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
steps: int64
elapsed_seconds: double
peak_memory_gb: double
seed: int64
solution: string
problem_id: string
input_ids: list<item: int64>
child 0, item: int64
answer: string
labels: list<item: int64>
child 0, item: int64
problem_source: string
prompt: string
selection_hash: string
source_row_index: int64
id: string
length: int64
target_length: int64
prompt_length: int64
split: string
problem: string
solution_selection_hash: string
to
{'answer': Value('string'), 'id': Value('string'), 'input_ids': List(Value('int64')), 'labels': List(Value('int64')), 'length': Value('int64'), 'problem': Value('string'), 'problem_id': Value('string'), 'problem_source': Value('string'), 'prompt': Value('string'), 'prompt_length': Value('int64'), 'selection_hash': Value('string'), 'solution': Value('string'), 'solution_selection_hash': Value('string'), 'source_row_index': Value('int64'), 'split': Value('string'), 'target_length': Value('int64')}
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
steps: int64
elapsed_seconds: double
peak_memory_gb: double
seed: int64
solution: string
problem_id: string
input_ids: list<item: int64>
child 0, item: int64
answer: string
labels: list<item: int64>
child 0, item: int64
problem_source: string
prompt: string
selection_hash: string
source_row_index: int64
id: string
length: int64
target_length: int64
prompt_length: int64
split: string
problem: string
solution_selection_hash: string
to
{'answer': Value('string'), 'id': Value('string'), 'input_ids': List(Value('int64')), 'labels': List(Value('int64')), 'length': Value('int64'), 'problem': Value('string'), 'problem_id': Value('string'), 'problem_source': Value('string'), 'prompt': Value('string'), 'prompt_length': Value('int64'), 'selection_hash': Value('string'), 'solution': Value('string'), 'solution_selection_hash': Value('string'), 'source_row_index': Value('int64'), 'split': Value('string'), 'target_length': Value('int64')}
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.
answer string | id string | input_ids list | labels list | length int64 | problem string | problem_id string | problem_source string | prompt string | prompt_length int64 | selection_hash string | solution string | solution_selection_hash string | source_row_index int64 | split string | target_length int64 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | e6db3d64fc795aa252f2703c70cb5b6fb9f24dc8e8736ac2e96da2c1ec53f25c | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
835,
33915,
1058,
557,
6780,
2808,
856,
3586,
23074,
968,
1058,
597,
522,
526,
513,
810,
779,
28826,
856,
1... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 693 | A polynomial $P(x)$ is divisible by $x-1$ and the remainder is $R$ when $P(x)$ is divided by $(x-1)^2$.
Find the sum of the remainders when the polynomial $P(x)$ is divided by $x^2+x+1$ and by $x^2-x+1$. | 910b13f86810e0dc4f1c25c94c30fe511033651fb8944b322a5f6d022d872362 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: A polynomial $P(x)$ is divisible by $x-1$ and the remainder is $R$ when $P(x)$ is divided by $(x-1)^2$.
Find the sum of the remainders when the polynomial $P(x)$ is divided by $x^2+x+1$ and by $x^2-x+1$.
Solution:
| 98 | 001e76ebf9ff7dc84aba53b98429b18d2177656d106f099f062caf4f43b199bb | Since $P(x)$ is divisible by $x-1$, we can write $P(x) = (x-1)Q(x)$ for some polynomial $Q(x)$.
Given that the remainder is $R$ when $P(x)$ is divided by $(x-1)^2$, we can write:
\[ P(x) = (x-1)^2S(x) + R \]
for some polynomial $S(x)$.
Notice that $P(1) = 0$ since $P(x)$ is divisible by $x-1$.
Also, the derivative ... | 707314cda32febe22b5244281f49bc0c4268c66b33c306ff83ee96d90d0c9ee2 | 576,508 | train | 595 |
100\sqrt{3} | 737d0b775f3428d49a89d2fea1507a7d6ec293a1c81ca7596cea5edff1262711 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
835,
4903,
27371,
810,
768,
9054,
5611,
768,
2624,
3392,
523,
941,
53936,
803,
779,
27371,
856,
730,
2198,
... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 426 | A regular triangle and a square share a common side. The perimeter of the triangle is 60 cm and that of the square is 52 cm. What is the area of the triangle? | 1f35ddcc24f37d07c0575016e0929c67b297a4bb53b8d3025c903e41dfc01aec | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: A regular triangle and a square share a common side. The perimeter of the triangle is 60 cm and that of the square is 52 cm. What is the area of the triangle?
Solution:
| 63 | 001e9738fd79b6fb5b8d4b7d1413b99095cb282c0b6005682eece9923ee819e9 | Let's call the side length of the square $s$. Then the perimeter of the square is $4s$, and we know this is equal to 52 cm:
\[ 4s = 52 \Rightarrow s = 52 : 4 \Rightarrow s = 13 \text{ cm} \]
Since the triangle and square share a common side, the length of this shared side is also $s = 13$ cm.
Let's call the side len... | 6cde6b77a13fdf562e53f76482acbcccea6efb12cf17423132ad037087810cbf | 578,552 | train | 363 |
\frac{15}{4} | 3cd10fd26442dc280bb44b50baed03511fb4481e3bc2a283d7852bb1a09c7d2b | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
15500,
779,
19359,
803,
768,
25417,
61409,
797,
768,
44292,
916,
768,
9054,
4077,
803,
3392,
5397,
730,
1386,... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 792 | Find the radius of a sphere inscribed in a pyramid with a square base of side length 15 cm, if the height of the pyramid is 10 cm. Express your answer in terms of $\pi$. | 33f04460575baccfc2785756859b5a8a1dc4912143b4eca654e855b3364ed1d4 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Find the radius of a sphere inscribed in a pyramid with a square base of side length 15 cm, if the height of the pyramid is 10 cm. Express your answer in terms of $\pi$.
Solution:
| 66 | 001eedadb4de588b9b75bb2561f596dc93f9e4c30219055eb23741ff23fe5091 | To solve this problem, we can use the fact that the radius of the inscribed sphere is equal to the ratio of the volume of the pyramid to the surface area of the pyramid.
The volume of the pyramid is given by the formula:
\[ V = \frac{1}{3} Bh \]
where $B$ is the area of the base, and $h$ is the height of the pyramid.
... | c950aa313b80f72f1aad893ef2ad203ddbbf39dea6629af0fa3086b5d516374f | 876,098 | train | 726 |
5 | 36bdc49edee89d52603d78ad652ae36848eea11d1cfe630b87d4773529b018d8 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
835,
3297,
4129,
13240,
768,
2021,
803,
17083,
40728,
56367,
23271,
797,
2204,
1847,
14870,
523,
941,
15586,
... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 722 | A certain company produces a set of identical-looking cylindrical containers in three different sizes. The dimensions of the containers are such that the volume of the largest one is 4 times the volume of the smallest one, and the volume of the middle-sized container is the geometric mean of the volumes of the two othe... | b9bc22f86180e8c4e0c15c2abd54a5647a54b2b2d755f3da636d92e5b49ecf59 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: A certain company produces a set of identical-looking cylindrical containers in three different sizes. The dimensions of the containers are such that the volume of the largest one is 4 times the volume of the smallest one, ... | 138 | 001f114ddccb29e49f2e9593fc18d64c97a7605cb9d6268009df988dacae98a2 | To find the number of different combinations of three containers that can be selected such that the total volume of the selected containers is equal to the volume of exactly 6 middle-sized containers, let's first denote the volumes of the small, middle, and large containers as $S$, $M$, and $L$ respectively.
From the ... | 72f335d02206a73a6e02943effcb042c95bf077596ecbc9de4a0fe55c9c6442f | 200,569 | train | 584 |
6 | 9bad7ce80861abca81bc931b94aff8fa9605662b252525c36df3d22bf7a024e3 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
3747,
856,
779,
6584,
4948,
21378,
1058,
586,
513,
1559,
896,
779,
7864,
811,
779,
10805,
1058,
597,
571,
5... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 519 | What is the largest positive integer $m$ such that the solutions to the equation $x^3 - x + 1 = 0$ are not $m^{\text{th}}$ roots of any integer? | e99c223783ea79f6c702eec456b83802494a61bc1e233eee8eed339342320feb | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: What is the largest positive integer $m$ such that the solutions to the equation $x^3 - x + 1 = 0$ are not $m^{\text{th}}$ roots of any integer?
Solution:
| 67 | 001f1df94cf2fe5ca2c0942038fb21459276b22125570400d22f3e37825acc05 | Suppose there exists an integer $n$ such that $x^m = n$. Let $x_1$, $x_2$, and $x_3$ be the roots of the given cubic equation.
\[ x^3 - x + 1 = 0 \Rightarrow (x - x_1)(x - x_2)(x - x_3) = 0 \]
Using Vieta's formulas for the sum and product of the roots of a cubic equation:
\[ x_1 + x_2 + x_3 = 0 \quad (1) \]
\[ x_1x_... | cbe048fa9102fd6609651d4bee95a0bb8e64df645e8733848aa3bf321aea3d97 | 531,592 | train | 452 |
360^\circ | d665a5b8b61957608e6c2b2a79f6771a59f93cddb6dd218ad617861beaeff962 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
5683,
1124,
779,
10805,
509,
38799,
3320,
6780,
6981,
3088,
600,
526,
2945,
597,
2046,
2103,
13322,
527,
6981... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 552 | Solve the equation
$$\left(x+\frac{1}{x}\right)^2+\left(x+\frac{1}{x}\right)-12=0.$$
Express the solutions in the form $x=e^{i\theta}$, where $0^\circ \leq \theta < 360^\circ$. What is the sum of all values of $\theta$? | c2bbe20f56902be984ac4a28d03d45276fe713bfafe3548d6dced5be1fed972b | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Solve the equation
$$\left(x+\frac{1}{x}\right)^2+\left(x+\frac{1}{x}\right)-12=0.$$
Express the solutions in the form $x=e^{i\theta}$, where $0^\circ \leq \theta < 360^\circ$. What is the sum of all values of $\theta$?
S... | 102 | 001f22110e138bd957c4abd77996d5f2c76d63aa76d51adcdcbfcaa121b931e9 | Let $y = x + \frac{1}{x}$. Then we have the quadratic equation:
\[ y^2 + y - 12 = 0 \]
Factor the quadratic equation:
\[ (y - 3)(y + 4) = 0 \]
This gives us two possible values for $y$:
\[ y = 3 \quad \text{or} \quad y = -4 \]
Now, we solve the equations $x + \frac{1}{x} = 3$ and $x + \frac{1}{x} = -4$ for $x$.
1. ... | f2bf7b8176b26faaf8219da750d25f8e3e0fb852474babffeed75a58a8698277 | 432,528 | train | 450 |
(-\infty, 1) \cup (1, \infty) | 0d7c5e023a853f58c7990408de79ebf463091f340edbbb52baf6a35a265b828d | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
15500,
779,
3795,
803,
779,
1947,
1763,
3088,
11903,
571,
527,
522,
529,
2945,
597,
571,
527,
520,
597,
522... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 745 | Find the range of the function $\frac{x^2-4}{x^2+x-2}$, and determine the intervals on which the function is increasing or decreasing. | 514c093ee6cf89ef77ad0103a15d0bc23b085dfa46a9051c3e2f0b138a756e16 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Find the range of the function $\frac{x^2-4}{x^2+x-2}$, and determine the intervals on which the function is increasing or decreasing.
Solution:
| 60 | 001f3673e2dc28dd03b4a527d7e40d9407535ffd271c9ec929aa10f520c51ab2 | To find the range of the function $\frac{x^2-4}{x^2+x-2}$, we can simplify it by factoring the numerator and the denominator.
The numerator can be factored as:
\[ x^2 - 4 = (x + 2)(x - 2) \]
The denominator can be factored as:
\[ x^2 + x - 2 = (x + 2)(x - 1) \]
So, the function can be rewritten as:
\[ f(x) = \frac{(... | 156bec518504e8f5b425deba0c80f92cba1ccda3356c6004d0745839c70df987 | 108,311 | train | 685 |
20 | f50a612d98864e7cfb01afae6f61be2349df110e690f4e3473406510b4f8061c | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
6635,
1090,
2992,
779,
8831,
876,
25260,
856,
1058,
597,
513,
13460,
1094,
8831,
523,
5651,
521,
797,
779,
... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 479 | Let's say the hourly wage is $x$ dollars per hour. Then, in the first week, Xenia earned $15x$ dollars, and in the second week, she earned $22x$ dollars. Since she earned $47.60 more in the second week, we can set up the equation:
$$22x - 15x = 47.60$$
Simplifying the equation, we get:
$$7x = 47.60$$
Dividing both ... | 570b2c1314709b9fae0323184b9bf3334185b02bda915061887134be6b0082fa | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Let's say the hourly wage is $x$ dollars per hour. Then, in the first week, Xenia earned $15x$ dollars, and in the second week, she earned $22x$ dollars. Since she earned $47.60 more in the second week, we can set up the eq... | 327 | 001f6584a77ac02a07d9af58668daa1890832745e579ec14d3080f8d0cddcb13 | To find out how many weeks it will take Tom to save enough money, we need to calculate how much he saves each week and how much more he needs to save.
Tom gets $5 per week but spends $2 per week, so he saves $5 - $2 = $3 per week.
Tom needs $180 - $120 = $60 more to buy the bike.
To find out how many weeks it will t... | a53a46b850f0a3e29171dba8f708b45f3d190f2fe530df3860fed42ea39bd541 | 793,300 | train | 152 |
17 | 846c56c97d660b0d053655b359ff5c753c7ec2b53a30dd17e6063b27ebfb6000 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
835,
9054,
6947,
803,
5743,
1178,
768,
1509,
9812,
18365,
803,
779,
1919,
46162,
1058,
527,
521,
529,
521,
... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 607 | A square array of numbers has a first row consisting of the even integers $2,4,6,\ldots,100$ in increasing order. Each row below the first has one more entry than the row above it, and the bottom row has $50$ entries. Each entry in any row after the top row equals the sum of the two entries directly above it in the row... | 3543912ec82f8686506b567df367fce9ee5cdca06f2e3987ce31109e42e07814 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: A square array of numbers has a first row consisting of the even integers $2,4,6,\ldots,100$ in increasing order. Each row below the first has one more entry than the row above it, and the bottom row has $50$ entries. Each ... | 118 | 001f8f3e8eb6c7cb1c76878bc0d548c0a60cf121766472fb0a0dce1bb33a79f8 | To find the number of entries that are multiples of 17, we can examine the pattern of the numbers in the array.
First row:
The numbers are 2, 4, 6,..., 100.
Second row:
The numbers are 2 + 4 = 6, 4 + 6 = 10, 6 + 8 = 14,..., 98 + 100 = 198.
Third row:
The numbers are 6 + 10 = 16, 10 + 14 = 24, 14 + 18 = 32,..., 192 +... | b3f120879f5cefd8d5507d43db0b7311833ccc5dedfe8b31fca979b4a10f4640 | 685,050 | train | 489 |
3 | 1d5cff91e86771ee530f95cc15d4024e27aac71ac5cf343edcdb85fd7954feb7 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
6635,
1763,
8479,
513,
874,
779,
4121,
2080,
1559,
896,
708,
38799,
8479,
21173,
538,
526,
12767,
905,
708,
... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 628 | Let $\omega$ be the complex number such that
$$\omega^n=1$$
and
$$\omega^{2n}+\omega^n+1=0$$
where $n$ is a positive integer.
Determine $n$ and $\omega$. | f47ded81535855dd27cb4abe19a134d18082ad912d347023f82ac44865ebefe5 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Let $\omega$ be the complex number such that
$$\omega^n=1$$
and
$$\omega^{2n}+\omega^n+1=0$$
where $n$ is a positive integer.
Determine $n$ and $\omega$.
Solution:
| 76 | 001fe20c70eb1e639f9b25492a27882dc28e08e47123ff14c074dee381174878 | We are given that $\omega^n=1$ and $\omega^{2n}+\omega^n+1=0$. Since $\omega^{2n} = (\omega^n)^2 = 1^2 = 1$, we can substitute this into the second equation:
\[ 1 + \omega^n + 1 = 0 \Rightarrow 2 + \omega^n = 0 \Rightarrow \omega^n = -2 \]
This contradicts the first equation $\omega^n = 1$ unless we consider the poss... | 94c4f60cde14876aa2a6abb6925d9eb0e9dcf9315c4db66f1516982e29742686 | 391,229 | train | 552 |
98\% | 31e6d22f62dd6d93b2a191020f3897f6fb1d67239b0875c2871b9fe581a35e74 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
835,
39389,
3506,
45853,
730,
1664,
25397,
1178,
1352,
4623,
6867,
811,
730,
2476,
514,
1720,
15941,
1429,
18... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 263 | A candy bar weighing 30 grams has its weight reduced to 70% after losing some water. Then, the candy is left outside for a day and its water content increases by 40%. What percentage of the original weight is the new weight of the candy? | 65d4da84af37ec8392ebfd223eee0bfeeadc199dfb2858949573dd1755a2b24a | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: A candy bar weighing 30 grams has its weight reduced to 70% after losing some water. Then, the candy is left outside for a day and its water content increases by 40%. What percentage of the original weight is the new weight... | 78 | 001ff16a5d65f07589aa0f1dc630e9640f9545999f9b2cfc0254447a0ef4460e | The candy bar initially weighs 30 grams. Its weight is reduced to 70% of the original weight, which is:
\[ 30 \cdot 0.7 = 21 \text{ grams} \]
Then, the candy's water content increases by 40%. This means the weight increases by 40% of the current weight:
\[ 21 \cdot 0.4 = 8.4 \text{ grams} \]
Add the weight increase t... | c8b0cd13fe7d8d78d192bce1500308780bab8fefa917f440283b1df9ec08723e | 625,634 | train | 185 |
6 | 98290818264bc96fb2ae3078df502a25921cb80c146ddc6764735073f68e1db2 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
63118,
779,
28826,
1403,
779,
33915,
1058,
597,
2139,
1374,
18082,
527,
597,
2139,
1632,
18082,
528,
597,
213... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 475 | Determine the remainder when the polynomial $x^{12}+2x^{11}+3x^{10}+...+10x+11$ is divided by $x^4-x^2+1$. | 49678f6f33a80e1c441707aff2c75ef4b8c7327d7ad032d87a0472f7dff658ea | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Determine the remainder when the polynomial $x^{12}+2x^{11}+3x^{10}+...+10x+11$ is divided by $x^4-x^2+1$.
Solution:
| 66 | 002000eafc78f75e0f27dfd469b0a6fc8db130ecbfdb6bdf6fb58dde415f4122 | We know that:
\[ x^6 + 1 = (x^2 + 1)(x^4 - x^2 + 1) \]
So, $x^6 \equiv -1 \pmod{x^4-x^2+1}$.
Now, we can express the given polynomial in terms of $x^6$:
\[ x^{12}+2x^{11}+3x^{10}+...+10x+11 = \]
\[ (x^{12} + x^6) + (2x^{11} + 2x^5) + (3x^{10} + 3x^4) + (4x^9 + 4x^3) + (5x^8 + 5x^2) + (6x^7 + 6x) + 11 \]
Using the fa... | 4f18a43034e2035f81ded54ebe9e45e164e9f3bf0bcaef4fe53c3236cbdcc907 | 380,061 | train | 409 |
r^2 \arccos \left( \sqrt{\frac{r^2 - 1}{2r^2}} \right) | cefd1f8a0ba999ff57ec12579e90fc3c40ea8f53905219c4753710fdc0ad4702 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
6635,
1058,
591,
513,
874,
779,
19359,
803,
768,
13561,
521,
810,
2944,
708,
569,
13261,
5722,
1078,
3088,
... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 432 | Let $r$ be the radius of a circle, and let
\[\cos \frac{\theta}{2} = \sqrt{\frac{r^2 - 1}{2r^2}}.\]Express the area of the circle sector with central angle $\theta$ in terms of $r.$ | 5bd7d98ac89aa6cfb8f187139ac8fbe064ee205111fd85db7888a723db2430f1 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Let $r$ be the radius of a circle, and let
\[\cos \frac{\theta}{2} = \sqrt{\frac{r^2 - 1}{2r^2}}.\]Express the area of the circle sector with central angle $\theta$ in terms of $r.$
Solution:
| 88 | 002011d22af84230935f6ae1f1c1c70ba89c1db09c6e122945650e6fd1837abd | To find the area of the circle sector with central angle $\theta$, we first need to express $\theta$ in terms of $r$.
We are given:
\[ \cos \frac{\theta}{2} = \sqrt{\frac{r^2 - 1}{2r^2}} \]
To find $\theta$, we can take the inverse cosine of both sides:
\[ \frac{\theta}{2} = \arccos \left( \sqrt{\frac{r^2 - 1}{2r^2}}... | 2cc372d88194392ab46e75c9d1308ce77887f82fe04633b2d5a7f8c39ea384f4 | 709,819 | train | 344 |
0 | 5ddb1dbed85a906bc7795997c77197a9087f3952e23789ddf546ea456ab98a56 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
835,
8556,
856,
5848,
906,
7353,
535,
1058,
575,
572,
526,
1051,
730,
527,
3871,
1058,
575,
572,
527,
1051,... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 839 | A sequence is defined as follows: $b_1 = 2$, $b_2 = 3$, and for all positive integers $n$, $b_{n+2} = b_{n+1}b_n + 1$. Given that $b_{15} = 2752556$ and $b_{16} = 1567645923$, find the remainder when $\displaystyle \sum_{k=1}^{15} \frac{1}{b_k}$ is divided by $\frac{1}{100}$. | dcd697c110e24271304ac52d54b122de44ded182da9886e837e6cdffc2e10361 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: A sequence is defined as follows: $b_1 = 2$, $b_2 = 3$, and for all positive integers $n$, $b_{n+2} = b_{n+1}b_n + 1$. Given that $b_{15} = 2752556$ and $b_{16} = 1567645923$, find the remainder when $\displaystyle \sum_{k=... | 135 | 002053cd53004c0e68681f765d8416dbc93a7df2ce5456a8813b36b2ed117845 | The problem provides the recursive sequence $b_{n+2} = b_{n+1}b_n + 1$ and the initial terms $b_1 = 2$, $b_2 = 3$. We are tasked with finding the remainder when $\displaystyle \sum_{k=1}^{15} \frac{1}{b_k}$ is divided by $\frac{1}{100}$.
Given the recursive definition, we can derive a relation between consecutive term... | 407cb0fc8941c1d49c9b43f940b66d26f254e27ed908ceaae2b3ce026b1fd498 | 520,873 | train | 704 |
1225 | 56a50af30c644399629cf064906ba29f616a034fe801f937f77a61189c866880 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
15500,
779,
24680,
4948,
21378,
1058,
587,
513,
875,
1144,
1763,
35515,
20186,
1078,
10801,
17816,
602,
1078,
... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 771 | Find the smallest positive integer $n$ for which $\lfloor \sqrt{n} \rfloor$ is divisible by 5 and $\lceil \sqrt[3]{n} \rceil$ is divisible by 7. | ad07ada50ead850c554f29e619f020e157b039b6eaeec0918132bb23b579fc1a | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Find the smallest positive integer $n$ for which $\lfloor \sqrt{n} \rfloor$ is divisible by 5 and $\lceil \sqrt[3]{n} \rceil$ is divisible by 7.
Solution:
| 74 | 00208dfb688b25af0b6f4a84784a1e5d8ac360161a70ea61621d4f3ffcf9f9e6 | ## Step 1: Understand the conditions for n
We need to find the smallest positive integer $n$ such that $\lfloor \sqrt{n} \rfloor$ is divisible by 5 and $\lceil \sqrt[3]{n} \rceil$ is divisible by 7.
## Step 2: Analyze the first condition
For $\lfloor \sqrt{n} \rfloor$ to be divisible by 5, $\sqrt{n}$ must be greater t... | 11699148b55111799b7e3c241549fbb19cb2c0c459aa5a690d4f6950078d08fa | 224,653 | train | 697 |
270 | ad6e3a4ce3aa10216d49cdf9dde99bd2f4c1f1d4dfead45b061ab9834eb011cf | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
835,
5354,
4575,
5541,
1178,
730,
530,
3967,
1316,
1842,
841,
7740,
521,
730,
529,
3967,
1316,
1842,
15187,
... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 617 | A computer science club has 5 members who know C++, 4 members who know Java, and 3 members who know Python. The club is planning to send a team of 4 members to a coding competition, where at least one member should know each language. In how many ways can the team be selected? | ad330e869bc14d30ab6b78f7b830b6cc0ceb7a4037be17a54eb917c9f6c60fb7 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: A computer science club has 5 members who know C++, 4 members who know Java, and 3 members who know Python. The club is planning to send a team of 4 members to a coding competition, where at least one member should know eac... | 90 | 0020917254f375fc7e4437dbd78675779c78fc6017e80d7dcde443c6c13dfb12 | To solve this problem, we need to count the number of ways to select a team of 4 members that includes at least one member who knows C++, at least one member who knows Java, and at least one member who knows Python.
First, we find the total number of ways to choose 4 members from the club without any restrictions:
\[ ... | 0ce688ac9ff7f0e5da4db5d0049327c48341a3257b0ef6970969385a36d72d5a | 749,087 | train | 527 |
10 | 22e66e0c28e467c058f5de5e651658770564cdd248c57ef1029929797719b978 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
15500,
779,
6584,
2080,
803,
4948,
521,
25031,
6447,
32933,
5977,
10133,
856,
2442,
1399,
933,
7449,
811,
730... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 896 | Find the largest number of positive, consecutive perfect squares whose sum is less than or equal to 400. | cd1f01fa79f3c4cecb21d91f7031a01f27e642cf46138e775ae96e31fe9f4e13 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Find the largest number of positive, consecutive perfect squares whose sum is less than or equal to 400.
Solution:
| 46 | 0020a5d3d6a08d6db4ad0d4b381ca0735050444dfdcca1f90a209d04c14c54f5 | We can write a sum of consecutive perfect squares starting from $1^2$ as follows:
\[ 1^2 + 2^2 + 3^2 + \ldots + n^2 \]
This sum can be expressed using the formula:
\[ \frac{n(n + 1)(2n + 1)}{6} \]
We want to find the largest value of $n$ such that the sum is less than or equal to 400. We can solve this inequality:
\[... | 3ce0f3e02be3c48c0dd5c9d02d18d65fe892cc88cbacc381ba11b2668bd1c34c | 991,213 | train | 850 |
17 | ab4b32011c9db21d3ce5b1a7cb09f618b2d76e68fc378619b8d124a5f4805d7b | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
6635,
1058,
599,
513,
874,
768,
4121,
2080,
1559,
896,
1058,
599,
1051,
768,
2011,
2955,
39621,
1695,
1058,
... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 882 | Let $z$ be a complex number such that $z = a + bi,$ where $a$ and $b$ are positive integers. If the imaginary part of $z^2$ is 8 times the real part, find $z \times \overline{z}.$ | 93c7d662c692d19874adea97421b84d3d7852b0ae8aa0bc4e3682ed2519832b3 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Let $z$ be a complex number such that $z = a + bi,$ where $a$ and $b$ are positive integers. If the imaginary part of $z^2$ is 8 times the real part, find $z \times \overline{z}.$
Solution:
| 83 | 0020a6df4551e862e6c1c2c30ef8b1a8967630be98d488928b16115502a674f5 | Given that $z = a + bi$, we can square $z$ to find $z^2$.
First, let's square $z$:
\[ z^2 = (a + bi)^2 = a^2 + 2abi + b^2 i^2 = a^2 + 2abi - b^2 = (a^2 - b^2) + 2abi \]
The real part of $z^2$ is $(a^2 - b^2)$ and the imaginary part is $2ab$.
We are given that the imaginary part of $z^2$ is 8 times the real part, so:... | 010934c19fcdbfae0d9b8114702eac379d36f92e9532ce3427de20f5320881fd | 441,074 | train | 799 |
x^3 - x^2 - 2x + 2 | b660c6e2d9fb26fc8a465ce797f7a1d82824974f873e9514fc04bb60dcc7c408 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
941,
6219,
803,
779,
17537,
1947,
1763,
3088,
15170,
6780,
25052,
590,
6780,
26350,
856,
4222,
4174,
521,
916... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 925 | The graph of the rational function $\frac{p(x)}{q(x)}$ is shown below, with a horizontal asymptote of $y = 1$ and a vertical asymptote of $x = 2$. If $q(x)$ is cubic, $p(0) = 0$, and $q(0) = 2$, find $p(x) + q(x)$.
[asy]
size(8cm);
import graph;
Label f;
f.p=fontsize(6);
real f(real x) {return (x-1)/((x-1)*(x+1)*(... | 28df9e46fc64ab53a503602a1ba69e317f59a34325e0af92b1b16af725bd3b6e | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: The graph of the rational function $\frac{p(x)}{q(x)}$ is shown below, with a horizontal asymptote of $y = 1$ and a vertical asymptote of $x = 2$. If $q(x)$ is cubic, $p(0) = 0$, and $q(0) = 2$, find $p(x) + q(x)$.
[asy]
s... | 465 | 0020b4a8cbda571d6ad265a24f47d6540642201bacd83dc7995ab11d2a8503ad | Since the graph has a horizontal asymptote at $y = 1$, the degrees of the polynomials $p(x)$ and $q(x)$ must be the same.
Given that $q(x)$ is cubic, we know $p(x)$ is also cubic.
The presence of a hole in the graph at $x = 1$ indicates a factor of $(x-1)$ in both $p(x)$ and $q(x)$.
The vertical asymptote at $x = 2$... | b9770ee78acebcdf5f91bfe7c53eb4bf2dbaebfb7c54a51a48571b2d0e3fdac1 | 134,160 | train | 460 |
0 | 0d4328cd9fbd7dd237cda0ea8f0415637e36cffef99c7f3e7cdbfb577cc917a6 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
6635,
1058,
579,
6780,
2808,
874,
768,
17537,
1947,
1559,
896,
1352,
6219,
1178,
768,
17924,
28056,
2176,
105... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 1,062 | Let $f(x)$ be a rational function such that its graph has a horizontal asymptote $y = 3$ and two vertical asymptotes $x = -1$ and $x = 2.$ Also, the graph of $y = f(x)$ intersects the line $y = x$ at exactly two points. Find $f(0).$ | beab699342052ecf8f3e8c4060d7181291d3182e9e8ef484a4f58aa9a0841654 | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: Let $f(x)$ be a rational function such that its graph has a horizontal asymptote $y = 3$ and two vertical asymptotes $x = -1$ and $x = 2.$ Also, the graph of $y = f(x)$ intersects the line $y = x$ at exactly two points. Fin... | 99 | 0020bc4684acd35dd6879232e1f38ca2b01c93d7c8f00ae3bd6b646d970031cb | Given the conditions, $f(x)$ can be written in the form:
\[ f(x) = \frac{3(x - r_1)(x - r_2)}{(x + 1)(x - 2)} \]
where $r_1$ and $r_2$ are the roots of $f(x) = x$.
Since the graph of $y = f(x)$ intersects the line $y = x$ at exactly two points, $f(x) = x$ has exactly two roots. This means that $f(x) - x$ has exactly ... | 06d584cd7fea26b9abbe9c9ebb83497dbdbd5b68e0d12298c8bb358a14d23f18 | 162,412 | train | 963 |
15 | d9174bb5dd30eb3a03102fded83b52489d79dc1b3b8525e4f562e808c74458a4 | [
64005,
779,
2773,
3220,
523,
13085,
1180,
23223,
810,
3151,
779,
2524,
5642,
797,
1078,
5428,
791,
600,
5350,
509,
61169,
535,
941,
10609,
803,
768,
27371,
938,
2204,
25031,
46162,
521,
810,
1352,
2768,
856,
1547,
38085,
779,
53936,
523... | [
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100,
-100... | 673 | The sides of a triangle are three consecutive integers, and its area is two-thirds the perimeter. Find the sum of all possible values of the perimeter. | e01c597504acdefa756d5043911148cea54c5c229ce92e92a6c5631db3a529cb | augmented_math | Solve the following problem. Show your reasoning and put the final answer in \boxed{}.
Problem: The sides of a triangle are three consecutive integers, and its area is two-thirds the perimeter. Find the sum of all possible values of the perimeter.
Solution:
| 55 | 0020f08bc722fd968a80ff09f98715a2abf085b2268c73c40dda236b476a5952 | Let the sides of the triangle be $x$, $x+1$, and $x+2$. The perimeter is $P = 3x + 3$.
The semi-perimeter is $s = \frac{P}{2} = \frac{3x + 3}{2}$.
Using Heron's formula, the area of the triangle is:
\[ A = \sqrt{s(s - x)(s - (x + 1))(s - (x + 2))} \]
We know that the area is two-thirds the perimeter:
\[ A = \frac{2}... | 764bbb82dbabced0d65e344dcd56e3ddfab68f64674bece24e5a7be54027a5a3 | 979,738 | train | 618 |
End of preview.