mihailgribov/olympiad_style_integer_math_problems

Olympiad Math Corpus

Version: v2.1.1
Release date: 2026-05-03

59,486 synthetically generated olympiad-style math problems with verified integer answers and formal computation graphs.

Loading

from datasets import load_dataset

ds = load_dataset("mihailgribov/olympiad_style_integer_math_problems", split="train")

lemma_applicability is stored as list[{lemma, status}] rather than a sparse dict (required for Arrow-based consumers). To convert to a dict for local use:

lemma_app = {x["lemma"]: x["status"] for x in row["lemma_applicability"]}

Overview

Each problem is generated from a computation graph (CG-Python DSL) that formally defines the mathematical structure and answer. Problem text is produced by an LLM from the graph. Every answer is verified by independent evaluation of the graph by a deterministic evaluator.

Domains are classified by graph structure, not by problem text.

Domain	Count	%	Description
NT	39,959	67.2%	Number theory
COMB	11,054	18.6%	Combinatorics
ALG	5,774	9.7%	Algebra
GEOM	2,699	4.5%	Geometry

Files

The release ships the full dataset in both Parquet and JSONL formats. load_dataset and the HuggingFace viewer use ready.parquet; ready.jsonl is provided for direct reading, grep, and streaming without the datasets library. Both files contain the same records.

File	Records	Description
ready.parquet	59,486	Full dataset, Parquet (used by load_dataset / HF viewer)
ready.jsonl	59,486	Full dataset, JSONL (same records; for direct streaming)
ready_sample_50.jsonl	50	Stratified sample for inspection (JSONL)
lemmas_used.jsonl	88	Lemmas used in the dataset (id, name, description, counts)

ready_sample_50.jsonl contains 50 problems sampled stratified by olympiad_level: allocations follow the dataset's OL distribution proportionally (largest-remainder method), with at least one problem from every non-empty OL. Train/validation/test splits are intentionally left to the consumer — the dataset ships as a single shuffled file.

---

Problem Schema (ready.parquet / ready.jsonl, identical)

Field	Type	Description
id	str	6-character hex hash of the graph field (SHA-256). Deterministic.
alias	str	Human-readable identifier. Encodes template, antilemma, seed.
problem_hash	str	6-character hex hash of problem text (SHA-256).
created_at	str	ISO 8601 timestamp (UTC).
problem	str	Problem text in natural language (LaTeX math).
answer	int	Correct answer. Integer in range [0, 99999].
graph	str	Computation graph in CG-Python DSL.
domain	str	Primary domain: NT, COMB, ALG, GEOM.
secondary_domain	str|null	Secondary domain, or null.
goal	str	Goal type: COMPUTE, COUNT, SUM, EXTREMUM.
evaluator_id	str	Symbolic engine used for verification: sympy or cpsat.
olympiad_level	int	Mathematical sophistication (0–9). See Olympiad Level.
irt_difficulty	object|null	IRT-1PL difficulty estimate: {lo, mid, hi}. See IRT Difficulty.
root_lemma	str|null	Root lemma (core mathematical identity).
lemma_paths	list[str]	Solver paths: /-separated lemma chains.
lemma_set	list[str]	Sorted unique lemma IDs from all paths.
num_lemmas	int	Number of unique lemmas in all solver paths.
num_spawns	int	Number of structural enrichment steps applied during generation.
seed_template_id	str	Seed template used for generation.
recipe_id	str	Hash of sorted lemma_paths. Same structure = same recipe.
ending_id	str	Specialized ending identifier.
generation_time	float	Time to generate and verify the problem (seconds).
verification	object	{verified, answer, timestamp}. See below.
lemma_applicability	list[object]	Lemma selection labels ([{lemma, status}, ...]). See below.
solution_status	int|null	Aggregate LLM correctness. See Solution Status.
llm_solvers	list[object]	LLM solver results. See LLM Solvers.
license	str	"CC BY 4.0"

Example record

{
  "id": "12be79",
  "alias": "nt_sum_divisors_mod_v1_124444284_4622",
  "problem": "Let $n$ be the number of positive integers less than or equal to $674$ that are relatively prime to $15$. Let $\\sigma$ be the sum of the positive divisors function. Compute $\\sigma(n) \\pmod{89}$.",
  "answer": 1170,
  "graph": "graphs = [\\n    Graph(\\n        let={\\n            \"n\": CountOverSet(...)\\n        },\\n        goal=\"result\"\\n    )\\n]",
  "domain": "NT",
  "secondary_domain": null,
  "goal": "COMPUTE",
  "evaluator_id": "sympy",
  "root_lemma": "C4",
  "lemma_paths": ["C4"],
  "recipe_id": "08d162",
  "seed_template_id": "nt_sum_divisors_mod_v1",
  "ending_id": null,
  "olympiad_level": 5,
  "num_spawns": 0,
  "lemma_set": ["C4"],
  "num_lemmas": 1,
  "generation_time": 0.001,
  "created_at": "2026-02-08T06:06:42.273144Z",
  "verification": {
    "verified": true,
    "answer": 1170,
    "timestamp": "2026-02-08T06:06:42.274633Z"
  },
  "problem_hash": "49f89c",
  "license": "CC BY 4.0",
  "llm_solvers": [
    {
      "id": 5,
      "model": "deepseek-ai/DeepSeek-V3.2",
      "answer": 1170,
      "score": 3,
      "correct": {"strict": true, "boxed": true, "relaxed": true},
      "usage": {"prompt_tokens": 102, "completion_tokens": 828},
      "timestamp": "2026-02-12T20:35:01.853Z"
    }
  ],
  "solution_status": 1,
  "lemma_applicability": [
    {"lemma": "C4", "status": "ok"},
    {"lemma": "K17", "status": "no"},
    {"lemma": "L3b", "status": "no"},
    {"lemma": "V3", "status": "no"}
  ],
  "irt_difficulty": {"lo": -10.0, "mid": -3.35, "hi": 4.18}
}

verification

Key	Type	Description
verified	bool	true if answer matches independent graph evaluation.
answer	int	Expected answer from the generator (same as the top-level answer field).
error	str	Error message (only when verified is false).
timestamp	str	ISO 8601 timestamp of verification.

lemma_applicability

List of {lemma, status} entries indicating whether each candidate lemma is the correct first solving step. Sorted by lemma for determinism. Empty list when lemma_paths is empty. Non-empty on 56,949 of 59,486 problems.

"lemma_applicability": [
  {"lemma": "K3", "status": "ok_later"},
  {"lemma": "K5", "status": "same_pattern_wrong"},
  {"lemma": "V1", "status": "ok"},
  {"lemma": "V7", "status": "no"}
]

Label	Meaning
ok	Correct first step — the lemma appears first in at least one solver path.
ok_later	Correct lemma but not as first step — appears later in a solver path.
same_pattern_wrong	Wrong choice that matches the exact same graph pattern as an ok lemma (e.g. Legendre vs digit-sum formula for v_p(n!)).
no	Wrong — a domain-compatible lemma that does not match the problem structure. Sampled (up to 5).

---

Lemma Catalog (lemmas_used.jsonl)

The file contains 88 mathematical lemmas (identities and reduction rules) that appear in at least one problem in the dataset. Each lemma is a named transformation used by the symbolic solver to reduce computation graphs. Lemma IDs are referenced by the lemma_set, lemma_paths, root_lemma, and lemma_applicability fields in problem records.

Field	Type	Description
id	str	Unique lemma identifier (e.g. K3, V1, LIN_FORM).
name	str	Human-readable snake_case name.
type	str	solver_lemma (from code registry) or dataset_only (found in data only).
domains	list[str]	Mathematical domains where the lemma applies (e.g. ["number_theory"]).
level	str	Complexity level: trivial, elementary, standard, olympiad, advanced.
track	str	Methodology track: core, olympiad_nt, olympiad_algebra, etc.
description	str	Plain-text description of the mathematical pattern.
description_latex	str	LaTeX formula for the identity (e.g. \sum_{d \mid n} \varphi(d) = n).
dataset_count	int	Number of problems in the dataset that use this lemma.
dataset_fraction	float	Fraction of total problems using this lemma.
dataset_as_root	int	Number of problems where this lemma is the root_lemma.

Example record

{
  "id": "LIN_FORM",
  "name": "linear_form_range_counting",
  "type": "solver_lemma",
  "domains": ["algebra"],
  "level": "olympiad",
  "track": "olympiad_nt",
  "description": "count of linear form values.",
  "description_latex": "|\\{x \\in [a,b] : \\exists\\, k,\\; x = \\alpha + k\\beta\\}|",
  "dataset_count": 15888,
  "dataset_fraction": 0.1951,
  "dataset_as_root": 13176
}

---

Statistics

Olympiad Level

Mathematical sophistication level assigned by GPT-5.1 from the computation graph alone (no problem text). Measures the rarity and non-obviousness of the required mathematical insight. This is distinct from solve difficulty: the correlation with empirical irt_difficulty is weak (Pearson r ≈ 0.23). Problems scoring below 2 are excluded from the dataset as trivial or invalid.

Scoring prompt used for olympiad_level

The label is the output of a single, deterministic LLM call. The system prompt (reproduced verbatim below) fixes the evaluation axis so scores are comparable across releases.

You are a mathematical olympiad problem evaluator.

You receive computation graphs written in a Python DSL (CG-Python).
You do NOT see the natural-language problem statement.

The `goal` node is the value to compute.
`Ref("x")` denotes a previously defined variable.
`Var("x")` denotes a bound variable inside set comprehensions.

Your task is to assess the olympiad level of each problem on a scale 0–9.

CORE PRINCIPLE
    Grade the problem by the mathematical insight that is logically required
    to solve it, as inferred from the graph structure.
    Do NOT reward graph size, depth, nesting, large constants,
    or routine mechanical transformations.
    Key question: What mathematical idea is necessary here,
    and how obvious is it?

GRAPH-ONLY EVALUATION
    Infer difficulty ONLY from the graph.
    - which mathematical concepts are involved,
    - whether the goal requires a single known theorem directly,
      or a non-obvious connection between concepts,
    - whether quantifiers, sets, or extrema introduce real reasoning
      rather than simple wrapping.
    Do NOT treat the mere presence of advanced-looking nodes
    (Factorial, Binomial, EulerPhi, MoebiusMu, Lucas, etc.)
    or deep nesting as insight.

LEVELS (0–9)
    0 — Invalid / non-interpretable graph.
    1 — Bare textbook definition lookup.
    2 — Trivial; one-line fact.
    3 — Direct application of a single theorem.
    4 — Exam-style; correct method is obvious.
    5 — Training olympiad; standard idea, execution-heavy.
    6 — School olympiad; exactly one non-obvious key insight.
    7 — Regional / strong olympiad; combine standard ideas non-trivially.
    8 — National / IMO-shortlist; requires seeing hidden structure.
    9 — Top international olympiad; genuine non-routine insight or
        construction, not reducible to recalling a named theorem.

COMBINATION RULE
    If solving the graph requires combining multiple standard ideas
    in a way that is not obvious from the graph alone, increase the
    level by +1 relative to using any one idea alone.
    Do NOT increase the level if ideas are applied only sequentially
    or mechanically.

WRAPPER PENALTY
    If a complex inner subgraph merely computes a known invariant
    and the outer computation is routine arithmetic, score the problem
    by the outer goal only. Wrapping a known result inside a trivial
    shell does NOT increase the level. Such problems typically remain
    level 2–4.

OUTPUT FORMAT
    Output TSV, one line per graph:   id<TAB>level
    No explanations, no headers, no code fences.

The user message is a concatenation of the graph strings, one per target problem. Temperature and other decoding parameters match the text_gen pipeline configuration.

Level	Label	Description
0	Invalid	Incorrect, inconsistent, or ambiguous graph.
1	Textbook	Definition lookup, no real mathematical content.
2	Trivial	One-line fact, solved instantly.
3	Direct application	Single known theorem applied mechanically.
4	Exam-style	Standard exam problem; correct method is obvious.
5	Training olympiad	Standard idea; difficulty is mainly in execution.
6	School olympiad	Exactly one non-obvious key insight required.
7	Regional olympiad	Multiple standard ideas combined non-trivially.
8	National / ISL	Hidden structure; insight unlikely without experience.
9	Top international	Genuine non-routine insight or construction.

Scoring rules: (1) wrapping a known result in trivial arithmetic does not increase the level; (2) combining multiple independent ideas non-trivially adds +1; (3) graph size, nesting depth, and large constants are not rewarded.

Level	Count	%
2	4,249	7.1%	███▌
3	10,312	17.3%	████████▌
4	13,703	23.0%	███████████▌
5	12,269	20.6%	██████████
6	12,503	21.0%	██████████▌
7	5,642	9.5%	████▌
8	794	1.3%	▌
9	14	0.0%	▏

IRT Difficulty

Calibrated difficulty estimate based on Item Response Theory (IRT-1PL / Rasch model). Present for 59,486 tasks attempted by at least one LLM solver.

The Rasch model defines the probability that model j solves task i as:

P(correct) = σ(θⱼ − βᵢ)

where σ is the logistic function, θⱼ is model skill, and βᵢ is task difficulty. Higher β = harder task. Model skills (θ) are estimated jointly on "core" tasks attempted by ≥3 models; task difficulties (β) are then estimated per-task with fixed θ.

"irt_difficulty": {"lo": -1.86, "mid": 2.91, "hi": 7.44}

Key	Type	Description
lo	float	Lower bound of the 95% confidence interval.
mid	float	Point estimate of difficulty (MLE for mixed results, midpoint of CI for perfect scores).
hi	float	Upper bound of the 95% confidence interval.

Confidence intervals are computed via profile likelihood: the set of β values where the log-likelihood is within χ²(1)/2 = 1.92 of the maximum. This gives finite, interpretable bounds even for tasks attempted by a single model, where Wald-type intervals (β ± 1.96·SE) would diverge. The interval width reflects estimation uncertainty: tasks attempted by more models have narrower intervals (typical width 4–6) than single-model tasks (width 15–18).

Answer Distribution

All answers are non-negative integers in [0, 99999]. The answer distribution is non-uniform but smooth, with full-range coverage and no abrupt discontinuities.

Range	Count	%
0–9,999	28,477	47.9%	███████████████████████▌
10,000–19,999	6,161	10.4%	█████
20,000–29,999	4,851	8.2%	████
30,000–39,999	4,324	7.3%	███▌
40,000–49,999	4,422	7.4%	███▌
50,000–59,999	3,752	6.3%	███
60,000–69,999	3,011	5.1%	██▌
70,000–79,999	2,138	3.6%	█▌
80,000–89,999	1,582	2.7%	█
90,000–99,999	752	1.3%	▌

The median answer is 11,741 and the mean is 1,482,189. The distribution is concentrated in the lower range (47.9% of answers fall in [0, 9,999]), reflecting the typical output magnitude of small integer-valued olympiad problems, but has smooth coverage across the full [0, 99,999] range without gaps. Very small values such as 0, 1, and 2 do not dominate the distribution and occur with comparable, low frequencies, so no special or degenerate cases collapse into these values.

LLM Solvers

Each model receives the problem text as the user message with the following system prompt:

You are a participant of the International Mathematical Olympiad. Solve the given problem. Think step by step. Write your final answer as a single integer inside \boxed{}, for example: \boxed{42}

All models are queried with temperature=0 and max_completion_tokens=32768. No few-shot examples or chain-of-thought scaffolding is provided beyond the system prompt above.

ID	Model	Attempted	Solved	Solve rate ¹
11	google/gemma-2-9b-it	48,280	5,419	11.2%
5	deepseek-ai/DeepSeek-V3.2	40,455	37,966	93.8%
8	mathstral	37,484	6,401	17.1%
17	meta-llama/Llama-3.3-70B-Instruct	30,160	8,347	27.7%
2	openai/gpt-oss-120b	26,350	23,877	90.6%
1	openai/gpt-oss-20b	21,524	17,533	81.5%
36	qwen2.5:3b-32k	19,311	3,645	18.9%
10	qwen2-math:7b	14,510	4,676	32.2%
16	Qwen/Qwen3-Next-80B-A3B-Thinking	5,427	4,739	87.3%
4	NousResearch/Hermes-4-405B	3,283	1,293	39.4%
15	Qwen/Qwen3-Coder-480B-A35B-Instruct	2,078	1,198	57.7%
29	Qwen/Qwen3-235B-A22B-Instruct-2507	1,749	1,417	81.0%
3	Qwen/Qwen3-235B-A22B-Thinking-2507	1,343	1,268	94.4%
38	google/gemma-3-27b-it	819	166	20.3%

¹ Solve rate is not a comparable measure of model capability across rows of this table. Each solver was run on a different subset of problems (cost, rate limits, and the pilot vs. continuous phase of a given model all shaped the attempt budget), so the "Attempted" column varies by more than 10× between models and the per-row solve rate is computed against a different task pool for each row. For capability comparisons that account for task difficulty, use irt_difficulty on the task side together with jointly-fit model skill (θ).

For a per-OL view of how each model degrades with task difficulty, see the figure below. The 9 solvers are ordered on a rough capability ladder (top-left = strongest, bottom-right = weakest). For each model, attempts are grouped by olympiad_level (OL=2..9) and stacked: green = correct.strict (last \boxed{} matches the expected answer), red = wrong. Cells with fewer than 30 attempts are faded (not statistically significant).

Overall solver coverage:

• Total solver attempts: 252,773
• Total correct (strict): 117,945
• Tasks with ≥1 correct solution: 58,634 (98.6%)
• Mean attempts / task: 4.25
• Mean correct solutions / task: 1.98

Each llm_solvers entry:

Key	Type	Description
id	int	Integer identifier of the solver (see table above).
model	str	Model identifier.
answer	int|null	Parsed answer from \boxed{}.
score	int	Correctness level (see below).
correct	object	{strict, boxed, relaxed} booleans (see below).
usage	object	{prompt_tokens, completion_tokens}.
timestamp	str	ISO 8601 timestamp.

Correctness levels (each level implies the ones below it):

Score	Level	Definition
3	strict	Last \boxed{} integer equals the expected answer.
2	boxed	Expected answer appears as a standalone number inside any \boxed{}.
1	relaxed	Expected answer appears as a standalone number anywhere in the response.
0	wrong	None of the above.

Solution Status

Aggregate LLM correctness per problem. Derived from individual llm_solvers entries. Only problems attempted by at least one LLM solver receive a status; the rest are null.

Value	Label	Description
2	All correct	Every LLM solver achieved a strict match.
1	Mixed	At least one solver correct and at least one wrong.
0	All wrong	No solver achieved a strict match.
null	Untested	No LLM solver data available.

Status	Label	Count
2	All correct	4,194
1	Mixed	54,440
0	All wrong	730
null	Untested	122

---

Novelty

Problems are constructed from deterministic symbolic structures without using existing problem texts; large language models are used only for natural-language rendering after the mathematical content is fixed. This avoids reuse of web-circulated problems or content memorized by language models.

Curation and Quality Control

The dataset is produced by an automated generation pipeline followed by filtering. The following checks are applied to improve the overall quality of the dataset.

• Seed verification. Each problem starts from a simple seed graph that is directly solvable by a symbolic engine (e.g. SymPy, Google OR-Tools). Before any further transformations, the seed is evaluated to confirm that:
  • the engine terminates rather than hangs;
  • it completes without errors;
  • the answer is correct and falls within the [0, 99999] integer range.
• Full problem verification. The final problem (after structural transformations that increase complexity) is independently verified by a dedicated solver that sequentially applies mathematical lemmas (normalisation, simplification, known identities) to reduce the graph to an expression evaluable by a symbolic engine. Mismatches with the stored answer are rejected.
• Deduplication.
  • By graph: a SHA-256 hash of the computation graph is computed; identical graphs are detected and removed.
  • By text: problems with identical natural-language statements (SHA-256 of problem text) are removed; only the first occurrence is kept.
• Minimum sophistication. The LLM that assigns olympiad_level also assesses problem correctness: invalid or malformed problems receive level 0, and trivial ones receive level 1. Problems with olympiad_level below 2 are excluded.
• LLM-based suspect detection. The release pipeline runs several consistency checks against how a pool of independent LLM solvers behaved on each task. These checks flag problems where solver outcomes suggest a bug in the stored answer or statement rather than genuine difficulty — for example, tasks that no strong model can solve, tasks where multiple solvers converge on the same wrong answer, or tasks where weaker models succeed while stronger ones fail. Flagged problems are filtered out of the main release; they are not published as part of this version. The thresholds are tuned conservatively so that genuinely hard but correct problems are preserved rather than over-filtered.
• Record size limit. Serialized records exceeding 100 KB are excluded.

Intended Use

The dataset supports multiple training and evaluation approaches:

• Supervised fine-tuning: Training models on multi-step mathematical reasoning with verified integer answers.

• Reinforcement learning (RLVR): Verified answers enable automatic reward signals; the llm_solvers field contains logged model answer attempts with associated verification-based scores, supporting outcome-level reinforcement learning setups.

• Lemma selection training: The lemma_applicability annotations provide positive and negative examples for learning which lemma to apply first in a given problem. same_pattern_wrong entries serve as hard negatives — plausible-but-incorrect lemmas that match the same graph pattern as the correct one — useful for contrastive lemma-selection training.

• Curriculum learning: Difficulty is represented along multiple independent dimensions — mathematical sophistication (olympiad_level), empirical solve rate (irt_difficulty), structural complexity (num_lemmas, num_spawns), and domain — enabling gradual difficulty progression along different axes.

• Model trajectory shaping: Each problem's llm_solvers entry records which reference models solved it, placing the problem at a specific point in a multi-dimensional capability space defined by the set of solvers. Training a new model can be framed as steering its trajectory through this space: by selecting tasks with particular solver profiles — easy for one model and hard for another, uniquely solved by a thinking model but not a base one, consistently solved across a chosen reference set — one can guide the learning process toward specific capability regions rather than relying on a single scalar difficulty axis. For example, weighting training examples by agreement with a target profile allows shaping a model to match DeepSeek-V3.2 on number theory while preserving gpt-oss-20b-level combinatorics performance.

• Generalization evaluation: Records carry enough structural metadata (recipe_id, seed_template_id, lemma_paths, root_lemma, olympiad_level, domain) for consumers to carve out their own structure-aware evaluation splits that test generalization to new combinations of lemmas rather than memorization of fixed solution templates.

Citation

@misc{gribov2026olympiad,
  author       = {Gribov, Mikhail},
  title        = {Olympiad Math Corpus},
  year         = {2026},
  version      = {v2.1.1},
  publisher    = {Hugging Face},
  howpublished = {\url{https://huggingface.co/datasets/mihailgribov/olympiad_style_integer_math_problems}},
  license      = {CC BY 4.0}
}

License

This dataset is released under the Creative Commons Attribution 4.0 International license (CC BY 4.0).

Each record contains "license": "CC BY 4.0".