Cross-Tokenizer On-Policy Distillation via Byte-Prefix Marginalization
Paper • 2607.22334 • Published
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"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nLet's... | code | taco:codeforces | MEDIUM_HARD | {
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the area of the region defined by the inequality $|3x-18|+|2y+7| \\leq 3$?\n\nRemember to ... | 3 | math | dapo-17k | null | {
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(x) = \\log_{10} \\left(\\sin(\\pi x) \\cdot \\sin(2 \\pi x) \\cdot \\sin (3 \\pi x) \\cdots... | 12 | math | dapo-17k | null | {
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n$A$, $B$, and $C$ are three piles of rocks. The mean weight of the rocks in $A$ is 40 pounds, the ... | 59 | math | dapo-17k | null | {
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram below, for each row except the bottom row, the number in each cell is determined by... | 13 | math | dapo-17k | null | {
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"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nBerla... | code | taco:codeforces | HARD | {
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $r_1, r_2, \\ldots, r_{47}$ be the roots of $x^{47} - 1 = 0$. Compute \\( \\sum_{i=1}^{47} r_i... | 0 | math | dapo-17k | null | {
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"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nGeral... | code | taco:codeforces | MEDIUM_HARD | {
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCalculate the value of \\(\\dfrac{3\\times 5}{9\\times 11}\\times \\dfrac{7\\times 9\\times 11}{3\... | 1 | math | dapo-17k | null | {
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"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nValer... | code | taco:codeforces | HARD | {
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"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"250 300 2\\n157 204\\n\", \"output\": \"1 2\\n1 3\\n1 4\\n1 5\\n1 6\\n1 7\\n1 8\\n1 9\\n1 10\\n1 11\\n1 12\\n1 13\\n1 14\\n1 15\\n1 16\\n1 17\\n1 18\\n1 19\\n1 20\\n1 21\\n1 22\\n1 23\\n1 24\\n1 25\\n1 26\\n1 ... | null | null | |
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"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nBarba... | code | taco:codeforces | VERY_HARD | {
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be positive real numbers satisfying\n$$a -12b = 11 -\\frac{100}{a} \\,\\,\\,\\,an... | 47 | math | dapo-17k | null | {
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"rm_type": "dapo"
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"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nThere... | code | taco:codeforces | EASY | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"2\\n1 2 2 1\\n4 8 9 2\\n\", \"output\": \"3\\n12\\n\"}, {\"input\": \"1\\n0 0 0 0\\n\", \"output\": \"0\\n\"}, {\"input\": \"1\\n0 0 0 0\\n\", \"output\": \"0\\n\"}, {\"input\": \"1\\n0 0 1 0\\n\", \"output\":... | null | null | |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the addition problem \n\\[ \\setlength{\\tabcolsep}{1mm}\\begin{tabular}{cccccc}& W & H & I & T... | 169069 | math | dapo-17k | null | {
"domain": "math",
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive integers are less than $200\\pi$?\n\nRemember to put your answer on its own line... | 628 | math | dapo-17k | null | {
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"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nAnd a... | code | taco:codeforces | EASY | {
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"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"2 1 1\\n0 100000\\n-100000 -100000\\n\", \"output\": \"2\\n\"}, {\"input\": \"55 11 44\\n0 3343 3387 3470 3825 3832 3971 4026 4043 4389 4492 4886 5015 5084 5161 5436 5595 5616 5677 5987 6251 6312 6369 6469 648... | null | null | |
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"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nPasse... | code | taco:codewars | EASY | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"fn\", \"cases\": [{\"input\": [432, 3554, 291, 28, 2], \"output\": [\"112.2\"]}, {\"input\": [5, 76, 4, 1, 0], \"output\": [\"158.3\"]}, {\"input\": [48, 192, 19, 2, 3], \"output\": [\"39.6\"]}, {\"input\": [1, 2, 1, 1, 0], \"output\": [\"118.8... | null | null | |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nYou a... | code | taco:geeksforgeeks | MEDIUM | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"n = 11\\r\\nnums = [1,1,1,0,0,0,1,1,1,1,0]\\r\\nk = 2\", \"output\": \"6\"}], \"fn_name\": null}",
"has_tests": true
} | null | null | |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nChill... | code | taco:codeforces | MEDIUM | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"1\\n\", \"output\": \"-1\\n\"}, {\"input\": \"5\\n\", \"output\": \"10080\"}, {\"input\": \"6\\n\", \"output\": \"100170\"}, {\"input\": \"4\\n\", \"output\": \"1050\"}, {\"input\": \"15\\n\", \"output\": \"10... | null | null | |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nA per... | code | taco:codeforces | MEDIUM_HARD | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"4 1\\n\", \"output\": \"1\\n\"}, {\"input\": \"4 2\\n\", \"output\": \"7\\n\"}, {\"input\": \"5 3\\n\", \"output\": \"31\\n\"}, {\"input\": \"5 4\\n\", \"output\": \"76\\n\"}, {\"input\": \"200 1\\n\", \"outpu... | null | null | |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nGiven... | code | taco:leetcode | EASY | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"fn\", \"cases\": [{\"input\": [\"\\\"aaaabbbbcccc\\\"\"], \"output\": \"\\\"abccba\\\"abccba\"}], \"fn_name\": \"sortString\"}",
"has_tests": true
} | null | null | |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nElect... | code | taco:codeforces | MEDIUM | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"6\\n11 6\\n10 4\\n01 3\\n00 3\\n00 7\\n00 9\\n\", \"output\": \"22\\n\"}, {\"input\": \"5\\n11 1\\n01 1\\n00 100\\n10 1\\n01 1\\n\", \"output\": \"103\\n\"}, {\"input\": \"6\\n11 19\\n10 22\\n00 18\\n00 29\\n1... | null | null | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOne angle of a triangle is twice another, and the sides opposite these angles have lengths 15 and ... | 16 | math | dapo-17k | null | {
"domain": "math",
"rm_type": "dapo"
} | null | null |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA sequence of numbers $x_{1},x_{2},x_{3},\\cdots,x_{100}$ has the property that, for every integer... | 173 | math | dapo-17k | null | {
"domain": "math",
"rm_type": "dapo"
} | null | null |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown ... | 4 | math | dapo-17k | null | {
"domain": "math",
"rm_type": "dapo"
} | null | null |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nYou'v... | code | taco:codeforces | MEDIUM_HARD | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"6 6\\n1 1 2 3 5 8\\n1 2\\n3 2\\n1 4\\n4 3\\n4 5\\n4 6\\n\", \"output\": \"3\\n\"}, {\"input\": \"5 6\\n4 2 5 2 4\\n1 2\\n2 3\\n3 1\\n5 3\\n5 4\\n3 4\\n\", \"output\": \"2\\n\"}, {\"input\": \"3 1\\n13 13 4\\n1... | null | null | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathbf{v}_1,$ $\\mathbf{v}_2,$ $\\dots,$ $\\mathbf{v}_k$ be unit vectors such that\n\\[\\ma... | 8 | math | dapo-17k | null | {
"domain": "math",
"rm_type": "dapo"
} | null | null |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), ... | 4 | math | dapo-17k | null | {
"domain": "math",
"rm_type": "dapo"
} | null | null |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTerry decides to practice his arithmetic by adding the numbers between $10$ and $99$ inclusive. Ho... | 59 | math | dapo-17k | null | {
"domain": "math",
"rm_type": "dapo"
} | null | null |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nGiven... | code | taco:leetcode | MEDIUM_HARD | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"fn\", \"cases\": [{\"input\": [\"\\\"1101\\\"\"], \"output\": 8}], \"fn_name\": \"numSteps\"}",
"has_tests": true
} | null | null | |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nA coo... | code | taco:codeforces | EASY | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"3\\n1 1\\n2 2\\n3 3\\n\", \"output\": \"-1\\n\"}, {\"input\": \"6\\n1 5\\n2 3\\n1 10\\n7 10\\n7 7\\n10 10\\n\", \"output\": \"3\\n\"}, {\"input\": \"4\\n1 5\\n2 2\\n2 4\\n2 5\\n\", \"output\": \"1\\n\"}, {\"in... | null | null | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest natural number $n$ such that the expression $n^2 - n + 11$ has exactly four prim... | 132 | math | dapo-17k | null | {
"domain": "math",
"rm_type": "dapo"
} | null | null |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nRead ... | code | taco:codechef | HARD | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"2 3\\n0 0 5\\n8 3 2\\n0\\n10\\n20\", \"output\": \"0\\n1\\n0\"}], \"fn_name\": null}",
"has_tests": true
} | null | null | |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nBerla... | code | taco:codeforces | VERY_HARD | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"46865942276811740149949176718949673344632458696505595472917789224885825949034661409971763949176343056701403524645790892802371117466746709730235969308113002256137529699677021858777002204698794034488631496662175... | null | null | |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nGiven... | code | taco:atcoder | MEDIUM_HARD | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"3 2\\n1 2 3\\n2 2 2\\n1 2\\n2 3\\n\", \"output\": \"Yes\\n\"}, {\"input\": \"1 0\\n5\\n5\\n\", \"output\": \"Yes\\n\"}, {\"input\": \"2 1\\n1 1\\n2 1\\n1 2\\n\", \"output\": \"No\\n\"}, {\"input\": \"17 9\\n-9... | null | null | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFive bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes ... | 270 | math | dapo-17k | null | {
"domain": "math",
"rm_type": "dapo"
} | null | null |
[
{
"role": "user",
"content": "You are given a competitive programming problem. Think step by step, then provide a single complete Python program in a ```python code block. If a starter code / function signature is given, implement it; otherwise read from standard input and write to standard output.\n\nA Nic... | code | taco:hackerearth | MEDIUM | {
"domain": "code",
"rm_type": "code_sandbox",
"tests": "{\"type\": \"stdin\", \"cases\": [{\"input\": \"100\\n30 43 6335\\n19170 26502 15725\\n6401 11480 26963\\n5706 24466 28146\\n16828 23283 9962\\n44 493 11943\\n609 4829 32392\\n3903 14606 154\\n77 294 17422\\n1002 18718 19896\\n5383 5449 14772\\n1870 11540 1... | null | null | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn a simple graph, there are two disjoint sets of vertices $A$ and $B$ where $A$ has $k$ vertices ... | 6 | math | dapo-17k | null | {
"domain": "math",
"rm_type": "dapo"
} | null | null |
End of preview. Expand in Data Studio
BPM Training Prompts (mix-20k)
arXiv:2607.22334 · Project page
Prompt corpus behind every reported result in BPM (Byte-Prefix Marginalization), a cross-tokenizer on-policy distillation method. Prompt-only.
| Domain | Rows | Source | Upstream license |
|---|---|---|---|
| Mathematics | 10,000 | dapo-17k |
Apache-2.0 |
| Code | 10,000 | taco:* |
Apache-2.0 |
Fields
| Field | Type | Notes |
|---|---|---|
prompt |
list of {role, content} |
Chat-format user turn |
label |
string | Reference answer for mathematics; empty for code |
domain |
string | math or code |
source |
string | e.g. dapo-17k, taco:codeforces |
metadata |
dict | rm_type (dapo / code_sandbox); code rows carry tests |
difficulty |
string | Code rows: EASY / MEDIUM / HARD |
idx, has_starter |
int, bool | Function-signature code rows |
from datasets import load_dataset
ds = load_dataset("K1zE/BPM", split="train")
Redistributed for research reproduction; items remain subject to their original sources' terms.
Citation
@misc{wang2026bpm,
title={Cross-Tokenizer On-Policy Distillation via Byte-Prefix Marginalization},
author={Hao Wang and Kun Yuan and Wenlin Zhong and Minglei Zhang and Han Xiao and Ming Sun and Honggang Qi},
year={2026},
archivePrefix={arXiv},
primaryClass={cs.LG},
eprint={2607.22334},
}
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