Datasets:
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Error code: DatasetGenerationCastError
Exception: DatasetGenerationCastError
Message: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 1 new columns ({'prompt'}) and 4 missing columns ({'pred_len', 'out_len', 'true_len', 'text'}).
This happened while the json dataset builder was generating data using
hf://datasets/dungnv/qwen36-27b-length-traces/benchmark_prompts.json (at revision a198ac9438475bd736dd76b898942b72789add61), ['hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_olympiad.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_short.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/benchmark_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/collect_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36_uncapped.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/longcot600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/manifest_L32.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/olympiad600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_aug.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_base.json'], ['hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_olympiad.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_short.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/benchmark_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/collect_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36_uncapped.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/longcot600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/manifest_L32.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/olympiad600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_aug.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_base.json']
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)
Traceback: Traceback (most recent call last):
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1848, in _prepare_split_single
writer.write_table(table)
~~~~~~~~~~~~~~~~~~^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/arrow_writer.py", line 765, in write_table
self._write_table(pa_table, writer_batch_size=writer_batch_size)
~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/arrow_writer.py", line 773, in _write_table
pa_table = table_cast(pa_table, self._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
prompt: string
dataset: string
to
{'text': Value('string'), 'true_len': Value('int64'), 'out_len': Value('int64'), 'dataset': Value('string'), 'pred_len': Value('int64')}
because column names don't match
During handling of the above exception, another exception occurred:
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 1850, in _prepare_split_single
raise DatasetGenerationCastError.from_cast_error(
...<4 lines>...
)
datasets.exceptions.DatasetGenerationCastError: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 1 new columns ({'prompt'}) and 4 missing columns ({'pred_len', 'out_len', 'true_len', 'text'}).
This happened while the json dataset builder was generating data using
hf://datasets/dungnv/qwen36-27b-length-traces/benchmark_prompts.json (at revision a198ac9438475bd736dd76b898942b72789add61), ['hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_olympiad.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_short.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/benchmark_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/collect_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36_uncapped.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/longcot600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/manifest_L32.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/olympiad600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_aug.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_base.json'], ['hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_olympiad.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_short.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/benchmark_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/collect_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36_uncapped.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/longcot600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/manifest_L32.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/olympiad600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_aug.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_base.json']
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)Need help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
text string | true_len int64 | out_len int64 | dataset string | pred_len int64 |
|---|---|---|---|---|
<|im_start|>user
$$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m... | 24,313 | 24,313 | olympiad | 10,011 |
<|im_start|>user
## Real Expansion
Consider a "real" monatomic gas consisting of $N$ atoms of negligible volume and mass $m$ in equilibrium inside a closed cubical container of volume $V$. In this "real" gas, the attractive forces between atoms is small but not negligible. Because these atoms have negligible volume,... | 20,205 | 20,205 | olympiad | 22,377 |
<|im_start|>user
Suppose that $n>5$ and that the numbers $t_{1}, t_{2}, t_{3}, \ldots, t_{n-2}, t_{n-1}, t_{n}$ form an arithmetic sequence with $n$ terms. If $t_{3}=5, t_{n-2}=95$, and the sum of all $n$ terms is 1000 , what is the value of $n$ ?
(An arithmetic sequence is a sequence in which each term after the firs... | 8,885 | 8,885 | olympiad | 5,889 |
<|im_start|>user
$$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m... | 7,676 | 7,676 | olympiad | 8,384 |
<|im_start|>user
Points $A, R, M$, and $L$ are consecutively the midpoints of the sides of a square whose area is 650. The coordinates of point $A$ are $(11,5)$. If points $R, M$, and $L$ are all lattice points, and $R$ is in Quadrant I, compute the number of possible ordered pairs $(x, y)$ of coordinates for point $R$... | 30,089 | 30,089 | olympiad | 25,261 |
<|im_start|>user
Consider the sequence $t_{1}=1, t_{2}=-1$ and $t_{n}=\left(\frac{n-3}{n-1}\right) t_{n-2}$ where $n \geq 3$. What is the value of $t_{1998}$ ?
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 7,964 | 7,964 | olympiad | 14,590 |
<|im_start|>user
## Real Expansion
Consider a "real" monatomic gas consisting of $N$ atoms of negligible volume and mass $m$ in equilibrium inside a closed cubical container of volume $V$. In this "real" gas, the attractive forces between atoms is small but not negligible. Because these atoms have negligible volume,... | 18,148 | 18,148 | olympiad | 22,960 |
<|im_start|>user
Compute the least positive integer $n$ such that the set of angles
$$
\left\{123^{\circ}, 246^{\circ}, \ldots, n \cdot 123^{\circ}\right\}
$$
contains at least one angle in each of the four quadrants.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assista... | 23,930 | 23,930 | olympiad | 22,687 |
<|im_start|>user
What is the value of $x$ such that $\log _{2}\left(\log _{2}(2 x-2)\right)=2$ ?
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 1,018 | 1,018 | olympiad | 3,016 |
<|im_start|>user
Define $\log ^{*}(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to 1 . For example, $\log ^{*}(1000)=2$ since $\log 1000=3$ and $\log (\log 1000)=\log 3=0.477 \ldots \leq 1$. Let $a$ be the smallest integer such that $\log ... | 25,883 | 25,883 | olympiad | 14,207 |
<|im_start|>user
Define a reverse prime to be a positive integer $N$ such that when the digits of $N$ are read in reverse order, the resulting number is a prime. For example, the numbers 5, 16, and 110 are all reverse primes. Compute the largest two-digit integer $N$ such that the numbers $N, 4 \cdot N$, and $5 \cdot N... | 25,163 | 25,163 | olympiad | 26,838 |
<|im_start|>user
In non-right triangle $A B C$, distinct points $P, Q, R$, and $S$ lie on $\overline{B C}$ in that order such that $\angle B A P \cong \angle P A Q \cong \angle Q A R \cong \angle R A S \cong \angle S A C$. Given that the angles of $\triangle A B C$ are congruent to the angles of $\triangle A P Q$ in so... | 24,259 | 24,259 | olympiad | 39,803 |
<|im_start|>user
Determine all real values of $x$ for which $\log _{2}\left(2^{x-1}+3^{x+1}\right)=2 x-\log _{2}\left(3^{x}\right)$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 13,302 | 13,302 | olympiad | 10,553 |
<|im_start|>user
The equation $2^{x+2} 5^{6-x}=10^{x^{2}}$ has two real solutions. Determine these two solutions.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 15,078 | 15,078 | olympiad | 6,582 |
<|im_start|>user
In equilateral hexagon $A B C D E F, \mathrm{~m} \angle A=2 \mathrm{~m} \angle C=2 \mathrm{~m} \angle E=5 \mathrm{~m} \angle D=10 \mathrm{~m} \angle B=10 \mathrm{~m} \angle F$, and diagonal $B E=3$. Compute $[A B C D E F]$, that is, the area of $A B C D E F$.
Please reason step by step, and put your fi... | 32,448 | 32,448 | olympiad | 29,510 |
<|im_start|>user
For some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 14,555 | 14,555 | olympiad | 10,111 |
<|im_start|>user
Let $T=16$. An isosceles trapezoid has an area of $T+1$, a height of 2 , and the shorter base is 3 units shorter than the longer base. Compute the sum of the length of the shorter base and the length of one of the congruent sides.
Please reason step by step, and put your final answer within \boxed{}.<|... | 2,624 | 2,624 | olympiad | 4,529 |
<|im_start|>user
Let $T=688$. Let $a$ be the least nonzero digit in $T$, and let $b$ be the greatest digit in $T$. In square $N O R M, N O=b$, and points $P_{1}$ and $P_{2}$ lie on $\overline{N O}$ and $\overline{O R}$, respectively, so that $O P_{1}=O P_{2}=a$. A circle centered at $O$ has radius $a$, and quarter-circ... | 19,931 | 19,931 | olympiad | 20,476 |
<|im_start|>user
Let $10^{y}$ be the product of all real numbers $x$ such that $\log x=\frac{3+\left\lfloor(\log x)^{2}\right\rfloor}{4}$. Compute $y$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 19,509 | 19,509 | olympiad | 18,829 |
<|im_start|>user
Particles from the Sun ${ }^{1}$
Photons from the surface of the Sun and neutrinos from its core can tell us about solar temperatures and also confirm that the Sun shines because of nuclear reactions.
Throughout this problem, take the mass of the Sun to be $M_{\odot}=2.00 \times 10^{30} \mathrm{~kg}... | 10,259 | 10,259 | olympiad | 12,701 |
<|im_start|>user
All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual ... | 19,759 | 19,759 | olympiad | 13,599 |
<|im_start|>user
Diagonal $W Y$ of square $W X Y Z$ has slope 2. Determine the sum of the slopes of $W X$ and $X Y$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 19,806 | 19,806 | olympiad | 11,297 |
<|im_start|>user
If $2 \leq x \leq 5$ and $10 \leq y \leq 20$, what is the maximum value of $15-\frac{y}{x}$ ?
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 4,383 | 4,383 | olympiad | 3,036 |
<|im_start|>user
Let $T\neq 0$. Suppose that $a, b, c$, and $d$ are real numbers so that $\log _{a} c=\log _{b} d=T$. Compute
$$
\frac{\log _{\sqrt{a b}}(c d)^{3}}{\log _{a} c+\log _{b} d}
$$
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 10,484 | 10,484 | olympiad | 14,565 |
<|im_start|>user
Let $T=3$. In $\triangle A B C, A C=T^{2}, \mathrm{~m} \angle A B C=45^{\circ}$, and $\sin \angle A C B=\frac{8}{9}$. Compute $A B$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 10,493 | 10,493 | olympiad | 9,579 |
<|im_start|>user
A list $a_{1}, a_{2}, a_{3}, a_{4}$ of rational numbers is defined so that if one term is equal to $r$, then the next term is equal to $1+\frac{1}{1+r}$. For example, if $a_{3}=\frac{41}{29}$, then $a_{4}=1+\frac{1}{1+(41 / 29)}=\frac{99}{70}$. If $a_{3}=\frac{41}{29}$, what is the value of $a_{1} ?$
P... | 9,730 | 9,730 | olympiad | 11,377 |
<|im_start|>user
4. A complex dance
In this problem, we will solve a number of differential equations corresponding to very different physical phenomena that are unified by the idea of oscillation. Oscillations are captured elegantly by extending our notion of numbers to include the imaginary unit number $i$, strange... | 24,669 | 24,669 | olympiad | 10,544 |
<|im_start|>user
## Toffee Pudding
A box of mass $m$ is at rest on a horizontal floor. The coefficients of static and kinetic friction between the box and the floor are $\mu_{0}$ and $\mu$ (less than $\mu_{0}$ ), respectively. One end of a spring with spring constant $k$ is attached to the right side of the box, and... | 24,704 | 24,704 | olympiad | 18,738 |
<|im_start|>user
David builds a circular table; he then carves one or more positive integers into the table at points equally spaced around its circumference. He considers two tables to be the same if one can be rotated so that it has the same numbers in the same positions as the other. For example, a table with the nu... | 23,355 | 23,355 | olympiad | 13,469 |
<|im_start|>user
Let $T=5$. Compute the smallest positive integer $n$ such that there are at least $T$ positive integers in the domain of $f(x)=\sqrt{-x^{2}-2 x+n}$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 10,643 | 10,643 | olympiad | 14,243 |
<|im_start|>user
An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are... | 5,348 | 5,348 | olympiad | 18,791 |
<|im_start|>user
Compute the number of ordered pairs of integers $(a, b)$ such that $1<a \leq 50,1<b \leq 50$, and $\log _{b} a$ is rational.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 32,420 | 32,420 | olympiad | 13,444 |
<|im_start|>user
Suppose that the functions $f(x)$ and $g(x)$ satisfy the system of equations
$$
\begin{aligned}
f(x)+3 g(x) & =x^{2}+x+6 \\
2 f(x)+4 g(x) & =2 x^{2}+4
\end{aligned}
$$
for all $x$. Determine the values of $x$ for which $f(x)=g(x)$.
Please reason step by step, and put your final answer within \boxed{}... | 5,200 | 5,200 | olympiad | 5,182 |
<|im_start|>user
A computer is programmed to choose an integer between 1 and 99, inclusive, so that the probability that it selects the integer $x$ is equal to $\log _{100}\left(1+\frac{1}{x}\right)$. Suppose that the probability that $81 \leq x \leq 99$ is equal to 2 times the probability that $x=n$ for some integer $... | 12,355 | 12,355 | olympiad | 14,840 |
<|im_start|>user
Given a positive integer $n \geq 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_{0}, a_{1}, \ldots, a_{N}$ such that
(1) $a_{0}+a_{1}=-\frac{1}{n}$, and
(2) $\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}$ for $1 \leq k \leq N-1$.
Plea... | 25,691 | 25,691 | olympiad | 26,775 |
<|im_start|>user
Each positive integer less than or equal to 2019 is written on a blank sheet of paper, and each of the digits 0 and 5 is erased. Compute the remainder when the product of the remaining digits on the sheet of paper is divided by 1000 .
Please reason step by step, and put your final answer within \boxed{... | 32,140 | 32,140 | olympiad | 20,495 |
<|im_start|>user
Particles from the Sun ${ }^{1}$
Photons from the surface of the Sun and neutrinos from its core can tell us about solar temperatures and also confirm that the Sun shines because of nuclear reactions.
Throughout this problem, take the mass of the Sun to be $M_{\odot}=2.00 \times 10^{30} \mathrm{~kg}... | 11,439 | 11,439 | olympiad | 12,318 |
<|im_start|>user
Suppose that neither of the three-digit numbers $M=\underline{4} \underline{A} \underline{6}$ and $N=\underline{1} \underline{B} \underline{7}$ is divisible by 9 , but the product $M \cdot N$ is divisible by 9 . Compute the largest possible value of $A+B$.
Please reason step by step, and put your final... | 20,028 | 20,028 | olympiad | 13,079 |
<|im_start|>user
If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possible ordered pairs $(a, b)$ which satisfy this relationship.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 2,611 | 2,611 | olympiad | 5,698 |
<|im_start|>user
All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual ... | 19,731 | 19,731 | olympiad | 14,185 |
<|im_start|>user
Ross starts with an angle of measure $8^{\circ}$ and doubles it 10 times until he obtains $8192^{\circ}$. He then adds up the reciprocals of the sines of these 11 angles. That is, he calculates
$$
S=\frac{1}{\sin 8^{\circ}}+\frac{1}{\sin 16^{\circ}}+\frac{1}{\sin 32^{\circ}}+\cdots+\frac{1}{\sin 4096^... | 20,962 | 20,962 | olympiad | 17,887 |
<|im_start|>user
Let $P(x)=x^{2}+T x+800$, and let $r_{1}$ and $r_{2}$ be the roots of $P(x)$. The polynomial $Q(x)$ is quadratic, it has leading coefficient 1, and it has roots $r_{1}+1$ and $r_{2}+1$. Find the sum of the coefficients of $Q(x)$.
Please reason step by step, and put your final answer within \boxed{}.<|i... | 9,769 | 9,769 | olympiad | 9,347 |
<|im_start|>user
In rectangle $P A U L$, point $D$ is the midpoint of $\overline{U L}$ and points $E$ and $F$ lie on $\overline{P L}$ and $\overline{P A}$, respectively such that $\frac{P E}{E L}=\frac{3}{2}$ and $\frac{P F}{F A}=2$. Given that $P A=36$ and $P L=25$, compute the area of pentagon $A U D E F$.
Please rea... | 28,397 | 28,397 | olympiad | 18,008 |
<|im_start|>user
Let $r=1$ and $R=5$. A circle with radius $r$ is centered at $A$, and a circle with radius $R$ is centered at $B$. The two circles are internally tangent. Point $P$ lies on the smaller circle so that $\overline{B P}$ is tangent to the smaller circle. Compute $B P$.
Please reason step by step, and put y... | 8,535 | 8,535 | olympiad | 13,975 |
<|im_start|>user
Let $T=5$. If $|T|-1+3 i=\frac{1}{z}$, compute the sum of the real and imaginary parts of $z$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 3,324 | 3,324 | olympiad | 8,430 |
<|im_start|>user
For some positive integers $k$, the parabola with equation $y=\frac{x^{2}}{k}-5$ intersects the circle with equation $x^{2}+y^{2}=25$ at exactly three distinct points $A, B$ and $C$. Determine all such positive integers $k$ for which the area of $\triangle A B C$ is an integer.
Please reason step by st... | 20,867 | 20,867 | olympiad | 19,817 |
<|im_start|>user
Problem T3. Protostar formation
Let us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser ... | 15,789 | 15,789 | olympiad | 21,218 |
<|im_start|>user
A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$.
Determ... | 16,475 | 16,475 | olympiad | 11,997 |
<|im_start|>user
Chinara starts with the point $(3,5)$, and applies the following three-step process, which we call $\mathcal{P}$ :
Step 1: Reflect the point in the $x$-axis.
Step 2: Translate the resulting point 2 units upwards.
Step 3: Reflect the resulting point in the $y$-axis.
As she does this, the point $(3,5... | 14,609 | 14,609 | olympiad | 8,294 |
<|im_start|>user
An Electrified Soap Bubble
A spherical soap bubble with internal air density $\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\gamma$, density $\rho_{s}$ and thickne... | 7,480 | 7,480 | olympiad | 15,098 |
<|im_start|>user
An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$, and the other focus is at $(d, 7)$. Compute $d$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 16,231 | 16,231 | olympiad | 22,839 |
<|im_start|>user
A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it travels 3 units down to $(2,-2)$. On move 4 , it travels 4 units to $(-2,-2)$. It continues in this fashion, so that on move $n$... | 10,246 | 10,246 | olympiad | 20,843 |
<|im_start|>user
Compute the sum of all positive two-digit factors of $2^{32}-1$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 25,467 | 25,467 | olympiad | 9,070 |
<|im_start|>user
Triangle $A B C$ is inscribed in circle $\omega$. The tangents to $\omega$ at $B$ and $C$ meet at point $T$. The tangent to $\omega$ at $A$ intersects the perpendicular bisector of $\overline{A T}$ at point $P$. Given that $A B=14, A C=30$, and $B C=40$, compute $[P B C]$.
Please reason step by step, a... | 31,097 | 31,097 | olympiad | 34,764 |
<|im_start|>user
Let $T=49$. Compute the last digit, in base 10, of the sum
$$
T^{2}+(2 T)^{2}+(3 T)^{2}+\ldots+\left(T^{2}\right)^{2}
$$
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 27,791 | 27,791 | olympiad | 9,329 |
<|im_start|>user
Digital images consist of a very large number of equally spaced dots called pixels The resolution of an image is the number of pixels/cm in each of the horizontal and vertical directions.
Thus, an image with dimensions $10 \mathrm{~cm}$ by $15 \mathrm{~cm}$ and a resolution of 75 pixels/cm has a total... | 7,547 | 7,547 | olympiad | 12,112 |
<|im_start|>user
Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b \text {. }
$$
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 20,022 | 20,022 | olympiad | 14,085 |
<|im_start|>user
A cylinder with radius $r$ and height $h$ has volume 1 and total surface area 12. Compute $\frac{1}{r}+\frac{1}{h}$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 7,859 | 7,859 | olympiad | 6,917 |
<|im_start|>user
## String Cheese
a. When a faucet is turned on, a stream of water flows down with initial speed $v_{0}$ at the spout. For this problem, we define $y$ to be the vertical coordinate with its positive direction pointing up.
Assuming the water speed is only affected by gravity as the water falls, find th... | 14,134 | 14,134 | olympiad | 10,331 |
<|im_start|>user
Particles and Waves
Wave-particle duality, which states that each particle can be described as a wave and vice versa, is one of the central concepts of quantum mechanics. In this problem, we will rely on this notion and just a few other basic assumptions to explore a selection of quantum phenomena cov... | 4,231 | 4,231 | olympiad | 9,579 |
<|im_start|>user
What are all values of $x$ such that
$$
\log _{5}(x+3)+\log _{5}(x-1)=1 ?
$$
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 2,969 | 2,969 | olympiad | 3,216 |
<|im_start|>user
$\quad$ Suppose that $p$ and $q$ are two-digit prime numbers such that $p^{2}-q^{2}=2 p+6 q+8$. Compute the largest possible value of $p+q$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 11,382 | 11,382 | olympiad | 13,258 |
<|im_start|>user
Let $T=184$. A group of $n$ friends goes camping; two of them are selected to set up the campsite when they arrive and two others are selected to take down the campsite the next day. Compute the smallest possible value of $n$ such that there are at least $T$ ways of selecting the four helpers.
Please r... | 15,919 | 15,919 | olympiad | 7,682 |
<|im_start|>user
For each positive integer $n$, let $T(n)$ be the number of triangles with integer side lengths, positive area, and perimeter $n$. For example, $T(6)=1$ since the only such triangle with a perimeter of 6 has side lengths 2,2 and 2 .
Determine the values of $T(10), T(11)$ and $T(12)$.
Please reason step ... | 21,039 | 21,039 | olympiad | 9,607 |
<|im_start|>user
Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. The speed of the current in the river is always the same. When Serge paddles, he always paddles at the same constant speed. On days when he paddles with the current, it takes him 18 minutes to get from $A$ to $B$. When he ... | 3,498 | 3,498 | olympiad | 5,167 |
<|im_start|>user
Compute all ordered pairs of real numbers $(x, y)$ that satisfy both of the equations:
$$
x^{2}+y^{2}=6 y-4 x+12 \quad \text { and } \quad 4 y=x^{2}+4 x+12
$$
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 6,011 | 6,011 | olympiad | 5,471 |
<|im_start|>user
Let $n$ be a positive integer such that $\frac{3+4+\cdots+3 n}{5+6+\cdots+5 n}=\frac{4}{11}$. Compute $\frac{2+3+\cdots+2 n}{4+5+\cdots+4 n}$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 20,549 | 20,549 | olympiad | 18,854 |
<|im_start|>user
Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that the equation
holds for all rational numbers $x$ and $y$.
$$
f(x f(x)+y)=f(y)+x^{2}
$$
Here, $\mathbb{Q}$ denotes the set of rational numbers.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_star... | 27,413 | 27,413 | olympiad | 15,660 |
<|im_start|>user
Particles from the Sun ${ }^{1}$
Photons from the surface of the Sun and neutrinos from its core can tell us about solar temperatures and also confirm that the Sun shines because of nuclear reactions.
Throughout this problem, take the mass of the Sun to be $M_{\odot}=2.00 \times 10^{30} \mathrm{~kg}... | 10,377 | 10,377 | olympiad | 12,944 |
<|im_start|>user
Hexagon $A B C D E F$ has vertices $A(0,0), B(4,0), C(7,2), D(7,5), E(3,5)$, $F(0,3)$. What is the area of hexagon $A B C D E F$ ?
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 17,104 | 17,104 | olympiad | 10,264 |
<|im_start|>user
Let $T=125$. Cube $\mathcal{C}_{1}$ has volume $T$ and sphere $\mathcal{S}_{1}$ is circumscribed about $\mathcal{C}_{1}$. For $n \geq 1$, the sphere $\mathcal{S}_{n}$ is circumscribed about the cube $\mathcal{C}_{n}$ and is inscribed in the cube $\mathcal{C}_{n+1}$. Let $k$ be the least integer such th... | 18,556 | 18,556 | olympiad | 14,817 |
<|im_start|>user
## Hot Pocket
a. It's winter and you want to keep warm. The temperature is $T_{0}=263 \mathrm{~K}$ outside and $T_{1}=290 \mathrm{~K}$ in your room. You have started a fire, which acts as a hot reservoir at temperature $T_{2}=1800 \mathrm{~K}$.
You want to add a small amount of heat $d Q_{1}$ to you... | 19,610 | 19,610 | olympiad | 18,403 |
<|im_start|>user
Linh is driving at $60 \mathrm{~km} / \mathrm{h}$ on a long straight highway parallel to a train track. Every 10 minutes, she is passed by a train travelling in the same direction as she is. These trains depart from the station behind her every 3 minutes and all travel at the same constant speed. What ... | 8,590 | 8,590 | olympiad | 6,587 |
<|im_start|>user
Paul was planning to buy 20 items from the ARML shop. He wanted some mugs, which cost $\$ 10$ each, and some shirts, which cost $\$ 6$ each. After checking his wallet he decided to put $40 \%$ of the mugs back. Compute the number of dollars he spent on the remaining items.
Please reason step by step, a... | 3,498 | 3,498 | olympiad | 2,983 |
<|im_start|>user
2. Trajectory of a point mass
A point mass on the ground is thrown with initial velocity $\vec{v}_{0}$ that makes an angle $\theta$ with the horizontal. Assuming that air friction is negligible,
Context question:
(a) What value of $\theta$ maximizes the range?
Context answer:
\boxed{$\frac{\pi}{4}$}
... | 23,580 | 23,580 | olympiad | 14,954 |
<|im_start|>user
5. Polarization and Oscillation
In this problem, we will understand the polarization of metallic bodies and the method of images that simplifies the math in certain geometrical configurations.
Throughout the problem, suppose that metals are excellent conductors and they polarize significantly faster... | 12,391 | 12,391 | olympiad | 12,456 |
<|im_start|>user
The four zeros of the polynomial $x^{4}+j x^{2}+k x+225$ are distinct real numbers in arithmetic progression. Compute the value of $j$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 14,984 | 14,984 | olympiad | 18,254 |
<|im_start|>user
Point $P$ is on the hypotenuse $\overline{E N}$ of right triangle $B E N$ such that $\overline{B P}$ bisects $\angle E B N$. Perpendiculars $\overline{P R}$ and $\overline{P S}$ are drawn to sides $\overline{B E}$ and $\overline{B N}$, respectively. If $E N=221$ and $P R=60$, compute $\frac{1}{B E}+\fr... | 14,744 | 14,744 | olympiad | 21,312 |
<|im_start|>user
Particles and Waves
Wave-particle duality, which states that each particle can be described as a wave and vice versa, is one of the central concepts of quantum mechanics. In this problem, we will rely on this notion and just a few other basic assumptions to explore a selection of quantum phenomena cov... | 6,178 | 6,178 | olympiad | 11,173 |
<|im_start|>user
$$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m... | 18,275 | 18,275 | olympiad | 9,654 |
<|im_start|>user
$$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m... | 5,440 | 5,440 | olympiad | 8,380 |
<|im_start|>user
Let $T=1$. Circles $L$ and $O$ are internally tangent and have radii $T$ and $4 T$, respectively. Point $E$ lies on circle $L$ such that $\overline{O E}$ is tangent to circle $L$. Compute $O E$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<thin... | 16,062 | 16,062 | olympiad | 19,907 |
<|im_start|>user
Let $T=\frac{9}{17}$. When $T$ is expressed as a reduced fraction, let $m$ and $n$ be the numerator and denominator, respectively. A square pyramid has base $A B C D$, the distance from vertex $P$ to the base is $n-m$, and $P A=P B=P C=P D=n$. Compute the area of square $A B C D$.
Please reason step by... | 11,195 | 11,195 | olympiad | 8,347 |
<|im_start|>user
Compute the value of
$$
\sin \left(6^{\circ}\right) \cdot \sin \left(12^{\circ}\right) \cdot \sin \left(24^{\circ}\right) \cdot \sin \left(42^{\circ}\right)+\sin \left(12^{\circ}\right) \cdot \sin \left(24^{\circ}\right) \cdot \sin \left(42^{\circ}\right) \text {. }
$$
Please reason step by step, and ... | 19,655 | 19,655 | olympiad | 17,096 |
<|im_start|>user
Let $T=19$. Let $\mathcal{S}_{1}$ denote the arithmetic sequence $0, \frac{1}{4}, \frac{1}{2}, \ldots$, and let $\mathcal{S}_{2}$ denote the arithmetic sequence $0, \frac{1}{6}, \frac{1}{3}, \ldots$ Compute the $T^{\text {th }}$ smallest number that occurs in both sequences $\mathcal{S}_{1}$ and $\math... | 15,415 | 15,415 | olympiad | 14,285 |
<|im_start|>user
Regular hexagon $A B C D E F$ and regular hexagon $G H I J K L$ both have side length 24 . The hexagons overlap, so that $G$ is on $\overline{A B}, B$ is on $\overline{G H}, K$ is on $\overline{D E}$, and $D$ is on $\overline{J K}$. If $[G B C D K L]=\frac{1}{2}[A B C D E F]$, compute $L F$.
Please rea... | 31,036 | 31,036 | olympiad | 40,000 |
<|im_start|>user
An Electrified Soap Bubble
A spherical soap bubble with internal air density $\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\gamma$, density $\rho_{s}$ and thickne... | 18,536 | 18,536 | olympiad | 16,061 |
<|im_start|>user
Let $T=5$. Hannah flips two fair coins, while Otto flips $T$ fair coins. Let $p$ be the probability that the number of heads showing on Hannah's coins is greater than the number of heads showing on Otto's coins. If $p=q / r$, where $q$ and $r$ are relatively prime positive integers, compute $q+r$.
Plea... | 17,222 | 17,222 | olympiad | 13,074 |
<|im_start|>user
Compute the least integer greater than 2023 , the sum of whose digits is 17 .
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 13,040 | 13,040 | olympiad | 8,915 |
<|im_start|>user
In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$... | 19,692 | 19,692 | olympiad | 15,404 |
<|im_start|>user
Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8}$ be a regular octagon. Let $\mathbf{u}$ be the vector from $A_{1}$ to $A_{2}$ and let $\mathbf{v}$ be the vector from $A_{1}$ to $A_{8}$. The vector from $A_{1}$ to $A_{4}$ can be written as $a \mathbf{u}+b \mathbf{v}$ for a unique ordered pair of re... | 27,433 | 27,433 | olympiad | 20,802 |
<|im_start|>user
Let $T=15$. For complex $z$, define the function $f_{1}(z)=z$, and for $n>1, f_{n}(z)=$ $f_{n-1}(\bar{z})$. If $f_{1}(z)+2 f_{2}(z)+3 f_{3}(z)+4 f_{4}(z)+5 f_{5}(z)=T+T i$, compute $|z|$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 30,487 | 30,487 | olympiad | 21,681 |
<|im_start|>user
Suppose that $a(x+b(x+3))=2(x+6)$ for all values of $x$. Determine $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 3,121 | 3,121 | olympiad | 5,214 |
<|im_start|>user
Large Hadron Collider
Please read the general instructions in the separate envelope before you start this problem.
In this task, the physics of the particle accelerator LHC (Large Hadron Collider) at CERN is discussed. CERN is the world's largest particle physics laboratory. Its main goal is to get i... | 8,197 | 8,197 | olympiad | 9,918 |
<|im_start|>user
Let $T=7$. Compute the number of positive perfect cubes that are divisors of $(T+10) !$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| 15,749 | 15,749 | olympiad | 9,982 |
<|im_start|>user
$$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m... | 32,368 | 32,368 | olympiad | 10,667 |
<|im_start|>user
A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$.
Determ... | 11,151 | 11,151 | olympiad | 8,923 |
<|im_start|>user
LIGO-GW150914
In 2015, the gravitational-wave observatory LIGO detected, for the first time, the passing of gravitational waves (GW) through Earth. This event, named GW150914, was triggered by waves produced by two black holes that were orbiting on quasi-circular orbits. This problem will make you est... | 6,735 | 6,735 | olympiad | 11,925 |
<|im_start|>user
The taxicab distance between points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$ is defined as $d(A, B)=$ $\left|x_{A}-x_{B}\right|+\left|y_{A}-y_{B}\right|$. Given some $s>0$ and points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$, define the taxicab ellipse with ... | 29,479 | 29,479 | olympiad | 15,971 |
<|im_start|>user
Let $T=19$. The first two terms of a sequence are $a_{1}=3 / 5$ and $a_{2}=4 / 5$. For $n>2$, if $n$ is odd, then $a_{n}=a_{n-1}^{2}-a_{n-2}^{2}$, while if $n$ is even, then $a_{n}=2 a_{n-2} a_{n-3}$. Compute the sum of the squares of the first $T-3$ terms of the sequence.
Please reason step by step, a... | 29,640 | 29,640 | olympiad | 22,949 |
Qwen3.6-27B generation-length prediction: heads, calibrations and workloads
Artifacts for conformal length-aware LLM scheduling on Qwen/Qwen3.6-27B — predicting a
request's remaining generation length from a hidden layer during decoding, wrapping it in a
split-conformal interval, and scheduling with SRPT inside vLLM. Extends TRAIL
(Don't Stop Me Now, ICLR'25) to a hybrid-attention reasoning model.
This repo contains the derived artifacts, not the raw activations. The 3250 decode-step traces are
322 GB and are not uploaded. Everything needed to use the predictors is here; everything needed to retrain them is reproducible with the commands below (20 GPU-hours).
Setup
| model | Qwen/Qwen3.6-27B (Qwen3_5ForConditionalGeneration) |
| probe layer | 32 of 64 |
| sampling | temperature 0.6, max_tokens 32768, max_model_len 40960 |
| engine | vLLM 0.17.0, enforce_eager=True |
| feature | last-token hidden state at layer 32 per decode step, fp16, 5120-dim |
Qwen3.6-27B is hybrid-attention: 64 layers with full_attention_interval=4, so only 16 carry a
token-proportional KV cache and 48 hold fixed-size recurrent state. Consequences: KV is 64 KiB/token
(vs 192 for a comparable dense 14B), but each request commits ~144 MiB of recurrent state at
admission — equal to ~2300 tokens of KV. Per-request memory therefore barely grows with age until
well past the median generation.
Files
Heads — all trained on the same 1000 base traces, so they differ only in target/readout.
Held-out metrics on 300 base traces (mean T = 4625):
| file | target | readout | MAE | Spearman |
|---|---|---|---|---|
abs.pt |
sqrt(remaining), SmoothL1 |
clamp(z,0,zcap)² |
1431 | 0.737 |
prog.pt |
p=(g+1)/T via sigmoid |
t(1−p)/p, capped |
1539 | 0.638 |
trail_mlp.pt |
bin index, CE (TRAIL) | softmax(z)·bin_mids |
1439 | 0.720 |
trail.pt |
as above, 20 bins over 8000 | — | — | |
psi_L32.pt |
prompt-only → total T |
2090 | 0.70 |
trail_mlp.pt is the faithful TRAIL baseline: the paper's Linear(h→512)→ReLU→Linear(512→bins)
with 80 bins over 32768. trail.pt keeps the original 20-bins-over-8000 ceiling, which saturates on
7.5% of the short workload (34% of MATH) and understates the baseline — prefer trail_mlp.pt.
Calibrations (split-conformal, α=0.5, q_t pooled per decode step):
| file | calib set | M0 | median q_t |
|---|---|---|---|
conf_abs_base_a5.npz |
495 base | 17185 | 2436 |
conf_prog_base_a5.npz |
495 base | 17185 | 3289 |
conf_abs_aug_a5.npz |
+ 236 Olympiad | 28750 | 2935 |
conf_prog_aug_a5.npz |
+ 236 Olympiad | 28750 | 4539 |
⚠️ q_by_step[0] is NaN in every file. The collector records only true decode steps (nst == 1),
so the first token — produced by the prefill forward — is never captured and step 0 has no residuals
to take a quantile over. Every waiting request has g = 0, so a scheduler that indexes q_by_step[g]
without a guard gets NaN, and NaN compares False against everything: sorted() silently leaves the
admission queue in arbitrary order. Note beta * q does not save you — 0.0 * nan = nan. Fall back
to the nearest finite step.
⚠️ The augmented calibrations are not exchangeable with base traffic. Olympiad's median
true_len is 13722 vs base's 1585, so its residuals inflate q_t by +20% (abs) / +38% (prog), and
the extra support between step 17185 and 28750 comes almost entirely from Olympiad traces. Use
*_base_* when serving base-distribution traffic; the augmented files are provided for the
M0-vs-exchangeability comparison, not as a drop-in improvement.
Splits — split_base.json (train 1000 / calib 495 / eval 300, base only),
split_aug.json (same train, calib 731 = 495 base + 236 Olympiad, eval 300 Olympiad).
manifest_L32.json maps every trace path to (T, dataset).
Workloads — served request files with text, true_len, pred_len, dataset:
| file | n | median true_len |
Spearman(pred_len, true_len) |
|---|---|---|---|
bench_short.json |
845 | 1573 | 0.695 |
bench_olympiad.json |
228 | 13722 | 0.690 |
pred_len comes from ψ (prompt-only prefill predictor) and is never a copy of true_len —
0.0% identical in both. It is packed into the vLLM request id as the static admission priority, so
setting it to true_len would hand the oracle key to every predictor policy.
Prompts — collect_prompts.json (1800: gsm8k 500, mmlu_pro 500, bbh 400, math 400),
benchmark_prompts.json (850, verified zero overlap with the collection pool),
olympiad600_prompts.jsonl (OlympiadBench OE_TO_*_en_COMP, text-only English),
longcot600_prompts.jsonl (LongHorizonReasoning/longcot, 5 domains × 3 difficulties).
Prefill features — prefill_{collect,bench,olymp}.pt, mean-pooled layer-32 prompt states, one
row per prompt, the input to ψ.
Layer sweep — layer_sweep_qwen36.json (all 100 traces) and ..._uncapped.json (59 uncapped).
Choosing the probe layer
8 layers swept, 4 full_attention / 4 linear_attention, depth 0.17–0.75, ranked by held-out
cross-request Spearman over 5 resampled splits:
| set | best | ρ | vs runner-up |
|---|---|---|---|
| all 100 traces | L32 | 0.623 ± 0.044 | +0.032 ± 0.012 (separable) |
| 59 uncapped only | L32 | 0.700 ± 0.033 | +0.009 ± 0.022 (indistinguishable) |
On clean data L32 sits on a plateau with {32, 35, 43} — the choice within it is not load-bearing. The depth-ratio heuristic (0.5625 → L36) would have been meaningfully worse; its nearest swept neighbour L40 ranks 6th of 8. Layer type mattered little (linear 0.663 vs full 0.644).
Trace schema (for reproduction)
Each .pt is {"L32": float16 (n_steps, 5120), "T": int, "g": int32 (n_steps,), "dataset": str, "prompt": str}.
gis the 0-based index of the token an activation produced. Remaining isT − (g+1). Recording starts atg=1because the first token comes from the prefill forward, which is skipped.- A trace is not a contiguous step sequence — always index through
g, neverarange(len(H)). - Capped traces are mislabelled: a generation that never emits
</think>storesT = cap. Base capped 1% (5/500); Olympiad ~22%. Drop them from calibration — the residual|R_true − R̂|usesR_true = T − t, so a capped trace injects a too-small residual and makesq_toverconfident exactly in the tail.
Reproduction
export SCHED_MODEL=Qwen/Qwen3.6-27B SCHED_LAYER=32
python collect_activations_vllm.py --layers 32 --prompt_file collected/collect_prompts.json \
--max_tokens 32768 --max_model_len 40960 --gpu_mem 0.85 --chunk 100 --out <traces>
python build_split.py --init --base-glob '<traces>/*.pt' --sizes 1000,500,300 --out split_base.json
python train_head.py --head absolute --arch linear --split split_base.json --out abs.pt
python calibrate.py --ckpt abs.pt --split split_base.json --alphas 0.5 --out conf_abs_base_a5.npz
Collection is ~20 GPU-hours on one 80–96 GB card. enforce_eager=True is mandatory — CUDA graphs
bypass forward hooks, so the collector silently records nothing without it.
Limitations
- Raw activations are not included (~322 GB).
- Metrics are single-seed; head training variance was not measured.
- The layer sweep used
max_tokens 2500, where 41% of traces capped; the full collection used 32768, where only 1% of base traces cap. - Scheduling results are not included in this repo and remain under evaluation. One measured
property worth stating: with
max_num_seqs=64this workload never saturates the concurrency cap (~41 of 64 concurrent), so no queue forms and scheduling cannot help; a queue requiresmax_num_seqs=24. Preemption was exactly zero across ~100k schedule calls, so TRAIL's limited-preemption parametercis inert here.
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