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@@ -58,3 +58,5 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
 # Video files - compressed
 *.mp4 filter=lfs diff=lfs merge=lfs -text
 *.webm filter=lfs diff=lfs merge=lfs -text
+ready.jsonl filter=lfs diff=lfs merge=lfs -text
+train.jsonl filter=lfs diff=lfs merge=lfs -text

README.md

@@ -1,28 +1,237 @@
----
-license: cc-by-4.0
-language:
-- en
-pretty_name: Olympiad-Style Integer Math Problems
-size_categories:
-- 10K<n<100K
----
-## Olympiad-Style Math Problems with Structured Representation
+# Olympiad Math Corpus
 
-### Key Properties
+**42,112** synthetically generated olympiad-style math problems with verified integer answers, formal computation graphs, and structure-aware train/valid/test splits.
 
-- **Minimal data leakage:**  Problems are generated from scratch via deterministic symbolic construction.  
-  Natural-language text is produced only after the mathematical structure is fixed.
-- **Deterministic verification:** Each problem is associated with a formal computation graph whose value is obtained by deterministic evaluation.
-- **Structural diversity:**  The dataset covers a wide range of distinct solution structures, since problems are constructed as compositions of formal lemmas rather than surface-level text variations.
+## Dataset 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.
 
-### Dataset Description
+Domains are classified by graph structure, not by problem text.
 
-This dataset contains machine-generated olympiad-style mathematical problems with integer answers in the range 0–99999, avoiding concentration at special or trivial values.
+| Domain | Count | % | Description |
+|--------|------:|--:|-------------|
+| NT | 32,048 | 76.1% | Number theory |
+| COMB | 4,606 | 10.9% | Combinatorics |
+| ALG | 4,394 | 10.4% | Algebra |
+| GEOM | 1,064 | 2.5% | Geometry |
 
-Each problem is provided in two aligned forms:
-– A natural-language problem statement with LaTeX.
-– A structured formal representation that specifies the underlying mathematics and determines the answer.
+## File Format
 
-These core representations are complemented by additional structured fields and metadata, supporting filtering, analysis, and controlled evaluation.
+All files are JSONL (one JSON object per line).
+
+| File | Records | Description |
+|------|--------:|-------------|
+| `ready.jsonl` | 42,112 | Full dataset (shuffled) |
+| `train.jsonl` | 38,013 | Training split (~90%) |
+| `valid.jsonl` | 2,051 | Validation split (~5%, hyperparameter tuning) |
+| `test.jsonl` | 2,048 | Test split (~5%, final evaluation) |
+| `ready_sample_10.jsonl` | 10 | Random sample for inspection |
+| `lemmas_used.jsonl` | 82 | Lemmas used in the dataset (id, name, description, counts) |
+
+Splits are **grouped by `recipe_id`** — no solution structure appears in more than one split. Approximate stratification by domain and olympiad level. See `splitting.md` for the full split methodology.
+
+## Data Fields
+
+| 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). Measures insight rarity, not solve difficulty. See below. |
+| `irt_difficulty` | `object\|null` | IRT-1PL difficulty estimate: `{lo, mid, hi}`. See below. |
+| `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` | Total number of unique lemmas used. |
+| `num_base_lemmas` | `int` | Number of base lemmas (excluding spawned ones). |
+| `num_spawns` | `int` | Number of spawned (derived) lemmas. |
+| `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). Measures graph construction, enrichment, and symbolic evaluation. |
+| `verification` | `object` | `{verified, answer, timestamp}`. See below. |
+| `lemma_applicability` | `object` | Lemma selection labels for the first solving step. See below. |
+| `llm_solvers` | `list[object]` | LLM solver results (when available). See below. |
+| `license` | `str` | `"CC BY 4.0"` |
+
+### `verification`
+
+| Key | Type | Description |
+|-----|------|-------------|
+| `verified` | `bool` | `true` if answer matches independent graph evaluation. |
+| `answer` | `int` | Expected answer from the generator. |
+| `error` | `str` | Error message (only when `verified` is `false`). |
+| `timestamp` | `str` | ISO 8601 timestamp of verification. |
+
+### `lemma_applicability`
+
+Maps lemma IDs to labels indicating whether each lemma is the correct first solving step. Present on 41,020 of 42,112 problems (absent when `lemma_paths` is empty).
+
+```json
+"lemma_applicability": {
+  "V1": "ok",
+  "K3": "ok_later",
+  "K5": "same_pattern_wrong",
+  "V7": "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!)`). |
+| `near_miss` | Wrong choice whose graph pattern is structurally similar to an `ok` lemma (same prefix group). |
+| `no` | Wrong — a domain-compatible lemma that does not match the problem structure. Sampled (up to 5). |
+
+### `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.
+
+| 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.
+
+Distribution in the dataset:
+
+| Level | Count | % | |
+|:-----:|------:|-----:|---|
+| 2 | 1,931 | 4.6% | `██` |
+| 3 | 5,295 | 12.6% | `██████` |
+| 4 | 8,433 | 20.0% | `██████████` |
+| 5 | 8,874 | 21.1% | `██████████▌` |
+| 6 | 10,720 | 25.5% | `████████████▌` |
+| 7 | 5,982 | 14.2% | `███████` |
+| 8 | 857 | 2.0% | `█` |
+| 9 | 20 | 0.0% | `▏` |
+
+### `irt_difficulty`
+
+Calibrated difficulty estimate based on Item Response Theory (IRT-1PL / Rasch model). Present for 5,649 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 the 685 "core" tasks attempted by ≥3 models; task difficulties (β) are then estimated per-task with fixed θ.
+
+```json
+"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).
+
+### `llm_solvers`
+
+Each entry:
+
+| Key | Type | Description |
+|-----|------|-------------|
+| `id` | `int` | Solver ID (1-6). |
+| `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. |
+
+## 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 | 18,956 | 45.0% | `██████████████████████` |
+| 10,000–19,999 | 4,518 | 10.7% | `█████` |
+| 20,000–29,999 | 3,726 | 8.8% | `████` |
+| 30,000–39,999 | 3,300 | 7.8% | `███▌` |
+| 40,000–49,999 | 3,193 | 7.6% | `███▌` |
+| 50,000–59,999 | 2,782 | 6.6% | `███` |
+| 60,000–69,999 | 2,234 | 5.3% | `██▌` |
+| 70,000–79,999 | 1,587 | 3.8% | `█▌` |
+| 80,000–89,999 | 1,212 | 2.9% | `█` |
+| 90,000–99,999 | 604 | 1.4% | `▌` |
+
+The median answer is 14,652 and the mean is 24,770, indicating broad coverage of the target range. Very small values such as 0, 1, and 2 do not dominate the distribution and occur with comparable, low frequencies, suggesting that no special or degenerate cases collapse into these values.
+
+## Novelty
+
+Problems are constructed from deterministic symbolic structures without using existing problem texts or large language models; natural-language statements are generated only after the mathematical content is fixed, which 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.** Problems that appear structurally simple yet are consistently unsolved by LLMs are likely to contain errors in the problem statement. Such problems are flagged and excluded if attempted by 2 or more LLM solvers and: (a) no solver achieved a strict match, combined with (b) trivially simple structure (at most 1 base lemma, at most 1 spawn, no nested paths) or (c) `olympiad_level` below 3.
+* **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.
+
+- **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:**
+  Logged solver performance allows treating training as a trajectory through a space of reference models (`llm_solvers`). By selecting tasks that are relatively easy or difficult for specific solvers, the learning process can be guided toward desired regions of this space.
+
+- **Generalization evaluation:**
+  The train/test split is structure-aware, ensuring that test performance reflects generalization to new combinations of lemmas rather than memorization of fixed solution templates.
+
+## License
+
+This dataset is released under the **Creative Commons Attribution 4.0 International** license ([CC BY 4.0](https://creativecommons.org/licenses/by/4.0/)).
+
+Each record contains `"license": "CC BY 4.0"`.

lemmas_used.jsonl

@@ -0,0 +1,82 @@
+{"id": "LIN_FORM", "name": "linear_form_range_counting", "type": "solver_lemma", "domains": ["algebra"], "level": "olympiad", "track": "olympiad_nt", "description": "count of linear form values.", "dataset_count": 7646, "dataset_fraction": 0.1816, "dataset_as_root": 6288}
+{"id": "B1", "name": "equal_variables_extremum", "type": "solver_lemma", "domains": ["algebra"], "level": "standard", "track": "olympiad_algebra", "description": "MinOverSet/MaxOverSet of symmetric expression with symmetric constraint.", "dataset_count": 2268, "dataset_fraction": 0.0539, "dataset_as_root": 1345}
+{"id": "B3", "name": "fixed_product_extremum", "type": "solver_lemma", "domains": ["algebra"], "level": "standard", "track": "olympiad_algebra", "description": "MinOverSet of sum with fixed product constraint.", "dataset_count": 6206, "dataset_fraction": 0.1474, "dataset_as_root": 3813}
+{"id": "VIETA_SUM", "name": "vieta_sum_of_roots", "type": "solver_lemma", "domains": ["algebra"], "level": "standard", "track": "olympiad_algebra", "description": "SumOverSet(SolutionsSet(x, Eq(polynomial, 0))) pattern.", "dataset_count": 915, "dataset_fraction": 0.0217, "dataset_as_root": 614}
+{"id": "K2", "name": "sum_totient_floor_to_triangular", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "olympiad_nt", "description": "Summation(Mul(EulerPhi(Var(k)), Floor(Div(n, Var(k)))), k, 1, n)", "dataset_count": 1883, "dataset_fraction": 0.0447, "dataset_as_root": 798}
+{"id": "K3", "name": "divisor_sum_totient_to_n", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "olympiad_nt", "description": "SumOverDivisors(n, var, EulerPhi(Var(var)))", "dataset_count": 1592, "dataset_fraction": 0.0378, "dataset_as_root": 1079}
+{"id": "K13", "name": "valuation_product", "type": "solver_lemma", "domains": ["number_theory"], "level": "standard", "track": "olympiad_nt", "description": "MaxKDivides(Mul(a, b, ...), p)", "dataset_count": 385, "dataset_fraction": 0.0091, "dataset_as_root": 494}
+{"id": "K14", "name": "valuation_power", "type": "solver_lemma", "domains": ["number_theory"], "level": "standard", "track": "olympiad_nt", "description": "MaxKDivides(Pow(a, k), p)", "dataset_count": 579, "dataset_fraction": 0.0137, "dataset_as_root": 354}
+{"id": "L3b", "name": "count_popcount_parity", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "aimo_nt", "description": "CountOverSet(SolutionsSet(var, And(bounds, Eq(Mod(DigitSum(var, 2), 2), k))))", "dataset_count": 763, "dataset_fraction": 0.0181, "dataset_as_root": 480}
+{"id": "L3c", "name": "count_div2_congruence", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "aimo_nt", "description": "CountOverSet(SolutionsSet(var, And(bounds, Congruent(var, Floor(Div(var, 2)), m))))", "dataset_count": 783, "dataset_fraction": 0.0186, "dataset_as_root": 474}
+{"id": "V1", "name": "valuation_factorial_digit_sum", "type": "solver_lemma", "domains": ["number_theory"], "level": "advanced", "track": "aimo_nt", "description": "MaxKDivides(Factorial(n), p) where n is symbolic.", "dataset_count": 1536, "dataset_fraction": 0.0365, "dataset_as_root": 926}
+{"id": "V5", "name": "min_valuation_factorial", "type": "solver_lemma", "domains": ["number_theory"], "level": "advanced", "track": "aimo_nt", "description": "MinOverSet({n : v_p(n!) >= k})", "dataset_count": 457, "dataset_fraction": 0.0109, "dataset_as_root": 295}
+{"id": "V7", "name": "valuation_binomial_kummer", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "olympiad_nt", "description": "MaxKDivides(Binom(n, k), p)", "dataset_count": 1051, "dataset_fraction": 0.025, "dataset_as_root": 625}
+{"id": "V8", "name": "count_odd_binomials", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "olympiad_nt", "description": "CountOverSet({k : 0 ≤ k ≤ n ∧ C(n,k) odd})", "dataset_count": 1708, "dataset_fraction": 0.0406, "dataset_as_root": 774}
+{"id": "MAX_VAL", "name": "max_valuation", "type": "solver_lemma", "domains": ["number_theory"], "level": "standard", "track": "olympiad_nt", "description": "pattern for maximum valuation query.", "dataset_count": 232, "dataset_fraction": 0.0055, "dataset_as_root": 143}
+{"id": "LTE_DIFF", "name": "lte_difference", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "olympiad_nt", "description": "MaxKDivides(Sub(Pow(a, n), Pow(b, n)), p)", "dataset_count": 544, "dataset_fraction": 0.0129, "dataset_as_root": 547}
+{"id": "LTE_DIFF_P2", "name": "lte_difference_p2", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "olympiad_nt", "description": "MaxKDivides(Sub(Pow(a, n), Pow(b, n)), 2)", "dataset_count": 126, "dataset_fraction": 0.003, "dataset_as_root": 74}
+{"id": "LTE_SUM", "name": "lte_sum", "type": "solver_lemma", "domains": ["number_theory"], "level": "olympiad", "track": "olympiad_nt", "description": "MaxKDivides(Sum(Pow(a, n), Pow(b, n)), p) where n is odd.", "dataset_count": 84, "dataset_fraction": 0.002, "dataset_as_root": 88}
+{"id": "MAX_PRIME_BELOW", "name": "max_prime_below", "type": "solver_lemma", "domains": ["number_theory"], "level": "standard", "track": "olympiad_nt", "description": "MaxOverSet(SolutionsSet(n, And(n >= 2, n <= upper, IsPrime(n)))).", "dataset_count": 4051, "dataset_fraction": 0.0962, "dataset_as_root": 2717}
+{"id": "COUNT_PRIMES", "name": "count_primes", "type": "solver_lemma", "domains": ["number_theory"], "level": "standard", "track": "olympiad_nt", "description": "CountOverSet(SolutionsSet(n, And(n >= 2, n <= upper, IsPrime(n)))).", "dataset_count": 1218, "dataset_fraction": 0.0289, "dataset_as_root": 569}
+{"id": "SUM_PRIMES", "name": "sum_primes", "type": "solver_lemma", "domains": ["number_theory"], "level": "standard", "track": "olympiad_nt", "description": "SumOverSet(SolutionsSet(n, And(n >= lower, n <= upper, IsPrime(n)))).", "dataset_count": 502, "dataset_fraction": 0.0119, "dataset_as_root": 117}
+{"id": "SUM_DIVISIBLE", "name": "sum_divisible", "type": "solver_lemma", "domains": ["number_theory"], "level": "standard", "track": "olympiad_nt", "description": "SumOverSet(SolutionsSet(n, And(n >= 1, n <= N, n mod d = 0))).", "dataset_count": 524, "dataset_fraction": 0.0124, "dataset_as_root": 346}
+{"id": "COUNT_FIB_DIVISIBLE", "name": "count_fib_divisible", "type": "solver_lemma", "domains": ["number_theory"], "level": "standard", "track": "olympiad_nt", "description": "CountOverSet(SolutionsSet(n, And(n >= 1, n <= N, Divides(d, Fib(n))))).", "dataset_count": 789, "dataset_fraction": 0.0187, "dataset_as_root": 582}
+{"id": "COUNT_CARTESIAN", "name": "cartesian_product_cardinality", "type": "solver_lemma", "domains": ["combinatorics"], "level": "elementary", "track": "olympiad_combinatorics", "description": "CountOverSet of CartesianProduct.", "dataset_count": 1113, "dataset_fraction": 0.0264, "dataset_as_root": 879}
+{"id": "COUNT_COPRIME_GRID", "name": "count_coprime_pairs_grid", "type": "solver_lemma", "domains": ["combinatorics"], "level": "standard", "track": "olympiad_combinatorics", "description": "CountOverSet({(i,j) ∈ [1,a]×[1,b] : gcd(i,j)=1}).", "dataset_count": 999, "dataset_fraction": 0.0237, "dataset_as_root": 709}
+{"id": "COUNT_SUM_EQUALS", "name": "count_pairs_sum_equals_target", "type": "solver_lemma", "domains": ["combinatorics"], "level": "standard", "track": "olympiad_combinatorics", "description": "CountOverSet({(i,j) ∈ [1,a]×[1,b] : i+j=t})", "dataset_count": 1472, "dataset_fraction": 0.035, "dataset_as_root": 772}
+{"id": "SUM_INDEPENDENT", "name": "sum_over_cartesian_independent_var", "type": "solver_lemma", "domains": ["combinatorics"], "level": "standard", "track": "olympiad_combinatorics", "description": "SumOverSet(MapOverSet(SolutionsSet((i,j), domain=A×B), f(i)))", "dataset_count": 225, "dataset_fraction": 0.0053, "dataset_as_root": 152}
+{"id": "SUM_FACTOR_CARTESIAN", "name": "sum_over_cartesian_factor_product", "type": "solver_lemma", "domains": ["combinatorics"], "level": "standard", "track": "olympiad_combinatorics", "description": "SumOverSet(MapOverSet(SolutionsSet((i,j), domain=A×B), Mul(f(i), g(j))))", "dataset_count": 465, "dataset_fraction": 0.011, "dataset_as_root": 302}
+{"id": "C2", "name": "count_multiples_in_range", "type": "solver_lemma", "domains": ["combinatorics"], "level": "elementary", "track": "olympiad_combinatorics", "description": "CountOverSet(SolutionsSet(var, And(bounds..., Divides...)))", "dataset_count": 795, "dataset_fraction": 0.0189, "dataset_as_root": 438}
+{"id": "C3", "name": "count_powers_in_range", "type": "solver_lemma", "domains": ["combinatorics"], "level": "standard", "track": "olympiad_nt", "description": "CountOverSet(SolutionsSet(var, And(lower ≤ Pow(var,e) ≤ upper)))", "dataset_count": 557, "dataset_fraction": 0.0132, "dataset_as_root": 355}
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+{"id": "LEMMA_MOBIUS_SQUAREFREE", "name": "mobius_squarefree", "type": "spawner", "value": 1, "domains": ["NT"], "complexity": 3, "description": "μ(n)² = 1 when n is squarefree", "dataset_count": 436, "dataset_fraction": 0.0104, "dataset_as_root": 128}
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ready_sample_10.jsonl

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+{"id": "93e381", "alias": "nt_count_primes_v1_1125832087_1331", "problem": "Let $n$ be a positive integer. Define $A$ as the set of all positive integers $p$ for which there exists a positive integer $q$ such that $p \\cdot q = 6$, $\\gcd(p, q) = 1$, and $p < q$. Let $k$ be the number of elements in $A$. Determine the number of prime numbers $n$ such that $n \\geq k$ and $n \\leq 27720$. Denote this count by $c$. Compute $$\\sum_{n=1}^{c} \\tau(n),$$ where $\\tau(n)$ denotes the number of positive divisors of $n$.", "answer": 24678, "graph": "graphs = [\n    Graph(\n        let={\n            \"upper\": Const(27720),\n            \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), CountOverSet(set=SolutionsSet(var=Var(\"p\"), condition=And(IsPositive(arg=Var(name='p')), Exists(var=Var(name='q'), condition=And(Eq(left=Mul(Var(name='p'), Var(name='q')), right=Const(value=6)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)), Lt(left=Var(name='p'), right=Var(name='q')))))))), Leq(Var(\"n\"), Ref(\"upper\")), IsPrime(Var(\"n\"))))),\n            \"Q\": Summation(var=\"n\", start=Const(1), end=Abs(arg=Ref(name='result')), expr=NumDivisors(n=Var(\"n\"))),\n        },\n        goal=Ref(\"Q\"),\n    )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "COPRIME_PAIRS", "lemma_paths": ["COPRIME_PAIRS"], "recipe_id": "2bb3aa", "seed_template_id": "nt_count_primes_v1", "ending_id": "", "olympiad_level": 5, "num_spawns": 0, "lemma_set": ["COPRIME_PAIRS"], "num_lemmas": 1, "num_base_lemmas": 1, "generation_time": 1.037, "created_at": "2026-02-08T03:41:25.299630Z", "verification": {"verified": true, "answer": 24678, "timestamp": "2026-02-08T03:41:26.336838Z"}, "problem_hash": "50dd45", "license": "CC BY 4.0", "lemma_applicability": {"COPRIME_PAIRS": "ok", "DS2": "no", "K15": "no", "K18": "no", "V8_SUM": "no", "VAL_SUM_EQ": "no"}}
+{"id": "186caa", "alias": "nt_count_gcd_equals_v1_458359167_2303", "problem": "Let $k$ be the minimum value of $x + y$ over all pairs of positive integers $(x, y)$ such that $xy = 33124$. Let $d = 7$ and $N$ be the number of positive integers $n$ such that $1 \\leq n \\leq 32400$ and $\\gcd(n, k) = d$. Compute the remainder when $32114 \\cdot N$ is divided by $99273$.", "answer": 97134, "graph": "graphs = [\n    Graph(\n        let={\n            \"upper\": Const(32400),\n            \"k\": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var(\"x\"), Var(\"y\")), Const(33124)))), expr=Sum(Var(\"x\"), Var(\"y\")))),\n            \"d\": Const(7),\n            \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(GCD(a=Var(\"n\"), b=Ref(\"k\")), Ref(\"d\"))))),\n            \"Q\": Mod(value=Mul(Const(32114), Ref(\"result\")), modulus=Const(99273)),\n        },\n        goal=Ref(\"Q\"),\n    )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "B3", "lemma_paths": ["B3"], "recipe_id": "0cd20d", "seed_template_id": "nt_count_gcd_equals_v1", "ending_id": "", "olympiad_level": 4, "num_spawns": 0, "lemma_set": ["B3"], "num_lemmas": 1, "num_base_lemmas": 1, "generation_time": 2.516, "created_at": "2026-02-08T05:18:13.347217Z", "verification": {"verified": true, "answer": 97134, "timestamp": "2026-02-08T05:18:15.862895Z"}, "problem_hash": "b319d1", "license": "CC BY 4.0", "lemma_applicability": {"B3": "ok", "COUNT_PRIMES": "no", "K18": "no", "LTE_DIFF_P2": "no", "LTE_SUM": "no", "V8_SUM": "no"}}
+{"id": "6f331e", "alias": "antilemma_product_of_sums_v1_677425708_480", "problem": "Let\n$$S_1 = \\sum_{k=1}^{13} k.$$Let $S_2$ be the sum of all products $ij$ taken over all ordered pairs $(i,j)$ of integers with $1\\le i\\le 3$ and $1\\le j\\le 9$.\nLet $x = S_1 S_2$.\n\nLet $p$ be the largest prime not exceeding $193$. Let $T$ be the number of integers $n$ such that $2\\le n\\le p$ and $n$ is prime.\n\nDefine $Q$ to be the remainder when\n$$x^2 + Tx + x$$\nis divided by $64992$.\n\nFind the value of $Q$.", "answer": 15425, "graph": "graphs = [\n    Graph(\n        let={\n            \"_m\": Const(2),\n            \"_n\": Const(2),\n            \"S1\": Summation(var=\"k\", start=Const(1), end=Const(13), expr=Var(\"k\")),\n            \"S2\": SumOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"i\"), Var(\"j\")]), condition=Const(1), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(3)), right=IntegerRange(start=Const(1), end=Const(9)))), expr=Mul(Var(\"i\"), Var(\"j\")))),\n            \"x\": Mul(Ref(\"S1\"), Ref(\"S2\")),\n            \"Q\": Mod(value=Sum(Pow(Ref(\"x\"), Const(2)), Mul(CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Ref(\"_n\")), Leq(Var(\"n\"), MaxOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Ref(\"_m\")), Leq(Var(\"n\"), Const(193)), IsPrime(Var(\"n\")))))), IsPrime(Var(\"n\"))))), Ref(\"x\")), Const(5)), modulus=Const(64992)),\n        },\n        goal=Ref(\"Q\"),\n    )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "MAX_PRIME_BELOW", "lemma_paths": ["MAX_PRIME_BELOW/COUNT_PRIMES/ENDING_quadratic_mod", "PRODUCT_OF_SUMS"], "recipe_id": "bcbd39", "seed_template_id": "antilemma_product_of_sums_v1", "ending_id": "", "olympiad_level": 5, "num_spawns": 0, "lemma_set": ["COUNT_PRIMES", "ENDING_quadratic_mod", "MAX_PRIME_BELOW", "PRODUCT_OF_SUMS"], "num_lemmas": 4, "num_base_lemmas": 4, "generation_time": 0.003, "created_at": "2026-02-08T03:33:41.277827Z", "verification": {"verified": true, "answer": 15425, "timestamp": "2026-02-08T03:33:41.280608Z"}, "problem_hash": "38e95a", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 39990, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 271, "completion_tokens": 3215}, "timestamp": "2026-02-08T20:39:00.898Z"}], "lemma_applicability": {"MAX_PRIME_BELOW": "ok", "PRODUCT_OF_SUMS": "ok", "COUNT_PRIMES": "ok_later", "ENDING_quadratic_mod": "ok_later", "K14": "no", "LTE_SUM": "no", "POLY_PADIC_VAL_CONST": "no", "V5": "no", "V7": "no"}, "irt_difficulty": {"lo": 4.08, "mid": 7.04, "hi": 10.0}}
+{"id": "b34118", "alias": "nt_count_divisible_and_v1_1742523217_4496", "problem": "Let $n$ be a positive integer such that $n \\leq 104700$, $n$ is divisible by 6, and $n$ is divisible by 10. Let $r$ be the number of such integers $n$. Let $s$ be the number of ordered pairs $(i,j)$ with $1 \\leq i \\leq 55$, $1 \\leq j \\leq 93$, and $\\gcd(i,j) = 1$. Compute the remainder when $r^2 + 16r + s$ is divided by 88548.", "answer": 19476, "graph": "graphs = [\n    Graph(\n        let={\n            \"_n\": Const(16),\n            \"upper\": Const(104700),\n            \"d1\": Const(6),\n            \"d2\": Const(10),\n            \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"d1\")), Const(0)), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"d2\")), Const(0))))),\n            \"Q\": Mod(value=Sum(Pow(Ref(\"result\"), Const(2)), Mul(Ref(\"_n\"), Ref(\"result\")), CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"i\"), Var(\"j\")]), condition=Eq(GCD(a=Var(\"i\"), b=Var(\"j\")), Const(1)), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(55)), right=IntegerRange(start=Const(1), end=Const(93)))))), modulus=Const(88548)),\n        },\n        goal=Ref(\"Q\"),\n    )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "COUNT_COPRIME_GRID", "lemma_paths": ["COUNT_COPRIME_GRID/ENDING_quadratic_mod"], "recipe_id": "b00957", "seed_template_id": "nt_count_divisible_and_v1", "ending_id": "", "olympiad_level": 5, "num_spawns": 0, "lemma_set": ["COUNT_COPRIME_GRID", "ENDING_quadratic_mod"], "num_lemmas": 2, "num_base_lemmas": 2, "generation_time": 4.863, "created_at": "2026-02-08T08:52:31.571743Z", "verification": {"verified": true, "answer": 19476, "timestamp": "2026-02-08T08:52:36.434334Z"}, "problem_hash": "a233ea", "license": "CC BY 4.0", "lemma_applicability": {"COUNT_COPRIME_GRID": "ok", "ENDING_quadratic_mod": "ok_later", "K17": "no", "MAX_PRIME_BELOW": "no", "MAX_VAL": "no", "MOD_FACTORIAL": "no", "MOD_MUL": "no"}}
+{"id": "6c17e9", "alias": "comb_count_derangements_v1_1520064083_1290", "problem": "Let $a_1 = 5$ and $b_1 = 5$. Define $n_2 = a_1 + b_1$ and $$f = \\sum_{k=0}^{n_2} (-1)^k \\binom{n_2}{k}.$$ Let $a = 5$, $b = 1 + f$, and $n_1 = a + b$. Define $$w = \\sum_{k=0}^{n_1} (-1)^k \\binom{n_1}{k},$$ and let $n = 8 + w$. Compute the value of $!n$, the number of derangements of $n$ elements.", "answer": 14833, "graph": "graphs = [\n    Graph(\n        let={\n            \"a1\": Const(5),\n            \"b1\": Const(5),\n            \"n2\": Sum(Ref(\"a1\"), Ref(\"b1\")),\n            \"f\": Summation(var=\"k\", start=Const(0), end=Ref(\"n2\"), expr=Mul(Pow(Const(-1), Var(\"k\")), Binom(n=Ref(\"n2\"), k=Var(\"k\")))),\n            \"a\": Const(5),\n            \"b\": Sum(Const(1), Ref(\"f\")),\n            \"n1\": Sum(Ref(\"a\"), Ref(\"b\")),\n            \"w\": Summation(var=\"k\", start=Const(0), end=Ref(\"n1\"), expr=Mul(Pow(Const(-1), Var(\"k\")), Binom(n=Ref(\"n1\"), k=Var(\"k\")))),\n            \"n\": Sum(Const(8), Ref(\"w\")),\n            \"result\": Subfactorial(arg=Ref(name='n')),\n        },\n        goal=Ref(\"result\"),\n    )\n]", "domain": "COMB", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "BINOMIAL_ALTERNATING", "lemma_paths": ["BINOMIAL_ALTERNATING"], "recipe_id": "c21569", "seed_template_id": "comb_count_derangements_v1", "ending_id": "", "olympiad_level": 4, "num_spawns": 2, "lemma_set": ["BINOMIAL_ALTERNATING"], "num_lemmas": 1, "num_base_lemmas": -1, "generation_time": 0.002, "created_at": "2026-02-08T03:54:41.788213Z", "verification": {"verified": true, "answer": 14833, "timestamp": "2026-02-08T03:54:41.789728Z"}, "problem_hash": "b4ddb2", "license": "CC BY 4.0", "lemma_applicability": {"BINOMIAL_ALTERNATING": "ok", "C2": "no", "COUNT_CARTESIAN": "no", "COUNT_INTEGER_RANGE": "no", "V8": "no", "V8_SUM": "no"}}
+{"id": "02cdb8", "alias": "modular_count_residue_v1_1742523217_1091", "problem": "Let $m$ be the minimum value of $x + y$ over all ordered pairs of positive integers $(x, y)$ such that $xy = 121$. Let $r$ be the minimum value of $x + y$ over all ordered pairs of positive integers $(x, y)$ such that $xy = 9$. Let $U = 53824$. Compute the number of positive integers $n$ such that $1 \\leq n \\leq U$ and $n \\equiv r \\pmod{m}$.", "answer": 2447, "graph": "graphs = [\n    Graph(\n        let={\n            \"_n\": Const(121),\n            \"upper\": Const(53824),\n            \"m\": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var(\"x\"), Var(\"y\")), Ref(\"_n\")))), expr=Sum(Var(\"x\"), Var(\"y\")))),\n            \"r\": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var(\"x\"), Var(\"y\")), Const(9)))), expr=Sum(Var(\"x\"), Var(\"y\")))),\n            \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"m\")), Ref(\"r\"))))),\n        },\n        goal=Ref(\"result\"),\n    )\n]", "domain": "ALG", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "B3", "lemma_paths": ["B3"], "recipe_id": "0cd20d", "seed_template_id": "modular_count_residue_v1", "ending_id": "", "olympiad_level": 4, "num_spawns": 0, "lemma_set": ["B3"], "num_lemmas": 1, "num_base_lemmas": 1, "generation_time": 5.277, "created_at": "2026-02-08T03:24:57.409896Z", "verification": {"verified": true, "answer": 2447, "timestamp": "2026-02-08T03:25:02.686536Z"}, "problem_hash": "67fc9c", "license": "CC BY 4.0", "llm_solvers": [{"id": 2, "model": "openai/gpt-oss-120b", "answer": 2447, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 226, "completion_tokens": 744}, "timestamp": "2026-02-09T10:53:54.912Z"}], "lemma_applicability": {"B3": "ok", "POLY_PADIC_VAL_CONST": "no"}, "irt_difficulty": {"lo": -10.0, "mid": 0.0, "hi": 10.0}}
+{"id": "aed7e7", "alias": "diophantine_product_count_v1_1742523217_3988", "problem": "Let $k = 480$ and let $u = 172$. Consider the set of all positive integers $x$ such that $1 \\leq x \\leq u$, $x$ divides $k$, and $\\frac{k}{x} \\leq u$. Compute the number of elements in this set.", "answer": 20, "graph": "graphs = [\n    Graph(\n        let={\n            \"k\": Const(480),\n            \"upper\": Const(172),\n            \"result\": CountOverSet(set=SolutionsSet(var=Var(\"x\"), condition=And(Geq(Var(\"x\"), Const(1)), Leq(Var(\"x\"), Ref(\"upper\")), Divides(divisor=Var(\"x\"), dividend=Ref(\"k\")), Leq(Div(Ref(\"k\"), Var(\"x\")), Ref(\"upper\"))))),\n        },\n        goal=Ref(\"result\"),\n    )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "B3", "lemma_paths": ["LIN_FORM", "ONE_PHI_2"], "recipe_id": "9858be", "seed_template_id": "diophantine_product_count_v1", "ending_id": "", "olympiad_level": 4, "num_spawns": 0, "lemma_set": ["LIN_FORM", "ONE_PHI_2"], "num_lemmas": 2, "num_base_lemmas": 2, "generation_time": 0.194, "created_at": "2026-02-08T06:10:19.715251Z", "verification": {"verified": true, "answer": 20, "timestamp": "2026-02-08T06:10:19.909096Z"}, "problem_hash": "eee8b4", "license": "CC BY 4.0", "lemma_applicability": {"LIN_FORM": "ok", "ONE_PHI_2": "ok", "DS2": "no", "K18": "no", "MOD_FACTORIAL": "no", "MOD_MUL": "no", "V1": "no"}}
+{"id": "fa7ee7", "alias": "nt_num_divisors_compute_v1_1520064083_4987", "problem": "Let $T$ be the set of all integers $t$ such that $24 \\leq t \\leq 10488$ and $t = 9a + 15b$ for some positive integers $a \\leq 517$ and $b \\leq 389$. Let $n = |T|$, and let $d(n)$ denote the number of positive divisors of $n$. Compute $20191 \\cdot d(n)$.", "answer": 60573, "graph": "graphs = [\n    Graph(\n        let={\n            \"n\": CountOverSet(set=SolutionsSet(var=Var(\"t\"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=517)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=389)), Geq(left=Var(name='t'), right=Const(value=24)), Leq(left=Var(name='t'), right=Const(value=10488)), Eq(left=Var(name='t'), right=Sum(Mul(Const(value=9), Var(name='a')), Mul(Const(value=15), Var(name='b'))))))))),\n            \"result\": NumDivisors(n=Ref(\"n\")),\n            \"_c\": Const(20191),\n            \"Q\": Mul(Ref(\"_c\"), Ref(\"result\")),\n        },\n        goal=Ref(\"Q\"),\n    )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "LIN_FORM", "lemma_paths": ["LIN_FORM"], "recipe_id": "7b2633", "seed_template_id": "nt_num_divisors_compute_v1", "ending_id": "", "olympiad_level": 5, "num_spawns": 0, "lemma_set": ["LIN_FORM"], "num_lemmas": 1, "num_base_lemmas": 1, "generation_time": 0.002, "created_at": "2026-02-08T06:32:34.319099Z", "verification": {"verified": true, "answer": 60573, "timestamp": "2026-02-08T06:32:34.320657Z"}, "problem_hash": "345ec3", "license": "CC BY 4.0", "lemma_applicability": {"LIN_FORM": "ok", "C2": "no", "COUNT_PRIMES": "no", "DS2": "no", "L3c": "no", "V3": "no"}}
+{"id": "b29015", "alias": "antilemma_v1_legendre_784195855_244", "problem": "Let $S$ be the set of all positive integers $p$ such that there exists a positive integer $q$ with $p < q$, $pq = 54$, and $\\gcd(p, q) = 1$. Let $m$ be the number of elements in $S$. Let $T$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 1747684$. For each such pair, compute $x + y$, and let $s$ be the smallest such sum. Let $x$ be the largest integer $k$ such that $m^k$ divides $s!$. Compute $x$.", "answer": 2639, "graph": "graphs = [\n    Graph(\n        let={\n            \"_n\": CountOverSet(set=SolutionsSet(var=Var(\"p\"), condition=And(IsPositive(arg=Var(name='p')), Exists(var=Var(name='q'), condition=And(Eq(left=Mul(Var(name='p'), Var(name='q')), right=Const(value=54)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)), Lt(left=Var(name='p'), right=Var(name='q'))))))),\n            \"x\": MaxKDivides(target=Factorial(MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var(\"x\"), Var(\"y\")), Const(1747684)))), expr=Sum(Var(\"x\"), Var(\"y\"))))), base=Ref(\"_n\")),\n        },\n        goal=Ref(\"x\"),\n    )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "COPRIME_PAIRS", "lemma_paths": ["COPRIME_PAIRS/B3/V1", "V1"], "recipe_id": "358b4f", "seed_template_id": "antilemma_v1_legendre", "ending_id": "", "olympiad_level": 7, "num_spawns": 0, "lemma_set": ["B3", "COPRIME_PAIRS", "V1"], "num_lemmas": 3, "num_base_lemmas": 3, "generation_time": 0.002, "created_at": "2026-02-08T03:03:23.782264Z", "verification": {"verified": true, "answer": 2639, "timestamp": "2026-02-08T03:03:23.783877Z"}, "problem_hash": "7a9161", "license": "CC BY 4.0", "lemma_applicability": {"COPRIME_PAIRS": "ok", "V1": "ok", "B3": "ok_later", "K5": "same_pattern_wrong", "COUNT_FIB_DIVISIBLE": "no", "DS2": "no", "LTE_SUM": "no", "MAX_PRIME_BELOW": "no", "V8": "no"}}
+{"id": "7e7901", "alias": "nt_count_with_divisor_count_v1_1918700295_3438", "problem": "Let $S$ be the set of all ordered pairs $(x,y)$ of positive integers such that $xy = 36$. Define $d$ to be the minimum value of $x + y$ over all pairs $(x,y) \\in S$. Now, let $N$ be the number of positive integers $n$ such that $1 \\le n \\le 90000$ and the number of positive divisors of $n$ is exactly $d$. Find the value of $N$.", "answer": 8978, "graph": "graphs = [\n    Graph(\n        let={\n            \"upper\": Const(90000),\n            \"div_count\": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var(\"x\"), Var(\"y\")), Const(36)))), expr=Sum(Var(\"x\"), Var(\"y\")))),\n            \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(NumDivisors(n=Var(\"n\")), Ref(\"div_count\"))))),\n        },\n        goal=Ref(\"result\"),\n    )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "B3", "lemma_paths": ["B3"], "recipe_id": "0cd20d", "seed_template_id": "nt_count_with_divisor_count_v1", "ending_id": "", "olympiad_level": 5, "num_spawns": 0, "lemma_set": ["B3"], "num_lemmas": 1, "num_base_lemmas": 1, "generation_time": 4.257, "created_at": "2026-02-08T08:37:50.215786Z", "verification": {"verified": true, "answer": 8978, "timestamp": "2026-02-08T08:37:54.472973Z"}, "problem_hash": "6e01a7", "license": "CC BY 4.0", "lemma_applicability": {"B3": "ok", "K15": "no", "L3c": "no", "V1": "no", "V3": "no", "V8_SUM": "no"}}

train.jsonl

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