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The dataset generation failed
Error code: DatasetGenerationError
Exception: CastError
Message: Couldn't cast
input: string
output: string
score: double
step: int64
reward_methods: string
outcome_score: double
ground_truth: string
truncated: double
final_reward_used: double
advantage_first_token: double
advantage_first_mid_token: double
advantage_middle_token: double
advantage_mid_last_token: double
advantage_last_token: double
uid: string
to
{'input': Value('string'), 'output': Value('string'), 'score': Value('float64'), 'step': Value('int64'), 'reward_methods': Value('string'), 'outcome_score': Value('float64'), 'ground_truth': Value('string'), 'final_reward_used': Value('float64'), 'advantage_first_token': Value('float64'), 'advantage_first_mid_token': Value('float64'), 'advantage_middle_token': Value('float64'), 'advantage_mid_last_token': Value('float64'), 'advantage_last_token': Value('float64'), 'uid': 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 1835, in _prepare_split_single
for key, table in generator:
^^^^^^^^^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 621, in wrapped
for item in generator(*args, **kwargs):
~~~~~~~~~^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/packaged_modules/json/json.py", line 361, 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 2381, in table_cast
return cast_table_to_schema(table, schema)
File "/usr/local/lib/python3.14/site-packages/datasets/table.py", line 2309, in cast_table_to_schema
raise CastError(
...<3 lines>...
)
datasets.table.CastError: Couldn't cast
input: string
output: string
score: double
step: int64
reward_methods: string
outcome_score: double
ground_truth: string
truncated: double
final_reward_used: double
advantage_first_token: double
advantage_first_mid_token: double
advantage_middle_token: double
advantage_mid_last_token: double
advantage_last_token: double
uid: string
to
{'input': Value('string'), 'output': Value('string'), 'score': Value('float64'), 'step': Value('int64'), 'reward_methods': Value('string'), 'outcome_score': Value('float64'), 'ground_truth': Value('string'), 'final_reward_used': Value('float64'), 'advantage_first_token': Value('float64'), 'advantage_first_mid_token': Value('float64'), 'advantage_middle_token': Value('float64'), 'advantage_mid_last_token': Value('float64'), 'advantage_last_token': Value('float64'), 'uid': 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 1379, 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 956, 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 1702, 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 1887, 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 | score float64 | step int64 | reward_methods string | outcome_score float64 | ground_truth string | final_reward_used float64 | advantage_first_token float64 | advantage_first_mid_token float64 | advantage_middle_token float64 | advantage_mid_last_token float64 | advantage_last_token float64 | uid string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
system
Please reason step by step and put the final answer in \boxed{}.
user
In the decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) participate, and digits may repeat. It is known that the sum of the digits of the number \( 2M \) is 31, and the sum of the digits of th... | Let's break this down into a series of logical steps:
1. **Understanding the Problem**:
- We are given an even number \( M \) which uses only the digits 0, 2, 4, 5, 7, and 9.
- The sum of the digits of \( 2M \) is 31.
- The sum of the digits of \( M / 2 \) is 28.
2. **Analyze the Relationship Between \( M \... | 0 | 1 | RULE_BASED | 0 | 29 | 0 | 0 | 0 | 0 | 0 | 0 | 474670c2-d09f-45cc-b530-ed1c231d86ee |
system
Please reason step by step and put the final answer in \boxed{}.
user
In the decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) participate, and digits may repeat. It is known that the sum of the digits of the number \( 2M \) is 31, and the sum of the digits of th... | Let's denote the sum of the digits of \( M \) as \( S(M) \). We know the following:
1. The sum of the digits of \( 2M \) is 31.
2. The sum of the digits of \( \frac{M}{2} \) is 28.
3. \( M \) is an even number.
Since the sum of the digits of \( 2M \) is 31, and each digit of \( M \) contributes to the sum of digits of... | 0 | 1 | RULE_BASED | 0 | 29 | 0 | 0 | 0 | 0 | 0 | 0 | 474670c2-d09f-45cc-b530-ed1c231d86ee |
system
Please reason step by step and put the final answer in \boxed{}.
user
There's a real number written on every field of a $n \times n$ chess board. The sum of all numbers of a "cross" (union of a line and a column) is $\geq a$ . What's the smallest possible sum of all numbers on the board¿ Let's think step by s... | We'll prove that the smallest possible sum is $\frac{n}{2} a (n + 1)$ (obviously, this would be the minimal case).
NdrFcphysics
If we prove this, we could conjecture that matrix $\Bold{b}$ with all positive entries (and having $\frac{a}{n} (n + 1)$ on each row and each column) would work.
뛩user
What does "à la K... | 0 | 1 | RULE_BASED | 0 | \frac{n^2}{2n-1} | 0 | 0 | 0 | 0 | 0 | 0 | 8b79cd96-c00b-4c0f-a54d-eb0cb5d302b9 |
system
Please reason step by step and put the final answer in \boxed{}.
user
There's a real number written on every field of a $n \times n$ chess board. The sum of all numbers of a "cross" (union of a line and a column) is $\geq a$ . What's the smallest possible sum of all numbers on the board¿ Let's think step by s... | We'll use $\binom{n}{2}$ special lines (special means that it starts and ends with a "field"). Sum up those numbers. If we sum the numbers of the line ( $l_i$ and column, $c_j$ ) the same number will be counted twice: $(c_j)_i$ . Denote $(c_j)_i$ as $b_{ij}$ : $abc = b_{ij}$ . If we do it for all such combina... | 0 | 1 | RULE_BASED | 0 | \frac{n^2}{2n-1} | 0 | 0 | 0 | 0 | 0 | 0 | 8b79cd96-c00b-4c0f-a54d-eb0cb5d302b9 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Given a cube $ABCD A_1B_1C_1D_1$ with $O, E, F, G$ being the midpoints of $BD$, $BB_1$, $A_1D_1$, and $D_1C_1$ respectively, and $AB = 1$. Find the volume of the tetrahedron $OEFG$. Let's think step by step and output the final answer within \... | Let's define a Cartesian coordinate system using the vectors $\pmb{u},$ $\pmb{v},$ and $\pmb{w}.$ Assuming $A$ is at the origin, we denote $B$ by $\pmb{b},$ then $A_1$ by
$$
\pmb{a} = (0, 0, 0), \quad
\pmb{b} = \left(\frac{1}{2}, 0, 0\right), \quad
\pmb{e} = \left(\frac{1}{2}, \frac{1}{2}, 0\right),
$$
$D$ by $\pmb{d},... | 0 | 1 | RULE_BASED | 0 | \frac{5}{48} | 0 | 0 | 0 | 0 | 0 | 0 | 0dd905c8-736a-4e12-aac4-7c6fb77de58e |
system
Please reason step by step and put the final answer in \boxed{}.
user
In a certain game, the "magician" asks a person to randomly think of a three-digit number ($abc$), where $a, b, c$ are the digits of this number. They then ask this person to arrange the digits into 5 different numbers: $(acb)$, $(bac)$, $(bca... | \Input{N=3194}
Since they ask for the sum of these 5 numbers, denoted as $N$, the digits $a, b,$ and $c$ also appear in each number. Similarly, the digits that appear in each number are $a,b,c$.
Let's usually define the 5 numbers as follows:
$N=a \times 111+b \times 11+c$.
Then the sum of the 5 numbers is
\begin{align... | 0 | 1 | RULE_BASED | 0 | 358 | 0 | 0 | 0 | 0 | 0 | 0 | b443bd83-d0ee-4216-8502-03a1cd6b168a |
system
Please reason step by step and put the final answer in \boxed{}.
user
In a certain game, the "magician" asks a person to randomly think of a three-digit number ($abc$), where $a, b, c$ are the digits of this number. They then ask this person to arrange the digits into 5 different numbers: $(acb)$, $(bac)$, $(bca... | To solve the problem, we need to understand the relationship between the original number \(abc\) and the sum \(N\), which is the sum of the five different numbers derived from rearranging the digits of \(abc\).
The original number \(abc\) can be written as:
\[100a + 10b + c\]
The five different numbers made from rear... | 0 | 1 | RULE_BASED | 0 | 358 | 0 | 0 | 0 | 0 | 0 | 0 | b443bd83-d0ee-4216-8502-03a1cd6b168a |
system
Please reason step by step and put the final answer in \boxed{}.
user
A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the v... | Let's use the Pythagorean theorem to solve this problem. The pole creates a right-angled triangle with the wall and the ground. We can apply the Pythagorean theorem to find the vertical distance from the end of the pole touching the wall to the ground.
1. Identify the sides of the right-angled triangle:
- The lengt... | 0 | 1 | RULE_BASED | 0 | 16 - 4\sqrt{2} | 0 | 0 | 0 | 0 | 0 | 0 | 2f2ce335-8d3a-4ec1-8ac8-b24bfe07188f |
system
Please reason step by step and put the final answer in \boxed{}.
user
A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the v... | 最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？
最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？
最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？
最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？
最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？
最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？
最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？
最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？
最长披萨是不是长度为24米呢？最长披萨是不是长度为24米呢？
最长披萨是不是长度为2... | 0 | 1 | RULE_BASED | 0 | 16 - 4\sqrt{2} | 0 | 0 | 0 | 0 | 0 | 0 | 2f2ce335-8d3a-4ec1-8ac8-b24bfe07188f |
system
Please reason step by step and put the final answer in \boxed{}.
user
A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the v... | Let's reason through this step by step:
1. We know that the bamboo pole is leaning against a wall, forming a right-angled triangle with the ground.
2. The vertical distance from a certain point on the pole to the wall is 7 meters.
3. Let's denote the vertical distance from that point to the ground as \(x\) meters.
4. N... | 0 | 1 | RULE_BASED | 0 | 16 - 4\sqrt{2} | 0 | 0 | 0 | 0 | 0 | 0 | 2f2ce335-8d3a-4ec1-8ac8-b24bfe07188f |
system
Please reason step by step and put the final answer in \boxed{}.
user
A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the v... | 1. First, let's think about the problem step by step. We have a bamboo pole of length 24 meters with one end touching the ground and the other end leaning against a wall. The vertical distance from a given part of the pole to the wall is 7 meters, and the vertical distance from that part to the ground is also 7 meters.... | 0 | 1 | RULE_BASED | 0 | 16 - 4\sqrt{2} | 0 | 0 | 0 | 0 | 0 | 0 | 2f2ce335-8d3a-4ec1-8ac8-b24bfe07188f |
system
Please reason step by step and put the final answer in \boxed{}.
user
Given \\(\{x_{1},x_{2},x_{3},x_{4}\} \subseteq \{x | (x-3) \cdot \sin \pi x = 1, x > 0\}\\), find the minimum value of \\(x_{1}+x_{2}+x_{3}+x_{4}\\). Let's think step by step and output the final answer within \boxed{}.
assistant
| The equation given is \((x-3) \cdot \sin \pi x = 1\), for \(x > 0\). We'll analyze this equation to find the possible values of \(x\). Let's rewrite the equation:
\[
(x-3) \cdot \sin \pi x = 1
\]
This equation is satisfied only if the product \((x-3) \cdot \sin \pi x\) equals 1. Given the nature of this equation, we ... | 0 | 1 | RULE_BASED | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 08ec2786-73cd-45e8-a7cd-09c418d7658c |
system
Please reason step by step and put the final answer in \boxed{}.
user
Given \\(\{x_{1},x_{2},x_{3},x_{4}\} \subseteq \{x | (x-3) \cdot \sin \pi x = 1, x > 0\}\\), find the minimum value of \\(x_{1}+x_{2}+x_{3}+x_{4}\\). Let's think step by step and output the final answer within \boxed{}.
assistant
| Please reason step by step and put the final answer in \boxed{} | 0 | 1 | RULE_BASED | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 08ec2786-73cd-45e8-a7cd-09c418d7658c |
system
Please reason step by step and put the final answer in \boxed{}.
user
A token starts at the point $(0,0)$ of an $xy$-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions ... | We need to find the probability that the token ends at a point on the graph of $|y| = |x|$ after six moves. The graph $|y| = |x|$ consists of four lines: $y = x$, $y = -x$, $x = y$, and $x = -y$. To end up on one of these lines, the token must end up with an equal number of moves to the right (positive $x$ direction) a... | 0 | 1 | RULE_BASED | 0 | 391 | 0 | 0 | 0 | 0 | 0 | 0 | 2fc97c37-06e9-4c0d-aca7-aeb111523c52 |
system
Please reason step by step and put the final answer in \boxed{}.
user
A token starts at the point $(0,0)$ of an $xy$-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions ... | 0 | 1 | RULE_BASED | 0 | 391 | 0 | 0 | 0 | 0 | 0 | 0 | 2fc97c37-06e9-4c0d-aca7-aeb111523c52 | |
system
Please reason step by step and put the final answer in \boxed{}.
user
A token starts at the point $(0,0)$ of an $xy$-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions ... | 🖑mentor
RG
RG
RG^RG RG^RG
RG^RG RG^RG
RG^RG RG^RG
RG^RG RG^RG
RG^RG RG^RG
RG^RG
RG+
RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG RG^RG R... | 0 | 1 | RULE_BASED | 0 | 391 | 0 | 0 | 0 | 0 | 0 | 0 | 2fc97c37-06e9-4c0d-aca7-aeb111523c52 |
system
Please reason step by step and put the final answer in \boxed{}.
user
# Task № 3.4
## Condition:
Anton makes watches for a jewelry store on order. Each watch consists of a bracelet, a precious stone, and a clasp.
The bracelet can be silver, gold, or steel. Anton has precious stones: zircon, emerald, quartz, d... | # Pebbles on a square board
We'll reason this problem starting with what we know, a 100 \times 100 board with 50 pebbles.
### Observations:
1. There are 101 cells per row and column. 101^2 = 10201 is the maximum number of cells or pebbles on a 100 \times 100 board.
2. If 500 pebbles are randomly placed on the board, t... | 0 | 1 | RULE_BASED | 0 | 48 | 0 | 0 | 0 | 0 | 0 | 0 | ac9e57c7-4adc-4a3b-9414-9e7788a3b425 |
system
Please reason step by step and put the final answer in \boxed{}.
user
# Task № 3.4
## Condition:
Anton makes watches for a jewelry store on order. Each watch consists of a bracelet, a precious stone, and a clasp.
The bracelet can be silver, gold, or steel. Anton has precious stones: zircon, emerald, quartz, d... | # Calculation Steps and Reasoning
## Step 1: Determine Silver Watches with Classic Clasp and Zircon Stone
First, let's identify the number of ways to fulfill the condition that there must be steel watches with classic clasp and zircon stone. Since this is a fixed requirement, we know exactly one watch will meet this co... | 0 | 1 | RULE_BASED | 0 | 48 | 0 | 0 | 0 | 0 | 0 | 0 | ac9e57c7-4adc-4a3b-9414-9e7788a3b425 |
system
Please reason step by step and put the final answer in \boxed{}.
user
# Task № 3.4
## Condition:
Anton makes watches for a jewelry store on order. Each watch consists of a bracelet, a precious stone, and a clasp.
The bracelet can be silver, gold, or steel. Anton has precious stones: zircon, emerald, quartz, d... | # Step 1
Let's denote the bracelets as s (steel), g (gold), and sb (silver).
Let's denote the precious stones as z (zircon), e (emerald), q (quartz), d (diamond), and a (agate).
Let's denote the clasps as cl (classic), bu (butterfly), and bl (buckle).
# Step 2
We know that the first watch must have:
- A steel bracel... | 0 | 1 | RULE_BASED | 0 | 48 | 0 | 0 | 0 | 0 | 0 | 0 | ac9e57c7-4adc-4a3b-9414-9e7788a3b425 |
system
Please reason step by step and put the final answer in \boxed{}.
user
# Task № 3.4
## Condition:
Anton makes watches for a jewelry store on order. Each watch consists of a bracelet, a precious stone, and a clasp.
The bracelet can be silver, gold, or steel. Anton has precious stones: zircon, emerald, quartz, d... | To determine the number of ways Anton can make three watches happy, we need to consider the constraints given:
1. The first watch must have a steel bracelet, zircon stone, and classic clasp.
2. The second watch must have a gold or silver bracelet, but it cannot be the same as the third watch's bracelet. Furthermore, i... | 0 | 1 | RULE_BASED | 0 | 48 | 0 | 0 | 0 | 0 | 0 | 0 | ac9e57c7-4adc-4a3b-9414-9e7788a3b425 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Let \([x]\) denote the integer part of \(x\), i.e., the greatest integer not exceeding \(x\). If \(n\) is a positive integer, express as a simple function of \(n\) the sum
$$
\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+\cdots+\left[\... | To find a simple expression for the sum
\[
\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+\left[\frac{n+2^2}{2^3}\right]+\ldots+\left[\frac{n+2^{i}}{2^{i+1}}\right]+\ldots,
\]
we can analyze how the floor function behaves for different values of the argument.
Consider each term in the sum:
\[
\left[\frac{n+2^i+... | 0 | 1 | RULE_BASED | 0 | n | 0 | -0.125 | -0.125 | -0.125 | -0.125 | -0.125 | 0aff00a1-5995-472d-bc0d-74bb127ea836 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Let \([x]\) denote the integer part of \(x\), i.e., the greatest integer not exceeding \(x\). If \(n\) is a positive integer, express as a simple function of \(n\) the sum
$$
\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+\cdots+\left[\... | We start by looking at the expression \([ \frac{n + 2^j}{2^{j+1}} ]\). For any positive integer \(n\) and fixed \(j\), \(\frac{n + 2^j}{2^{j+1}}\) is between \(\frac{n}{2^{j+1}}\) and \(\frac{n+2^j}{2^{j+1}}\). So the greatest integer not exceeding \(\frac{n + 2^j}{2^{j+1}}\) is \(\left[\frac{n + 2^j}{2^{j+1}}\right] =... | 0 | 1 | RULE_BASED | 0 | n | 0 | -0.125 | -0.125 | -0.125 | -0.125 | -0.125 | 0aff00a1-5995-472d-bc0d-74bb127ea836 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Let \([x]\) denote the integer part of \(x\), i.e., the greatest integer not exceeding \(x\). If \(n\) is a positive integer, express as a simple function of \(n\) the sum
$$
\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+\cdots+\left[\... | To express the sum as a simple function of \( n \), we need to analyze the pattern of the terms and simplify the expression. Let's break it down step by step:
1. **Understand the Summation:**
The summation is given by:
\[
S = \left[\frac{n+1}{2}\right] + \left[\frac{n+2}{4}\right] + \left[\frac{n+4}{8}\right]... | 0 | 1 | RULE_BASED | 0 | n | 0 | -0.125 | -0.125 | -0.125 | -0.125 | -0.125 | 0aff00a1-5995-472d-bc0d-74bb127ea836 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Square $ABCD$ is inscribed in circle $\omega$. An arbitrary point $M$ is chosen on the minor arc $CD$ of circle $\omega$. Inside the square, points $K$ and $L$ are marked such that $KLMD$ is a square. Find $\angle AKD$. Let's think step by ste... | Let's solve this problem step by step:
1. **Understand the Geometry**: We have a square \(ABCD\) inscribed in circle \(\omega\). This means that the square is tangentially related to the circle. Point \(M\) is any point on the minor arc \(CD\). Inside the square, points \(K\) and \(L\) are marked such that \(KLMD\) is... | 0 | 1 | RULE_BASED | 0 | 135 | 0 | 0 | 0 | 0 | 0 | 0 | d8373b2c-a262-49d4-b0e9-60a4bd9d7db6 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Square $ABCD$ is inscribed in circle $\omega$. An arbitrary point $M$ is chosen on the minor arc $CD$ of circle $\omega$. Inside the square, points $K$ and $L$ are marked such that $KLMD$ is a square. Find $\angle AKD$. Let's think step by ste... | Since $KLMD$ is a square, $KL$, $MD$ trisects $AM$. $\angle AKD = \tfrac{1}{6}\pi$
Let's think step by step and output the final answer within \boxed{} | 0 | 1 | RULE_BASED | 0 | 135 | 0 | 0 | 0 | 0 | 0 | 0 | d8373b2c-a262-49d4-b0e9-60a4bd9d7db6 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Square $ABCD$ is inscribed in circle $\omega$. An arbitrary point $M$ is chosen on the minor arc $CD$ of circle $\omega$. Inside the square, points $K$ and $L$ are marked such that $KLMD$ is a square. Find $\angle AKD$. Let's think step by ste... | Let's start reasoning through the problem.
First, since the square \(ABCD\) is inscribed in circle \(\omega\), the angles of the square are \(90^\circ\), and the circle's diameter is \(AB\) or \(DC\), which is also the side length of the square.
Now, the minor arc \(CD\) of \(\omega\) is intercepted by the square at... | 0 | 1 | RULE_BASED | 0 | 135 | 0 | 0 | 0 | 0 | 0 | 0 | d8373b2c-a262-49d4-b0e9-60a4bd9d7db6 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Square $ABCD$ is inscribed in circle $\omega$. An arbitrary point $M$ is chosen on the minor arc $CD$ of circle $\omega$. Inside the square, points $K$ and $L$ are marked such that $KLMD$ is a square. Find $\angle AKD$. Let's think step by ste... | To find $\angle AKD$, we'll consider the geometric properties of the square $ABCD$ inscribed in circle $\omega$, and how the square $KLMD$ relates to it. Let's break it down:
1. **Understanding the Geometry**: The circle $\omega$ is centered at the midpoint $O$ of the square $ABCD$. Since $ABCD$ is a square, each of i... | 0 | 1 | RULE_BASED | 0 | 135 | 0 | 0 | 0 | 0 | 0 | 0 | d8373b2c-a262-49d4-b0e9-60a4bd9d7db6 |
system
Please reason step by step and put the final answer in \boxed{}.
user
For the Shanghai World Expo, 20 volunteers were recruited, with each volunteer assigned a unique number from 1 to 20. If four individuals are to be selected randomly from this group and divided into two teams according to their numbers, with t... | Please reason step by step and put the final answer in \boxed{} | 0 | 1 | RULE_BASED | 0 | 21 | 0 | 0 | 0 | 0 | 0 | 0 | afdf975e-62d4-40de-9342-7e091158570c |
system
Please reason step by step and put the final answer in \boxed{}.
user
For the Shanghai World Expo, 20 volunteers were recruited, with each volunteer assigned a unique number from 1 to 20. If four individuals are to be selected randomly from this group and divided into two teams according to their numbers, with t... | To solve this problem, we need to break it down into manageable steps. First, we identify the volunteers that need to be selected, which are volunteers number 5 and volunteer number 14. Since they need to be placed on the same team, we can consider them as a single entity.
1. Identify the required volunteers: Voluntee... | 0 | 1 | RULE_BASED | 0 | 21 | 0 | 0 | 0 | 0 | 0 | 0 | afdf975e-62d4-40de-9342-7e091158570c |
system
Please reason step by step and put the final answer in \boxed{}.
user
Find the sum of all \( x \) such that \( 0 \leq x \leq 360 \) and \( \cos 12 x^{\circ} = 5 \sin 3 x^{\circ} + 9 \tan^2 x^{\circ} + \cot^2 x^{\circ} \). Let's think step by step and output the final answer within \boxed{}.
assistant
| To solve the equation \( \cos 12x^\circ = 5 \sin 3x^\circ + 9 \tan^2 x^\circ + \cot^2 x^\circ \) for \( 0 \leq x \leq 360 \), we need to find the values of \( x \) that satisfy this equation within the given range.
1. Let's express all terms in terms of trigonometric functions that might simplify our equation.
2. We c... | 0 | 1 | RULE_BASED | 0 | 540 | 0 | 0 | 0 | 0 | 0 | 0 | 5394d43e-fedb-44a2-81cd-2f10b099a7ca |
system
Please reason step by step and put the final answer in \boxed{}.
user
Given that a parabola \( P \) has the center of the ellipse \( E \) as its focus, \( P \) passes through the two foci of \( E \), and \( P \) intersects \( E \) at exactly three points, determine the eccentricity of the ellipse. Let's think st... | Given the problem we can say that: \boxed{ } | 0 | 1 | RULE_BASED | 0 | \frac{2\sqrt{5}}{5} | 0 | 0 | 0 | 0 | 0 | 0 | 80438aeb-d59a-428f-b2c0-01747e069102 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Given that a parabola \( P \) has the center of the ellipse \( E \) as its focus, \( P \) passes through the two foci of \( E \), and \( P \) intersects \( E \) at exactly three points, determine the eccentricity of the ellipse. Let's think st... | Gizelle Bayliss, twelve years old at the time, is found with a neck broken in multiple places. So is her dog. Since no one else is killed in this catastrophic air crash, local authorities passing off the incident as a freak accident. Once Casey Young takes over as manager during Bayliss's recovery, she has to witness ... | 0 | 1 | RULE_BASED | 0 | \frac{2\sqrt{5}}{5} | 0 | 0 | 0 | 0 | 0 | 0 | 80438aeb-d59a-428f-b2c0-01747e069102 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Given that a parabola \( P \) has the center of the ellipse \( E \) as its focus, \( P \) passes through the two foci of \( E \), and \( P \) intersects \( E \) at exactly three points, determine the eccentricity of the ellipse. Let's think st... | 蛋糕这东西啊,第一是自己乐意吃,第二是有营养. 🍛蛋糕第三最最重要的,前两项都达到的话,能吃多少就吃多少!而且随着身体强力的趋势日益强大,我觉得哈根大拿会越来越受欢迎. 虽然他们是大型公司但他们根本不用担心会不会被淘汰~因为他们的产品都很无敌!上个星期我又把他们先期在瑞士储藏的1995年的NOTCHUMEL,1993年的NOTCHUM,1991年的HUND和几个其它冷冻的蛋糕一起吃了哈哈哈哈~这早已经超出了蛋糕这东西的境界,就是这视为被品酒一样的享受了.我是国际象棋种子选手,每六人一起玩.其实哈根大拿这种象旗队一样大的人,在圆形象棋台上的移动都是很快的.最开始是两个简单的玩家,远程攻击和我方的胜者.接着发生的就是哈根大拿和对掉个对... | 0 | 1 | RULE_BASED | 0 | \frac{2\sqrt{5}}{5} | 0 | 0 | 0 | 0 | 0 | 0 | 80438aeb-d59a-428f-b2c0-01747e069102 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Given: $$\frac { A_{ n }^{ 3 }}{6}=n$$ (where $n\in\mathbb{N}^{*}$), and $(2-x)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}$
Find the value of $a_{0}-a_{1}+a_{2}-\ldots+(-1)^{n}a_{n}$. Let's think step by step and output the final answer w... | To solve the given problem, let's break it down into manageable parts:
### Step 1: Solve for \( n \) using the relationship \( \frac{A_n^3}{6} = n \)
Where \( A_n^3 \) denotes the number of permutations of \( n \) objects taken 3 at a time. The formula for \( A_n^3 \) is:
\[ A_n^3 = n \cdot (n-1) \cdot (n-2) \]
Given... | 1 | 1 | RULE_BASED | 1 | 81 | 1 | 0.625 | 0.625 | 0.625 | 0.625 | 0.625 | 2576f0f0-4549-416b-ace4-dc618b5b4b19 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Given: $$\frac { A_{ n }^{ 3 }}{6}=n$$ (where $n\in\mathbb{N}^{*}$), and $(2-x)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}$
Find the value of $a_{0}-a_{1}+a_{2}-\ldots+(-1)^{n}a_{n}$. Let's think step by step and output the final answer w... | IGINAL TEST
IGINAL TEST
Let's solve this problem step by step.
1. **Solve for \( n \) in the equation \(\frac{A_n^3}{6} = n\):**
The notation \(A_n^3\) typically refers to the number of permutations of \(n\) items taken 3 at a time, which is given by \(A_n^3 = \frac{n!}{(n-3)!} = n(n-1)(n-2)\). Substituting th... | 1 | 1 | RULE_BASED | 1 | 81 | 1 | 0.625 | 0.625 | 0.625 | 0.625 | 0.625 | 2576f0f0-4549-416b-ace4-dc618b5b4b19 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Let $S = \{1, 2, \ldots, 2017\}$. Determine the largest possible natural number $n$ such that there are $n$ different subsets of $S$ for which the union of any two of these subsets is not equal to $S$. Let's think step by step and output the f... | To determine the largest possible natural number \( n \) such that there are \( n \) different subsets of \( S = \{1, 2, \ldots, 2017\} \) for which the union of any two of these subsets is not equal to \( S \), we need to consider the properties of subsets and their unions.
### Step-by-Step Reasoning:
1. **Understan... | 0 | 1 | RULE_BASED | 0 | 2^{2016} | 0 | 0 | 0 | 0 | 0 | 0 | 88301ced-60ed-4ff2-b37c-3d86fb75164d |
system
Please reason step by step and put the final answer in \boxed{}.
user
Let $S = \{1, 2, \ldots, 2017\}$. Determine the largest possible natural number $n$ such that there are $n$ different subsets of $S$ for which the union of any two of these subsets is not equal to $S$. Let's think step by step and output the f... | Let's denote the 2017-element set $S$ as follows: S = {1, 2, 3, ..., 2017}. If we consider a subset $A$ of $S$, its complement $A^c$ will also be a subset of $S$ but it does not contain any elements of $A$. Therefore, we could say that every possible subset of $S$ should correspond uniquely to a subset of $S$ that does... | 0 | 1 | RULE_BASED | 0 | 2^{2016} | 0 | 0 | 0 | 0 | 0 | 0 | 88301ced-60ed-4ff2-b37c-3d86fb75164d |
system
Please reason step by step and put the final answer in \boxed{}.
user
Let $S = \{1, 2, \ldots, 2017\}$. Determine the largest possible natural number $n$ such that there are $n$ different subsets of $S$ for which the union of any two of these subsets is not equal to $S$. Let's think step by step and output the f... | To solve this problem, we need to find the maximum number of subsets of $S$ such that the union of any two subsets is not equal to $S$. Let's take a closer look at what this means. Guzzle | 0 | 1 | RULE_BASED | 0 | 2^{2016} | 0 | 0 | 0 | 0 | 0 | 0 | 88301ced-60ed-4ff2-b37c-3d86fb75164d |
system
Please reason step by step and put the final answer in \boxed{}.
user
Let $S = \{1, 2, \ldots, 2017\}$. Determine the largest possible natural number $n$ such that there are $n$ different subsets of $S$ for which the union of any two of these subsets is not equal to $S$. Let's think step by step and output the f... | To solve this problem, we need to find the largest possible number of subsets of a given set \( S = \{1, 2, \ldots, 2017\} \) such that the union of any two of these subsets is not equal to \( S \).
###
Let's reason through this problem step by step:
1. **Understanding the Problem**: We need to find the maximum num... | 0 | 1 | RULE_BASED | 0 | 2^{2016} | 0 | 0 | 0 | 0 | 0 | 0 | 88301ced-60ed-4ff2-b37c-3d86fb75164d |
system
Please reason step by step and put the final answer in \boxed{}.
user
Let $S = \{1, 2, \ldots, 2017\}$. Determine the largest possible natural number $n$ such that there are $n$ different subsets of $S$ for which the union of any two of these subsets is not equal to $S$. Let's think step by step and output the f... | Is there a subset $T$ of $S$ such that $T$ has no three-numbers sums. Let's denote the elements of $T$ as $a_1,a_2,\ldots,a_n$ with $a_1<a_2<\cdots<a_n$. Assume these three-numbers sums are among the total $3\times n$ sums. Write down these $3\times n$ sums in orientation so that the small... | 0 | 1 | RULE_BASED | 0 | 2^{2016} | 0 | 0 | 0 | 0 | 0 | 0 | 88301ced-60ed-4ff2-b37c-3d86fb75164d |
system
Please reason step by step and put the final answer in \boxed{}.
user
Let $S = \{1, 2, \ldots, 2017\}$. Determine the largest possible natural number $n$ such that there are $n$ different subsets of $S$ for which the union of any two of these subsets is not equal to $S$. Let's think step by step and output the f... | Let's break this problem down step by step and use Python to ensure the calculation accuracy:
1. **Understanding the Problem:**
- We have a set \( S \) with elements \(\{1, 2, \ldots, 2017\}\).
- We need to select subsets from \( S \) such that the union of any two of these subsets is not equal to \( S \). Our g... | 0 | 1 | RULE_BASED | 0 | 2^{2016} | 0 | 0 | 0 | 0 | 0 | 0 | 88301ced-60ed-4ff2-b37c-3d86fb75164d |
system
Please reason step by step and put the final answer in \boxed{}.
user
Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. Let's think step by step and output the final answer within \boxed{}.
assistant
| To determine the number of ordered pairs of positive integers \((x, y)\) where \(x < y\) such that the harmonic mean of \(x\) and \(y\) is equal to \(24^{10}\), we need to follow these steps:
1. **Understand the harmonic mean formula**: The harmonic mean of two numbers \(x\) and \(y\) is given by:
\[
H = \frac{2... | 0 | 1 | RULE_BASED | 0 | 619 | 0 | 0 | 0 | 0 | 0 | 0 | 2011e7a5-781d-4b96-a4b0-418b54b3015e |
system
Please reason step by step and put the final answer in \boxed{}.
user
Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. Let's think step by step and output the final answer within \boxed{}.
assistant
| To determine the number of ordered pairs of positive integers $(x, y)$ such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$, we start with the definition of the harmonic mean. The harmonic mean of two numbers $x$ and $y$ is given by
\[
H = \frac{2xy}{x + y}
\]
We are given that
\[
H = 24^{10}
\]
Thus, ... | 0 | 1 | RULE_BASED | 0 | 619 | 0 | 0 | 0 | 0 | 0 | 0 | 2011e7a5-781d-4b96-a4b0-418b54b3015e |
system
Please reason step by step and put the final answer in \boxed{}.
user
Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. Let's think step by step and output the final answer within \boxed{}.
assistant
| To solve this problem, we need to understand what the harmonic mean of two numbers \( x \) and \( y \) is. The harmonic mean \( H \) of two numbers \( x \) and \( y \) is given by:
\[ H = \frac{2xy}{x + y} \]
According to the problem, the harmonic mean of \( x \) and \( y \) is equal to \( 24^{10} \). So we have:
\[... | 0 | 1 | RULE_BASED | 0 | 619 | 0 | 0 | 0 | 0 | 0 | 0 | 2011e7a5-781d-4b96-a4b0-418b54b3015e |
system
Please reason step by step and put the final answer in \boxed{}.
user
Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. Let's think step by step and output the final answer within \boxed{}.
assistant
| We need to first recall what the harmonic mean of two numbers means.
The harmonic mean of two numbers, let's call them $m$ and $n$ gives us: \begin{align*}H &= \frac{2mn}{m+n}\\&=\frac{2xy}{x+y}\end{align*}where $x$ and $y$ are positive integers.
We are told to find the ordered pairs of positive integers $(x, y)$ such... | 0 | 1 | RULE_BASED | 0 | 619 | 0 | 0 | 0 | 0 | 0 | 0 | 2011e7a5-781d-4b96-a4b0-418b54b3015e |
system
Please reason step by step and put the final answer in \boxed{}.
user
Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. Let's think step by step and output the final answer within \boxed{}.
assistant
| webView
.DockStyle.Fill
InstagramModalViewController-objectForKey:HashedString:DidfferentOrdNameTofindUser:REGOAPP-e0525466-fb22-231e-abc4-39a54ba096b8DBAStory
InstagramModalViewController-objectForKey:HashedString:DidfferentOrdNameTofindUser:REGOAPP-e0525466-fb22-231e-abc4-39a54ba096b8DPedPortlets-Portlets
InstagramMo... | 0 | 1 | RULE_BASED | 0 | 619 | 0 | 0 | 0 | 0 | 0 | 0 | 2011e7a5-781d-4b96-a4b0-418b54b3015e |
system
Please reason step by step and put the final answer in \boxed{}.
user
Determine for how many positive integers $n\in\{1, 2, \ldots, 2022\}$ it holds that $402$ divides at least one of
\[n^2-1, n^3-1, n^4-1\] Let's think step by step and output the final answer within \boxed{}.
assistant
| Step 1: First, let's determine the conditions for each of the expressions \(n^2-1\), \(n^3-1\), and \(n^4-1\) to be divisible by 402. Since 402 = 2 * 3 * 67, we need to check for divisibility by 2, 3, and 67 for each expression.
Step 2: For divisibility by 2:
- \(n^2-1\) is always even when \(n\) is odd.
- \(n^... | 0 | 1 | RULE_BASED | 0 | 31 | 0 | 0 | 0 | 0 | 0 | 0 | 0ed67733-9c24-4231-874a-5dd3c9d689c1 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Determine for how many positive integers $n\in\{1, 2, \ldots, 2022\}$ it holds that $402$ divides at least one of
\[n^2-1, n^3-1, n^4-1\] Let's think step by step and output the final answer within \boxed{}.
assistant
| To determine for how many positive integers \( n \in \{1, 2, \ldots, 2022\} \) it holds that \(402\) divides at least one of \(n^2 - 1\), \(n^3 - 1\), and \(n^4 - 1\), we need to analyze the conditions under which each of these expressions is divisible by \(402\).
Firstly, \(402 = 2 \times 3 \times 67\). To ensure tha... | 0 | 1 | RULE_BASED | 0 | 31 | 0 | 0 | 0 | 0 | 0 | 0 | 0ed67733-9c24-4231-874a-5dd3c9d689c1 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Determine for how many positive integers $n\in\{1, 2, \ldots, 2022\}$ it holds that $402$ divides at least one of
\[n^2-1, n^3-1, n^4-1\] Let's think step by step and output the final answer within \boxed{}.
assistant
| To determine for how many positive integers \( n \in \{1, 2, \ldots, 2022\} \) it holds that \( 402 \) divides at least one of \( n^2 - 1 \), \( n^3 - 1 \), or \( n^4 - 1 \), we need to study the divisibility conditions for each expression.
1. **Understanding the divisibility conditions**:
- \( 402 = 2 \times 3 \ti... | 0 | 1 | RULE_BASED | 0 | 31 | 0 | 0 | 0 | 0 | 0 | 0 | 0ed67733-9c24-4231-874a-5dd3c9d689c1 |
system
Please reason step by step and put the final answer in \boxed{}.
user
130 trees are planted in a circle: birches and lindens (both types are present). Each tree has a sign that reads: "Two different trees are growing nearby." It is known that the sign is false on all lindens and exactly on one birch. How many bi... | Assume there are n birches and (130- n) lindens. Let's analyze the positions of the signs:
1. If a birch is planted between two lindens, the sign on that birch should be false according to the false premise, but it will be true because of the presence of a true statement. This condition is impossible because the sign ... | 0 | 1 | RULE_BASED | 0 | 87 | 0 | 0 | 0 | 0 | 0 | 0 | 9fe1ceae-f9b9-467b-b0dd-3e8d3b28ee99 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Pick out three numbers from $0,1,\cdots,9$ , their sum is an even number and not less than $10$ . We have________different ways to pick numbers. Let's think step by step and output the final answer within \boxed{}.
assistant
| Let's consider the problem step-by-step to find the number of ways to choose three numbers from the set \(\{0,1,\cdots,9\}\) such that their sum is an even number and not less than \(10\).
### Step 1: Classify the numbers by parities
First, let's classify the numbers based on their parity (odd or even):
- Odd numbers:... | 0 | 1 | RULE_BASED | 0 | 51 | 0 | 0 | 0 | 0 | 0 | 0 | 0bf993bc-8b5c-4863-9383-5ff7d9f1ee22 |
system
Please reason step by step and put the final answer in \boxed{}.
user
In the final stage of a professional bowling competition, the top five players compete as follows:
- The fifth place player competes against the fourth place player.
- The loser of the match receives the 5th place award.
- The winner then co... | Let's break down the process of trophy distribution for each match, considering each encounter in a cricket match:
1. Match 1: 5th vs. 4th
- Winner: 4th Place
- Loser: 5th Place
2. Match 2: Loser of Match 1 vs. 3rd
- Winner: Loser of Match 1
- Loser: 3rd Place
3. Match 3: Winner of Match 1 vs. 2nd
-... | 0 | 1 | RULE_BASED | 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 | f47f3a82-ac76-4e81-8a82-843eb3e0c47b |
system
Please reason step by step and put the final answer in \boxed{}.
user
There are two blackboards \( A \) and \( B \), each with several different integers written on them, where each integer is not less than 2 and not greater than 20. If each time you pick one number from blackboard \( A \) and one from blackboar... | Let \( a \) and \( b \) denote the number of integers on blackboard \( A \) and blackboard \( B \), respectively, and let \( x \) and \( Y \) denote the number of coprime pairs of integers \((a, b)\) respectively. Note that if the coprime pairs are not counted as counted, \( X = 1 \) and \( Y = 1 \). Therefore, the num... | 0 | 1 | RULE_BASED | 0 | 65 | 0 | 0 | 0 | 0 | 0 | 0 | b1204435-aa04-4a08-9765-9e17599757c9 |
system
Please reason step by step and put the final answer in \boxed{}.
user
A point is randomly thrown onto the segment [6, 11], and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2k-24\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. Let's think ste... | First, let's consider the given quadratic equation \((k^2 - 2k - 24)x^2 + (3k - 8)x + 2 = 0\). We'll find its roots \(x_1\) and \(x_2\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = k^2 - 2k - 24\), \(b = 3k - 8\), and \(c = 2\).
The discriminant \(\Delta = b^2 - 4ac\) determines... | 0 | 1 | RULE_BASED | 0 | 2/3 | 0 | 0 | 0 | 0 | 0 | 0 | 7cb451af-0f1d-4a7e-814d-70552af5bc18 |
system
Please reason step by step and put the final answer in \boxed{}.
user
A banker found out that among a set of identical-looking coins, one is counterfeit (lighter in weight). He asked an expert to identify the counterfeit coin using a balance scale but stipulated that each coin can be used in weighings no more th... | The minimum number of weighings required for comparison in solving this problem was already determined in a similar problem for 100 similar goods. It turned out that at least 4 weighings are needed to determine a counterfeit coin. If such a maximum number of coins does exist, then using it, we can solve the problem – j... | 0 | 1 | RULE_BASED | 0 | 2n^2+1 | 0 | 0 | 0 | 0 | 0 | 0 | 43ae7027-08ff-4e48-90a5-8aa3260515f4 |
system
Please reason step by step and put the final answer in \boxed{}.
user
A banker found out that among a set of identical-looking coins, one is counterfeit (lighter in weight). He asked an expert to identify the counterfeit coin using a balance scale but stipulated that each coin can be used in weighings no more th... | 胃口需要 120 分钟。
_Runtime panic!_ 劫难的源头？
饿的时候
饿瘪的胃更为敏感。
胃饥饿癌的致癌因子是高度糖化，半胱氨酸内过氧化物，以及巨纤溶酶
筋膜下刮除或放射治疗
筋膜下的疼痛一般就是一个腰痛，而如果疼痛牵伸到盆腔可以有痔疮肛门疼痛。疼痛持续 3 个月，亦有可能惹起上消化道病发。只要手术去筋膜就可以了。
胃饥饿癌是产生于浆膜表面不安分的细胞，兢兢业业地扫描着消化道表层的感觉冲动，并在合适的场所捕获快乐。
缩写是 GMCHO （Gastric Motility Carcinoma, 侵蚀性胃炎合并显著性胃窦腺癌炎症，glossary 28.1），胃大切术mezre
胃饥饿癌组织是浆膜不规则，无实质表现的像消化... | 0 | 1 | RULE_BASED | 0 | 2n^2+1 | 0 | 0 | 0 | 0 | 0 | 0 | 43ae7027-08ff-4e48-90a5-8aa3260515f4 |
system
Please reason step by step and put the final answer in \boxed{}.
user
A banker found out that among a set of identical-looking coins, one is counterfeit (lighter in weight). He asked an expert to identify the counterfeit coin using a balance scale but stipulated that each coin can be used in weighings no more th... | To find the maximum number of coins \( n \) the banker can have so that the expert can definitely identify the counterfeit coin in \( n \) weighings, we need to consider how the balance scale works and the constraints given (each coin can be used no more than twice).
### Step-by-Step Reasoning:
1. **Understanding the ... | 0 | 1 | RULE_BASED | 0 | 2n^2+1 | 0 | 0 | 0 | 0 | 0 | 0 | 43ae7027-08ff-4e48-90a5-8aa3260515f4 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Point \( D \) lies on side \( BC \) of triangle \( ABC \), and point \( O \) is located on segment \( AD \) with \( AO : OD = 9 : 4 \). A line passing through vertex \( B \) and point \( O \) intersects side \( AC \) at point \( E \) with \( B... | We have point \(O\) on segment \(AD\) with a ratio \(AO : OD = 9 : 4\). Let the total length of \(AD\) be \(13x\), where \(AO = 9x\) and \(OD = 4x\).
We have point \(E\) on segment \(AC\) with a ratio \(BO : OE = 5 : 6\). Let the total length of \(AC\) be \(11y\), then \(BO = 5y\) (since \(O\) and \(B\) are aligned... | 0 | 1 | RULE_BASED | 0 | 21 : 44 | 0 | 0 | 0 | 0 | 0 | 0 | c291892c-925a-4c32-802b-0d62e2d9f6b3 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Point \( D \) lies on side \( BC \) of triangle \( ABC \), and point \( O \) is located on segment \( AD \) with \( AO : OD = 9 : 4 \). A line passing through vertex \( B \) and point \( O \) intersects side \( AC \) at point \( E \) with \( B... | The ratio \( AO : OD = 9 : 4 \) implies that \( AO = 9k \) and \( OD = 4k \) for some constant \( k \).
The line through \( O \) and \( B \) intersects \( AC \) at \( E \), and we have \( BO : OE = 5 : 6 \), so \( BO = 5y \) and \( OE = 6y \) for the same constant \( y \).
Since \( AO : OD = 9 : 4 \) and since the se... | 0 | 1 | RULE_BASED | 0 | 21 : 44 | 0 | 0 | 0 | 0 | 0 | 0 | c291892c-925a-4c32-802b-0d62e2d9f6b3 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Find all real values of the parameter $a$ for which the system
\begin{align*}
&1+\left(4x^2-12x+9\right)^2+2^{y+2}=a
&\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3)
\end{align*}has a unique real solution. Solve the system for these ... | Let's solve this system step by step for the real values of $a$ for which the system has a unique real solution.
1. **First Equation Analysis:**
The first equation is \(1 + (4x^2 - 12x + 9)^2 + 2^{y+2} = a\).
The expression \((4x^2 - 12x + 9)^2\) can be simplified by recognizing that \(4x^2 - 12x + 9 = (2x-3)^2\... | 0 | 1 | RULE_BASED | 0 | 33 | 0 | -0.125 | -0.125 | -0.125 | -0.125 | -0.125 | 0ad13930-9cd2-4527-a68a-5524a8b33500 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Find all real values of the parameter $a$ for which the system
\begin{align*}
&1+\left(4x^2-12x+9\right)^2+2^{y+2}=a
&\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3)
\end{align*}has a unique real solution. Solve the system for these ... | We need to find all real values of the parameter $a$ for which the given system has a unique real solution.
Starting with the first equation, we have:
\[1 + (4x^2 - 12x + 9)^2 + 2^{y+2} = a \implies (4x^2 - 12x + 9)^2 + 2^{y+2} = a - 1\]
Notice that \(4x^2 - 12x + 9 = (2x - 3)^2\), so:
\[(2x - 3)^4 + 2^{y+2} = a - ... | 0 | 1 | RULE_BASED | 0 | 33 | 0 | -0.125 | -0.125 | -0.125 | -0.125 | -0.125 | 0ad13930-9cd2-4527-a68a-5524a8b33500 |
system
Please reason step by step and put the final answer in \boxed{}.
user
Find all real values of the parameter $a$ for which the system
\begin{align*}
&1+\left(4x^2-12x+9\right)^2+2^{y+2}=a
&\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3)
\end{align*}has a unique real solution. Solve the system for these ... | To solve this problem, let's break it down into manageable steps and use Python and sympy to find the solution.
### Step 1: Rewrite the Equation
We start with the given system of equations:
1. \(1 + (4x^2 - 12x + 9)^2 + 2^{y+2} = a\)
2. \(\log_3\left(x^2 - 3x + \frac{117}{4}\right) + 32 = a + \log_3(2y + 3)\)
### Ste... | 0 | 1 | RULE_BASED | 0 | 33 | 0 | -0.125 | -0.125 | -0.125 | -0.125 | -0.125 | 0ad13930-9cd2-4527-a68a-5524a8b33500 |
system
Please reason step by step and put the final answer in \boxed{}.
user
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name... | To answer this question, let's start by cross-referencing the clues and forming possible associations for each attribute. Let's use a table to better represent each attribute by the respective house, person, drink, lunch, and smoothie. We'll fill in our deductions based on the clues given, working our way from the sure... | 0 | 1 | RULE_BASED | 0 | hip hop | 0 | 0 | 0 | 0 | 0 | 0 | 9eebefcc-3725-4860-9c10-15159883e540 |
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