inequality $|x-2| \\geq 1$ implies that the distanc... |
DeepScaleR | [
{
"role": "user",
"content": "\\angle 1 + \\angle 2 = 180^\\circ\n\\angle 3 = \\angle 4\nFind \\angle 4."
}
] | math | {
"style": "rule",
"ground_truth": "35^\\circ"
} | {
"index": 79,
"solution": "1. **Using the left triangle's angle sum property**: The sum of angles in any triangle is $180^\\circ$. Given $\\angle 1 + 70^\\circ + 40^\\circ = 180^\\circ$, we can solve for $\\angle 1$:\n \\[\n \\angle 1 + 110^\\circ = 180^\\circ\n \\]\n \\[\n \\angle 1 = 180^\\circ - 110^\... |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of an isosceles right triangle is $2p$. Its area is:"
}
] | math | {
"style": "rule",
"ground_truth": "$(3-2\\sqrt{2})p^2$"
} | {
"index": 80,
"solution": "1. **Identify the properties of the triangle**: We are given an isosceles right triangle. In such a triangle, the two legs are equal, and the hypotenuse is $\\sqrt{2}$ times the length of a leg. Let the length of each leg be $x$.\n\n2. **Write the equation for the perimeter**: The perime... |
DeepScaleR | [
{
"role": "user",
"content": "If $3^{2x}+9=10\\left(3^{x}\\right)$, then the value of $(x^2+1)$ is "
}
] | math | {
"style": "rule",
"ground_truth": "1 or 5"
} | {
"index": 81,
"solution": "1. **Rewrite the equation**: Start by rewriting the given equation:\n \\[\n 3^{2x} + 9 = 10 \\cdot 3^x\n \\]\n Rearrange the terms to form a quadratic equation in terms of $3^x$:\n \\[\n 3^{2x} - 10 \\cdot 3^x + 9 = 0\n \\]\n\n2. **Substitute $u$ for $3^x$**: Let $u = 3^x$.... |
DeepScaleR | [
{
"role": "user",
"content": "How many unordered pairs of edges of a given cube determine a plane?"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 82,
"solution": "1. **Understanding the Problem**: We need to find how many unordered pairs of edges in a cube determine a plane. Two edges determine a plane if they are either parallel or intersecting (not skew).\n\n2. **Total Number of Edges in a Cube**: A cube has 12 edges.\n\n3. **Choosing One Edge**... |
DeepScaleR | [
{
"role": "user",
"content": "The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if:"
}
] | math | {
"style": "rule",
"ground_truth": "c = a"
} | {
"index": 83,
"solution": "1. **Start by normalizing the quadratic equation**: Given the quadratic equation $ax^2 + bx + c = 0$, we can divide each term by $a$ (assuming $a \\neq 0$) to simplify the equation:\n \\[\n x^2 + \\frac{b}{a}x + \\frac{c}{a} = 0\n \\]\n\n2. **Identify the roots**: Let the roots of ... |
DeepScaleR | [
{
"role": "user",
"content": "How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?"
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 84,
"solution": "1. **Calculate the lower bound**: We start by calculating $2^8 + 1$:\n \\[\n 2^8 = 256 \\quad \\text{so} \\quad 2^8 + 1 = 257.\n \\]\n We need to find the smallest perfect cube greater than 257. We know $6^3 = 216$ and $7^3 = 343$. Since $343 > 257$, the smallest cube greater tha... |
DeepScaleR | [
{
"role": "user",
"content": "The root(s) of $\\frac {15}{x^2 - 4} - \\frac {2}{x - 2} = 1$ is (are):"
}
] | math | {
"style": "rule",
"ground_truth": "-3 \\text{ and } 5"
} | {
"index": 85,
"solution": "1. **Identify the common denominator and simplify the equation:**\n Given the equation:\n \\[\n \\frac{15}{x^2 - 4} - \\frac{2}{x - 2} = 1\n \\]\n We recognize that $x^2 - 4 = (x-2)(x+2)$. Thus, the common denominator for the fractions is $x^2 - 4$. Rewriting the second term wi... |
DeepScaleR | [
{
"role": "user",
"content": "What fraction of the large $12$ by $18$ rectangular region is shaded?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 86,
"solution": "1. **Identify the total area of the large rectangle**: \n The large rectangle has dimensions $12$ by $18$. Therefore, its area is calculated as:\n \\[\n \\text{Area} = 12 \\times 18 = 216 \\text{ square units}\n \\]\n\n2. **Determine the area of the quarter rectangle**:\n Since... |
DeepScaleR | [
{
"role": "user",
"content": "If $b>1$, $x>0$, and $(2x)^{\\log_b 2}-(3x)^{\\log_b 3}=0$, then $x$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 87,
"solution": "Given the equation:\n\\[\n(2x)^{\\log_b 2} - (3x)^{\\log_b 3} = 0\n\\]\n\nWe can rewrite the terms using the properties of exponents:\n\\[\n(2x)^{\\log_b 2} = 2^{\\log_b 2} \\cdot x^{\\log_b 2} = b^{\\log_b 2} \\cdot x^{\\log_b 2} = 2 \\cdot x^{\\log_b 2}\n\\]\n\\[\n(3x)^{\\log_b 3} = 3^... |
DeepScaleR | [
{
"role": "user",
"content": "What is the value of $\\sqrt{(3-2\\sqrt{3})^2}+\\sqrt{(3+2\\sqrt{3})^2}$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 88,
"solution": "1. **Define the expression and simplify each square root:**\n Let $x = \\sqrt{(3-2\\sqrt{3})^2} + \\sqrt{(3+2\\sqrt{3})^2}$.\n\n Since squaring a real number and then taking the square root gives the absolute value of the original number, we have:\n \\[\n \\sqrt{(3-2\\sqrt{3})^2}... |
DeepScaleR | [
{
"role": "user",
"content": "How many whole numbers between 1 and 1000 do not contain the digit 1?"
}
] | math | {
"style": "rule",
"ground_truth": "728"
} | {
"index": 89,
"solution": "1. **Identify the Range and Condition**: We need to find how many whole numbers between 1 and 1000 do not contain the digit 1.\n\n2. **Consider the Number of Digits**: Numbers between 1 and 1000 can have 1, 2, or 3 digits. We will consider each case separately.\n\n3. **Counting 1-Digit N... |
DeepScaleR | [
{
"role": "user",
"content": "What is the greatest number of consecutive integers whose sum is $45?$"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 90,
"solution": "To find the greatest number of consecutive integers whose sum is $45$, we need to consider sequences of integers, both positive and negative.\n\n1. **Understanding the sum of consecutive integers**:\n The sum of $N$ consecutive integers starting from $a$ can be expressed as:\n \\[\n ... |
DeepScaleR | [
{
"role": "user",
"content": "What is equivalent to $\\sqrt{\\frac{x}{1-\\frac{x-1}{x}}}$ when $x < 0$?"
}
] | math | {
"style": "rule",
"ground_truth": "-x"
} | {
"index": 91,
"solution": "To solve the problem, we first simplify the expression inside the square root:\n\n\\[\n\\sqrt{\\frac{x}{1-\\frac{x-1}{x}}}\n\\]\n\n1. Simplify the denominator:\n \\[\n 1 - \\frac{x-1}{x} = 1 - \\left(\\frac{x}{x} - \\frac{1}{x}\\right) = 1 - 1 + \\frac{1}{x} = \\frac{1}{x}\n \\]\n\... |
DeepScaleR | [
{
"role": "user",
"content": "If $p$ is a prime and both roots of $x^2+px-444p=0$ are integers, then"
}
] | math | {
"style": "rule",
"ground_truth": "31 < p \\le 41"
} | {
"index": 92,
"solution": "1. **Identify the requirement for integer roots**: The quadratic equation given is $x^2 + px - 444p = 0$. For the roots of this equation to be integers, the discriminant must be a perfect square. The discriminant $\\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by $\\Delta ... |
DeepScaleR | [
{
"role": "user",
"content": "For how many integers $n$ is $\\frac n{20-n}$ the square of an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 93,
"solution": "To solve the problem, we need to find the integers $n$ for which $\\frac{n}{20-n}$ is a perfect square.\n\n1. **Examine the range of $n$:**\n - If $n < 0$ or $n > 20$, the fraction $\\frac{n}{20-n}$ is negative, and thus cannot be a perfect square.\n - If $n = 20$, the fraction is un... |
DeepScaleR | [
{
"role": "user",
"content": "The limit of $\\frac {x^2-1}{x-1}$ as $x$ approaches $1$ as a limit is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 94,
"solution": "To find the limit of the function $\\frac{x^2-1}{x-1}$ as $x$ approaches 1, we start by simplifying the expression.\n\n1. **Factorize the Numerator**: \n The numerator $x^2 - 1$ is a difference of squares, which can be factored as:\n \\[\n x^2 - 1 = (x + 1)(x - 1)\n \\]\n\n2. **S... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\frac {1}{2 - \\frac {1}{2 - \\frac {1}{2 - \\frac12}}}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 95,
"solution": "To solve the expression $\\frac {1}{2 - \\frac {1}{2 - \\frac {1}{2 - \\frac12}}}$, we will evaluate it from the innermost fraction outward.\n\n1. **Evaluate the innermost fraction:**\n \\[\n 2 - \\frac{1}{2} = 2 - 0.5 = 1.5\n \\]\n So, the expression simplifies to:\n \\[\n \... |
DeepScaleR | [
{
"role": "user",
"content": "How many positive integer factors of $2020$ have more than $3$ factors?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 96,
"solution": "To solve this problem, we first need to find the prime factorization of $2020$. We have:\n\\[ 2020 = 2^2 \\times 5 \\times 101. \\]\n\nNext, we use the formula for the number of divisors of a number given its prime factorization. If $n = p^a \\times q^b \\times r^c \\times \\ldots$, then... |
DeepScaleR | [
{
"role": "user",
"content": "If $\\frac{x}{x-1} = \\frac{y^2 + 2y - 1}{y^2 + 2y - 2},$ then $x$ equals"
}
] | math | {
"style": "rule",
"ground_truth": "y^2 + 2y - 1"
} | {
"index": 97,
"solution": "1. Start with the given equation:\n \\[\n \\frac{x}{x-1} = \\frac{y^2 + 2y - 1}{y^2 + 2y - 2}\n \\]\n\n2. Cross multiply to eliminate the fractions:\n \\[\n x(y^2 + 2y - 2) = (x-1)(y^2 + 2y - 1)\n \\]\n\n3. Expand both sides of the equation:\n \\[\n xy^2 + 2xy - 2x = xy^2... |
DeepScaleR | [
{
"role": "user",
"content": "If $a \\otimes b = \\dfrac{a + b}{a - b}$, then $(6\\otimes 4)\\otimes 3 =$"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 98,
"solution": "1. **Calculate $6 \\otimes 4$:**\n \\[\n 6 \\otimes 4 = \\frac{6 + 4}{6 - 4} = \\frac{10}{2} = 5\n \\]\n\n2. **Calculate $5 \\otimes 3$:**\n \\[\n 5 \\otimes 3 = \\frac{5 + 3}{5 - 3} = \\frac{8}{2} = 4\n \\]\n\n3. **Combine the results:**\n Since $(6 \\otimes 4) \\otimes 3 ... |
DeepScaleR | [
{
"role": "user",
"content": "The number of diagonals that can be drawn in a polygon of 100 sides is:"
}
] | math | {
"style": "rule",
"ground_truth": "4850"
} | {
"index": 99,
"solution": "To find the number of diagonals in a polygon, we use the formula:\n\\[\n\\frac{n(n-3)}{2}\n\\]\nwhere $n$ is the number of sides of the polygon. This formula arises because each vertex of the polygon can connect to $n-3$ other vertices to form a diagonal (it cannot connect to itself or i... |
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