Dataset Preview
The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
The dataset generation failed
Error code: DatasetGenerationError
Exception: CastError
Message: Couldn't cast
input: string
output: string
gts: string
gt_hash: string
score: double
step: int64
uid: string
data_source: string
reward: double
acc: bool
verifier_status: string
verifier_error: string
num_cases: int64
num_passed: int64
testtype: string
question_id: string
num_evaluated: int64
case_status_counts_json: string
pred: string
to
{'input': Value('string'), 'output': Value('string'), 'gts': Value('string'), 'score': Value('float64'), 'step': Value('int64'), 'reward': Value('float64'), 'acc': Value('bool'), 'pred': 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
input: string
output: string
gts: string
gt_hash: string
score: double
step: int64
uid: string
data_source: string
reward: double
acc: bool
verifier_status: string
verifier_error: string
num_cases: int64
num_passed: int64
testtype: string
question_id: string
num_evaluated: int64
case_status_counts_json: string
pred: string
to
{'input': Value('string'), 'output': Value('string'), 'gts': Value('string'), 'score': Value('float64'), 'step': Value('int64'), 'reward': Value('float64'), 'acc': Value('bool'), 'pred': 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.
input string | output string | gts string | score float64 | step int64 | reward float64 | acc bool | pred string |
|---|---|---|---|---|---|---|---|
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $a$ and $b$ be relatively prime positive integers such that $\log_a b = \log_{200}(10000b)$. Then $a+b$ is
\[\textbf{(A) }15\qquad \textbf{(B) }25\qquad \textbf{(C) }35\qquad \textbf{(D) }45\qquad \textbf{(E) }55\qquad \]
Let's solve each problem step-by-step using Python and SymPy.
### Problem 1
Given the system ... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | 443 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $f(x) = x^2 - 2x$. Find the number of real solutions to the equation $f(f(f(x))) = 3$. Please reason step by step, and put your final answer within \boxed{}. Shay
Let $f(x) = x^2 - 2x$. Find the number of real solutions to the equation $f(f(f(x))) = 3$. Please reason step by step, and put your final answer within \... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $a, b, c, d, e$ be positive integers such that $a + b + c + d + e = 2010$. Find the least possible value of the expression
\[\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d} + \sqrt{e}\]
Please reason step by step, and put your final answer within \boxed{}. | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | 26 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | The $2007$th term and the $2008$th term of a sequence are -328 and -253, respectively. Each term after the first is obtained by subtracting the preceding term from the term before that. What is the first term in the sequence?
Let's solve each problem step by step using Python and SymPy.
### Problem 1: Find the largest... | 33 | 0 | 0 | 0 | false | 114 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | To solve the given system of equations, we can start by expressing the logarithmic equations in terms of the variables \(x\), \(y\), and \(z\).
The given system of equations is:
\[
\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}
\]
\[
\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}
\]
\[
\log_2\left(\frac{z}{xy}\right) = ... | 33 | 0 | 0 | 0 | false | 149 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $a,b,$ and $c$ be positive real numbers that satisfy $2 \log_a(b) = 1 + \log_a(b + c)$ and $2\log_b(c) = 1 + \log_b(c + a)$. Find the value of $\log_a(c)$ if $\log_a(c) = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. Please reason step by step, and put your final answer with... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Given that $x$, $y$, $z$, and $w$ are positive real numbers satisfying the system of equations
\[\log_x(y) = 4\]
\[\log_y(z) = 2\]
\[\log_z(w) = 3\]
\[\log_w(x) = 5\]
Then the value of $\log_x(z) + \log_y(w) + \log_z(x) + \log_w(y)$ is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $a = \log_2 3$ and $b = \log_5 3$. Then $15^{(a-b)/(2a)}$ can be written in the form $\sqrt[c]{d}$ for positive integers $c$ and $d$, and with $c$ as small as possible. Find $c+d$. Please reason step by step, and put your final answer within \boxed{}.犹如
(Original text) Let's solve each problem step by step.
### Pr... | 33 | 0 | 0 | 0 | false | 3 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x$ and $y$ be real numbers that satisfy the equation
\[2^x + 2^y = 8.\]
There are real numbers $a$ and $b$ such that for all ordered pairs $(x,y)$ satisfying the equation $2^x + 2^y = 8$, the expression $2^{2x+3} + 2^{2y+3}$ may be written as $a(2^{x+y+b} + 1)$. Find the value of $a+b$. Please reason step by step... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | To solve the given system of equations, we can start by converting the logarithmic equations into exponential form. The given system of equations is:
\[
\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}
\]
\[
\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}
\]
\[
\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}
\]
Converting t... | 33 | 0 | 0 | 0 | false | 14 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x, y,$ and $z$ be positive real numbers satisfying the system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Determine the value of $\left|\log_2(x^4y^3z^2)\right|$. Please provide a detailed explan... | 33 | 0 | 0 | 0 | false | 13 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | To solve the given system of logarithmic equations, we can use the properties of logarithms. Let's break down the problem step by step.
1. **Rewrite the Equations:**
\[
\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}
\]
\[
\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}
\]
\[
\log_2\left(\frac{z}{x... | 33 | 0 | 0 | 0 | false | 8 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 1 | 0 | 1 | true | 33 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Given the system of equations:
\[
\log_2\left({x \over yz}\right) = {1 \over 2}
\]
\[
\log_2\left({y \over xz}\right) = {1 \over 3}
\]
\[
\log_2\left({z \over xy}\right) = {1 \over 4}
\]
We can rewrite these equations as:
\[
\log_2(x) - \log_2(yz) = \frac{1}{2}
\]
\[
\log_2(y) - \log_2(xz) = \frac{1}{3}
\]
\[
\log_2(z)... | 33 | 0 | 0 | 0 | false | 27 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | 3 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | To solve this problem, let's first analyze the given system of logarithmic equations. We have:
\[
\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}
\]
\[
\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}
\]
\[
\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}
\]
Using the properties of logarithms, we can rewrite these equations ... | 33 | 0 | 0 | 0 | false | 7 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 1 | 0 | 1 | true | 33 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $a, b, c$ and $d$ be positive real numbers such that $a + b + c + d = 1$. Find the minimum value of
\[\frac{a + b}{ab} + \frac{a + c}{ac} + \frac{a + d}{ad} + \frac{b + c}{bc} + \frac{b + d}{bd} + \frac{c + d}{cd}.\]
Please reason step by step, and put your final answer within \boxed{}. NUIT
Let $f(x) = x^2 - 2x +... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $a, b,$ and $c$ be nonzero real numbers such that $a + b + c = 0$ and $a^3 + b^3 + c^3 = a^5 + b^5 + c^5$. Find the value of $a^2 + b^2 + c^2$. Please reason step by step, and put your final answer within \boxed{}. 1. **Find the value of \( x \) in the equation \( 16^{x-1} = 2^{4x+2} \).**
First, express both s... | 33 | 0 | 0 | 0 | false | \frac{13}{4} |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | To solve this problem, we need to first convert the given logarithmic equations into algebraic equations and then solve for \(x\), \(y\), and \(z\). Here are the steps:
1. Convert the logarithmic equations to algebraic form:
\[
\log_2\left(\frac{x}{yz}\right) = \frac{1}{2} \implies \frac{x}{yz} = 2^{\frac{1}{2}}... | 33 | 1 | 0 | 1 | true | 33 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | 21 |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$... | 33 | 0 | 0 | 0 | false | |
system
Please reason step by step, and put your final answer within \boxed{}.
user
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over ... | Let $a, b,$ and $c$ be positive integers such that $a + b + c = 20$. Find the maximum value of $abc$. Please reason step by step, and put your final answer within \boxed{}. Shay
Let $a, b,$ and $c$ be positive integers such that $a + b + c = 20$. Find the maximum value of $abc$. Please reason step by step, and put your... | 33 | 0 | 0 | 0 | false |
End of preview.
No dataset card yet
- Downloads last month
- -