Publish AIME++ sample v0.1.0
Browse files- CHANGELOG.md +18 -0
- CHECKSUMS.sha256 +4 -0
- CITATION.cff +20 -0
- LICENSE +21 -0
- Makefile +10 -0
- QUALITY_REPORT.md +51 -0
- README.md +228 -0
- VERSION +1 -0
- data/aime-graduate.jsonl +20 -0
- data/aime-hard.jsonl +98 -0
- data/aime-researcher.jsonl +5 -0
- data/aime.jsonl +34 -0
- docs/COMMERCIAL_ACCESS.md +54 -0
- schema/dataset.schema.json +37 -0
- scripts/score.py +156 -0
- scripts/validate_dataset.py +199 -0
CHANGELOG.md
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# Changelog
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All notable changes to the AIME++ sample package are documented here.
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## [Unreleased]
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- Documented that all problems are internally created and rights-held by Ulam AI.
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- Reframed the release as an MIT-licensed sample of Ulam AI's 100,000+ problem database.
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- Added commercial inventory for the 24,700+ AIME family, 1,000+ AIME-Graduate, and 100+ AIME-Researcher collections.
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- Clarified that each supplied integer answer is the authoritative golden solution for its problem.
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## [0.1.0] - 2026-08-26
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- Normalized 157 source records into Hugging Face-compatible JSONL.
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- Added four named difficulty configs and a combined default config.
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- Added stable IDs, zero-padded answer strings, and tier labels.
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- Added a versioned JSON Schema, deterministic validator, and exact-match scorer.
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- Added dataset-card metadata, quality disclosure, citation metadata, and commercial-access documentation.
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CHECKSUMS.sha256
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cb5b268e360b6ee36175b3febcdffed5a4d66b5cd761b28505ab6ab619422e61 data/aime.jsonl
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adbee93720b809430fade33ce3bee42060038adc835a47fc1eda413d980b3db6 data/aime-hard.jsonl
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18e3c877b829823522906e402d87cee4f6cfc659c02435b13901ba192beda65f data/aime-graduate.jsonl
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1a95633529051920c64194dbcad3aa40c07bd8211154c9cd2d4cbf7d09dfa2b2 data/aime-researcher.jsonl
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CITATION.cff
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cff-version: 1.2.0
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message: "If you use this dataset, please cite it using the metadata below."
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title: "AIME++ Sample"
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type: dataset
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authors:
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- name: "Ulam AI"
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version: 0.1.0
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repository-code: "https://huggingface.co/datasets/ulamai/AIME-Plus-Plus"
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url: "https://ulam.ai/"
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license: MIT
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abstract: >-
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A sample of 157 exact-answer mathematical reasoning problems spanning AIME,
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AIME Hard, AIME-Graduate, and AIME-Researcher difficulty tiers. Every answer
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is an integer from 0 through 999 and supports deterministic exact-match
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evaluation.
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keywords:
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- mathematical reasoning
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- benchmark
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- exact-match evaluation
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- verifiable rewards
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LICENSE
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MIT License
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Copyright (c) 2026 Ulam AI
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Permission is hereby granted, free of charge, to any person obtaining a copy
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of this software and associated documentation files (the "Software"), to deal
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in the Software without restriction, including without limitation the rights
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to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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copies of the Software, and to permit persons to whom the Software is
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furnished to do so, subject to the following conditions:
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The above copyright notice and this permission notice shall be included in all
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copies or substantial portions of the Software.
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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SOFTWARE.
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Makefile
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.PHONY: validate quality score-help
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validate:
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python3 scripts/validate_dataset.py
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quality: validate
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python3 scripts/validate_dataset.py --markdown
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score-help:
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python3 scripts/score.py --help
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QUALITY_REPORT.md
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# AIME++ Sample Quality Report
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**Package version:** 0.1.0
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**Audit date:** 2026-08-26
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**Records audited:** 157
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## Dataset scope and provenance
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All sample problems were created internally by Ulam AI, and Ulam AI holds the dataset rights. The release is intentionally answer-only: records contain problem statements and authoritative final integer answers, not worked derivations or reasoning traces.
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For the AIME++ task, the supplied answer is the golden solution. The reference scorer treats any different normalized integer as incorrect and awards no partial credit.
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## Deterministic package checks
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| Check | Result |
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|---|---:|
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| JSONL records parsed | 157 / 157 |
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| Records matching the five-field schema | 157 / 157 |
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| Integer answers in `0–999` | 157 / 157 |
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| Correct zero-padded `answer_str` values | 157 / 157 |
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| Unique stable IDs | 157 / 157 |
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| Exact or whitespace-normalized duplicate problems | 0 |
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| Empty problem statements | 0 |
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| ASCII control characters in problem statements | 0 |
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| Problems with odd dollar-sign delimiter counts | 0 |
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## Composition
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| Tier | Records | Min chars | Mean chars | Max chars |
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|---|---:|---:|---:|---:|
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| AIME | 34 | 57 | 149.7 | 290 |
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| AIME Hard | 98 | 321 | 584.9 | 842 |
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| AIME-Graduate | 20 | 387 | 549.1 | 1,011 |
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| AIME-Researcher | 5 | 577 | 825.4 | 1,258 |
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| **Total** | **157** | | | |
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## Reproduce the audit
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```bash
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python3 scripts/validate_dataset.py
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```
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The validator uses only the Python standard library and verifies every release data file against [CHECKSUMS.sha256](CHECKSUMS.sha256).
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## Ground-truth policy
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The `answer` field is the canonical numeric ground truth. The `answer_str` field is the same ground truth rendered as exactly three decimal digits. These fields must agree, and the validator enforces that invariant for every record.
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Worked derivations are outside the dataset format and are not required to determine correctness. A model submission is correct exactly when its normalized integer equals the supplied answer.
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This exact-answer contract makes evaluation deterministic, reproducible, and independent of a model judge.
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README.md
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---
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language:
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| 3 |
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- en
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| 4 |
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license: mit
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multilinguality:
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- monolingual
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pretty_name: AIME++ Sample
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size_categories:
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- n<1K
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source_datasets:
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- original
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task_categories:
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- question-answering
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- text-generation
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tags:
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- mathematics
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- mathematical-reasoning
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- reasoning
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- benchmark
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- evaluation
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| 21 |
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- rlvr
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- exact-match
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- latex
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- text
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| 25 |
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configs:
|
| 26 |
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- config_name: all
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| 27 |
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default: true
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| 28 |
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data_files:
|
| 29 |
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- split: test
|
| 30 |
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path:
|
| 31 |
+
- data/aime.jsonl
|
| 32 |
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- data/aime-hard.jsonl
|
| 33 |
+
- data/aime-graduate.jsonl
|
| 34 |
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- data/aime-researcher.jsonl
|
| 35 |
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- config_name: aime
|
| 36 |
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data_files:
|
| 37 |
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- split: test
|
| 38 |
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path: data/aime.jsonl
|
| 39 |
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- config_name: aime-hard
|
| 40 |
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data_files:
|
| 41 |
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- split: test
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| 42 |
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path: data/aime-hard.jsonl
|
| 43 |
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- config_name: aime-graduate
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| 44 |
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data_files:
|
| 45 |
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- split: test
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| 46 |
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path: data/aime-graduate.jsonl
|
| 47 |
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- config_name: aime-researcher
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| 48 |
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data_files:
|
| 49 |
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- split: test
|
| 50 |
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path: data/aime-researcher.jsonl
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| 51 |
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---
|
| 52 |
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|
| 53 |
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# AIME++ Sample
|
| 54 |
+
|
| 55 |
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**AIME++** is Ulam AI's exact-answer mathematical reasoning environment. It keeps one of the most useful properties of AIME-style evaluation—a compact, deterministic answer in the integer range `0–999`—and extends it across four levels of mathematical depth, from competition-style problems to research-level challenges.
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| 56 |
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|
| 57 |
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This repository contains a **157-problem, MIT-licensed sample** of Ulam AI's much larger problem catalog. Every problem has a canonical integer answer and a zero-padded three-digit representation, making evaluation inexpensive, reproducible, and free of judge-model variance.
|
| 58 |
+
|
| 59 |
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Ulam AI's broader database contains **100,000+ problems**. The commercially available AIME++ collections include **24,700+ AIME-family problems** spanning AIME and AIME Hard, **1,000+ AIME-Graduate problems**, and **100+ AIME-Researcher problems**. This repository is designed to let teams inspect the format and difficulty range before licensing a production-scale collection.
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| 60 |
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|
| 61 |
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> **Name clarification:** In this dataset, AIME means **AI Mathematical Environment**. “AIME-style” describes the `0–999` answer format. This project is not affiliated with or endorsed by the Mathematical Association of America or its competitions, and the sample does not claim to contain official competition problems.
|
| 62 |
+
|
| 63 |
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## Why AIME++
|
| 64 |
+
|
| 65 |
+
- **Verifier-friendly:** exact-match rewards are deterministic and require no subjective rubric or model judge.
|
| 66 |
+
- **One interface, four depths:** the answer contract stays fixed while the mathematical demands increase.
|
| 67 |
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- **Evaluation-ready:** stable IDs, explicit tiers, a versioned JSON Schema, and a reference scorer are included.
|
| 68 |
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- **Useful for capability profiling:** results can be compared overall and per tier without conflating output-format changes with problem difficulty.
|
| 69 |
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- **Frictionless sample:** the MIT license allows teams to test the data in their own training and evaluation stacks before discussing a larger license.
|
| 70 |
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| 71 |
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The dataset is intentionally **answer-only**: each record contains a problem and its authoritative golden answer in `0–999`, not a worked derivation or chain-of-thought trace. For the AIME++ task, that final answer is the golden solution. A different normalized integer is incorrect.
|
| 72 |
+
|
| 73 |
+
## Difficulty tiers
|
| 74 |
+
|
| 75 |
+
| Config | Tier | Description | Records |
|
| 76 |
+
|---|---|---|---:|
|
| 77 |
+
| `aime` | **AIME** | Standard AIME-style mathematical problems | 34 |
|
| 78 |
+
| `aime-hard` | **AIME Hard** | Harder AIME-style mathematical problems | 98 |
|
| 79 |
+
| `aime-graduate` | **AIME-Graduate** | Graduate-level problems with AIME-style answers | 20 |
|
| 80 |
+
| `aime-researcher` | **AIME-Researcher** | Research-level problems with AIME-style answers | 5 |
|
| 81 |
+
| `all` | All four tiers | Default combined evaluation config | **157** |
|
| 82 |
+
|
| 83 |
+
All configurations expose a single `test` split. This is an evaluation sample, not a train/test partition.
|
| 84 |
+
|
| 85 |
+
## From sample to full catalog
|
| 86 |
+
|
| 87 |
+
| Collection | This sample | Larger Ulam collection | Scope |
|
| 88 |
+
|---|---:|---:|---|
|
| 89 |
+
| AIME family | 132 | 24,700+ | Standard and harder AIME-style problems |
|
| 90 |
+
| AIME-Graduate | 20 | 1,000+ | Graduate-level mathematics with AIME-style answers |
|
| 91 |
+
| AIME-Researcher | 5 | 100+ | Research-level mathematics with AIME-style answers |
|
| 92 |
+
| Broader Ulam problem database | — | 100,000+ | Mathematical reasoning problems across Ulam collections |
|
| 93 |
+
|
| 94 |
+
The sample exposes only a small fraction of the commercial inventory while providing enough material to test parsing, training, RLVR rewards, evaluation code, and tier-level behavior.
|
| 95 |
+
|
| 96 |
+
## Quick start
|
| 97 |
+
|
| 98 |
+
When the repository is published on Hugging Face, load the complete sample with:
|
| 99 |
+
|
| 100 |
+
```python
|
| 101 |
+
from datasets import load_dataset
|
| 102 |
+
|
| 103 |
+
dataset = load_dataset("ulamai/AIME-Plus-Plus", "all", split="test")
|
| 104 |
+
print(dataset[0])
|
| 105 |
+
```
|
| 106 |
+
|
| 107 |
+
Load a single tier by replacing `all` with `aime`, `aime-hard`, `aime-graduate`, or `aime-researcher`.
|
| 108 |
+
|
| 109 |
+
The release files can also be loaded before publication:
|
| 110 |
+
|
| 111 |
+
```python
|
| 112 |
+
from datasets import load_dataset
|
| 113 |
+
|
| 114 |
+
dataset = load_dataset(
|
| 115 |
+
"json",
|
| 116 |
+
data_files={"test": "data/*.jsonl"},
|
| 117 |
+
split="test",
|
| 118 |
+
)
|
| 119 |
+
```
|
| 120 |
+
|
| 121 |
+
## Data schema
|
| 122 |
+
|
| 123 |
+
Each JSONL row has five fields:
|
| 124 |
+
|
| 125 |
+
| Field | Type | Description |
|
| 126 |
+
|---|---|---|
|
| 127 |
+
| `id` | string | Stable identifier, such as `aimepp-aime-0001` |
|
| 128 |
+
| `problem` | string | English problem statement with LaTeX markup |
|
| 129 |
+
| `answer` | integer | Canonical answer in `0–999` |
|
| 130 |
+
| `answer_str` | string | The same answer zero-padded to exactly three digits |
|
| 131 |
+
| `tier` | string | One of the four human-readable difficulty tiers |
|
| 132 |
+
|
| 133 |
+
Example:
|
| 134 |
+
|
| 135 |
+
```json
|
| 136 |
+
{
|
| 137 |
+
"id": "aimepp-aime-0001",
|
| 138 |
+
"problem": "Let $X,Y,Z>1$ and $W>1$ satisfy $\\log_X W=54$, $\\log_Y W=24$, and $\\log_{X^{3}Y^{2}Z^{2}}W=6$. Determine $\\log_Z W$.",
|
| 139 |
+
"answer": 72,
|
| 140 |
+
"answer_str": "072",
|
| 141 |
+
"tier": "AIME"
|
| 142 |
+
}
|
| 143 |
+
```
|
| 144 |
+
|
| 145 |
+
The normative record definition is [schema/dataset.schema.json](schema/dataset.schema.json).
|
| 146 |
+
|
| 147 |
+
## Evaluation
|
| 148 |
+
|
| 149 |
+
Create a JSONL file with one prediction per problem:
|
| 150 |
+
|
| 151 |
+
```json
|
| 152 |
+
{"id":"aimepp-aime-0001","prediction":"072"}
|
| 153 |
+
{"id":"aimepp-aime-0002","prediction":314}
|
| 154 |
+
```
|
| 155 |
+
|
| 156 |
+
Then run:
|
| 157 |
+
|
| 158 |
+
```bash
|
| 159 |
+
python3 scripts/score.py predictions.jsonl
|
| 160 |
+
```
|
| 161 |
+
|
| 162 |
+
Use `--config aime-hard` (or another config name) to score only one tier. The default protocol accepts an integer or a string containing only a one-to-three-digit integer. `72` and `"072"` are equivalent. Missing, malformed, or out-of-range predictions are incorrect. The scorer reports overall and per-tier accuracy, always using the full selected gold set as the denominator.
|
| 163 |
+
|
| 164 |
+
For systems that emit worked reasoning, `--allow-boxed` also accepts the last `\boxed{N}` in a string. Report which parsing mode was used whenever publishing results.
|
| 165 |
+
|
| 166 |
+
## Intended uses
|
| 167 |
+
|
| 168 |
+
- Exact-answer evaluation of mathematical reasoning systems
|
| 169 |
+
- Reinforcement learning with deterministic verifiable rewards
|
| 170 |
+
- Test-time-compute and inference-strategy comparisons
|
| 171 |
+
- Per-tier capability profiling and regression testing
|
| 172 |
+
- Technical inspection by prospective data or evaluation partners
|
| 173 |
+
|
| 174 |
+
## Out-of-scope uses
|
| 175 |
+
|
| 176 |
+
- Treating this public-answer sample as a hidden or contamination-resistant benchmark
|
| 177 |
+
- Claiming broad mathematical, scientific, or safety capability from this sample alone
|
| 178 |
+
- Comparing scores produced with different prompts, tool policies, budgets, or answer parsers as if they were directly equivalent
|
| 179 |
+
|
| 180 |
+
## Creation and rights
|
| 181 |
+
|
| 182 |
+
All problems in this sample were created internally by Ulam AI, and Ulam AI holds the rights to the dataset. The records are original Ulam AI material rather than official competition questions. The dataset does not contain personal data or user-contributed content.
|
| 183 |
+
|
| 184 |
+
Worked derivations and reasoning traces are not part of this dataset's product format. The final integer supplied with each problem is its golden solution and the normative target used by the scorer.
|
| 185 |
+
|
| 186 |
+
## Quality and validation
|
| 187 |
+
|
| 188 |
+
The packaged files pass deterministic checks for schema conformance, stable and unique IDs, non-empty problems, answer range, answer-string consistency, duplicate problems, control characters, and balanced dollar-sign LaTeX delimiters.
|
| 189 |
+
|
| 190 |
+
Run the checks locally with no third-party dependencies:
|
| 191 |
+
|
| 192 |
+
```bash
|
| 193 |
+
make validate
|
| 194 |
+
```
|
| 195 |
+
|
| 196 |
+
See [QUALITY_REPORT.md](QUALITY_REPORT.md) for measured results. The supplied answers are the authoritative ground truth for AIME++ exact-match evaluation.
|
| 197 |
+
|
| 198 |
+
## Reproducible reporting
|
| 199 |
+
|
| 200 |
+
Every reported result should include:
|
| 201 |
+
|
| 202 |
+
- dataset version or immutable commit hash;
|
| 203 |
+
- config and record count;
|
| 204 |
+
- system and model version;
|
| 205 |
+
- prompt template;
|
| 206 |
+
- tool-access policy;
|
| 207 |
+
- sampling parameters and number of attempts;
|
| 208 |
+
- token or compute budget;
|
| 209 |
+
- strict or boxed answer parsing mode; and
|
| 210 |
+
- overall plus per-tier accuracy.
|
| 211 |
+
|
| 212 |
+
Once answers are distributed, those records should be treated as inspection, development, or training data rather than a private holdout. Ulam AI can create separately governed evaluation material for commercial partners.
|
| 213 |
+
|
| 214 |
+
## License and commercial access
|
| 215 |
+
|
| 216 |
+
The sample is distributed under the permissive [MIT License](LICENSE), allowing teams to inspect, evaluate, train on, modify, and redistribute the sample subject to the license terms.
|
| 217 |
+
|
| 218 |
+
The larger 24,700+ AIME-family, 1,000+ AIME-Graduate, and 100+ AIME-Researcher collections are available separately, as are custom difficulty mixes and private evaluation services. See [docs/COMMERCIAL_ACCESS.md](docs/COMMERCIAL_ACCESS.md) or visit [ulam.ai](https://ulam.ai/).
|
| 219 |
+
|
| 220 |
+
## Citation
|
| 221 |
+
|
| 222 |
+
Citation metadata is provided in [CITATION.cff](CITATION.cff). Until a paper or technical report is published, cite the dataset by organization, title, version, and repository URL.
|
| 223 |
+
|
| 224 |
+
## Version
|
| 225 |
+
|
| 226 |
+
This package is version `0.1.0`. See [CHANGELOG.md](CHANGELOG.md) for release notes.
|
| 227 |
+
|
| 228 |
+
Copyright © 2026 Ulam AI.
|
VERSION
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|
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|
|
|
|
|
|
| 1 |
+
0.1.0
|
data/aime-graduate.jsonl
ADDED
|
@@ -0,0 +1,20 @@
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|
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|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
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|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
| 1 |
+
{"id":"aimepp-graduate-0001","problem":"Let $q=e^{2\\pi i\\tau}$ and $D=q\\dfrac{d}{dq}$. Define $E_4(q)=1+240\\sum_{n\\ge1}\\sigma_3(n)q^n$, $E_6(q)=1-504\\sum_{n\\ge1}\\sigma_5(n)q^n$, and $\\Delta=(E_4^3-E_6^2)/1728$. Put $\\mathcal R=\\displaystyle\\sum_{r=0}^{5}(-1)^r\\binom{8}{5-r}\\binom{10}{r}(D^rE_4)(D^{5-r}E_6)$. If $\\dfrac{E_4\\mathcal R}{\\Delta^2}=\\sum_{n\\ge-1}c_nq^n$, find the least nonnegative residue of $c_{63}$ modulo $1000$.","answer":920,"answer_str":"920","tier":"AIME-Graduate"}
|
| 2 |
+
{"id":"aimepp-graduate-0002","problem":"Let $X\\subset\\mathbb{CP}^{11}$ be a smooth complete intersection of multidegree $(2,3,4,4,5)$, with its complex orientation, and let $h\\in H^2(X;\\mathbb Z)$ be the restricted hyperplane class. For the underlying real tangent bundle, write $p(TX_{\\mathbb R})=1+p_1+p_2+p_3=\\dfrac{(1+h^2)^{12}}{(1+4h^2)(1+9h^2)(1+16h^2)^2(1+25h^2)}$ through degree $12$. Define $\\widehat A(X)=\\dfrac1{967680}\\int_X(-31p_1^3+44p_1p_2-16p_3)$ and $\\sigma(X)=\\dfrac1{945}\\int_X(2p_1^3-13p_1p_2+62p_3)$. Find the least nonnegative residue of $37\\widehat A(X)+53\\sigma(X)$ modulo $1000$.","answer":166,"answer_str":"166","tier":"AIME-Graduate"}
|
| 3 |
+
{"id":"aimepp-graduate-0003","problem":"Let $\\Sigma_A$ be the two-sided shift of finite type on the symbols $\\{0,1,2,3,4\\}$ with adjacency matrix $A=\\begin{pmatrix}1&1&0&1&0\\\\1&0&1&1&0\\\\0&1&1&0&1\\\\1&0&1&1&1\\\\0&1&0&1&1\\end{pmatrix}$. For a shift-orbit of least period exactly $84$, choose a representative $x_0x_1\\cdots x_{83}$ and read subscripts modulo $84$. Put $U\\equiv\\sum_{j=0}^{83}(x_j^2+3x_j+1)\\pmod7$ and $V\\equiv\\sum_{j=0}^{83}(x_jx_{j+1}+x_j+2x_{j+1}+1)\\pmod5$. Let $N_{a,b}$ be the number of distinct such shift-orbits with $(U,V)=(a,b)$. Find the least nonnegative residue of $N_{3,3}+17N_{4,4}$ modulo $1000$.","answer":724,"answer_str":"724","tier":"AIME-Graduate"}
|
| 4 |
+
{"id":"aimepp-graduate-0004","problem":"Let $V:L^2([0,1])\\to L^2([0,1])$ be the Volterra operator $(Vf)(x)=\\int_0^x f(t)\\,dt$, and put $K=V^*V$. Let $T=(\\bigwedge^2K)\\oplus(\\operatorname{Sym}^3K)$ on $\\bigwedge^2L^2([0,1])\\oplus\\operatorname{Sym}^3L^2([0,1])$. Write the Fredholm determinant as $\\det(I+tT)=\\sum_{m\\ge0}a_mt^m$. If $a_9=p/q$ in lowest terms with $q>0$, find the least nonnegative residue of $p+q$ modulo $1000$.","answer":347,"answer_str":"347","tier":"AIME-Graduate"}
|
| 5 |
+
{"id":"aimepp-graduate-0005","problem":"Let $A\\in M_{102}(\\mathbb F_3)$. Regard $\\mathbb F_3^{102}$ as an $\\mathbb F_3[t]$-module by $t\\cdot v=Av$, and suppose it is isomorphic to $\\displaystyle\\bigoplus_{r\\in\\lambda}\\mathbb F_3[t]/((t-1)^r)\\ \\oplus\\!\\bigoplus_{r\\in\\mu}\\mathbb F_3[t]/((t^2+1)^r)\\ \\oplus\\!\\bigoplus_{r\\in\\nu}\\mathbb F_3[t]/((t^3+2t+1)^r)$, where the parts are counted with multiplicity and $\\lambda=(9,8,6,6,4,2,2,2)$, $\\mu=(5,4,3)$, and $\\nu=(4,3,3,2,1)$. Let $C$ be the number of matrices $B\\in\\operatorname{GL}_{102}(\\mathbb F_3)$ satisfying $AB=BA$. Find the least nonnegative residue of $C$ modulo $1000$.","answer":328,"answer_str":"328","tier":"AIME-Graduate"}
|
| 6 |
+
{"id":"aimepp-graduate-0006","problem":"Give ordinal intervals the order topology and products the product topology. Set $\\alpha_1=\\omega^3+2\\omega^2+5\\omega+7$, $\\alpha_2=2\\omega^3+\\omega^2+4\\omega+9$, $\\alpha_3=3\\omega^2+6\\omega+11$, $\\alpha_4=\\omega^3+4\\omega^2+2\\omega+8$, $\\beta_1=3\\omega^3+2\\omega^2+8\\omega+10$, $\\beta_2=\\omega^3+5\\omega^2+3\\omega+12$, and $\\beta_3=3\\omega^2+6\\omega+13$. Let $X=\\bigl([0,\\omega^{\\alpha_1}\\!\\cdot2]\\times[0,\\omega^{\\alpha_2}\\!\\cdot3]\\times[0,\\omega^{\\alpha_3}\\!\\cdot5]\\times[0,\\omega^{\\alpha_4}\\!\\cdot7]\\bigr)\\sqcup\\bigl([0,\\omega^{\\beta_1}\\!\\cdot11]\\times[0,\\omega^{\\beta_2}\\!\\cdot13]\\times[0,\\omega^{\\beta_3}\\!\\cdot17]\\bigr)$. Define the Cantor--Bendixson derivatives by $X^{(0)}=X$, $X^{(\\gamma+1)}=(X^{(\\gamma)})'$, and $X^{(\\lambda)}=\\bigcap_{\\gamma<\\lambda}X^{(\\gamma)}$ for limit $\\lambda$. Let $\\rho$ be the largest ordinal with $X^{(\\rho)}\\ne\\varnothing$, write $\\rho=a\\omega^3+b\\omega^2+c\\omega+d$, and put $m=|X^{(\\rho)}|$. Find the least nonnegative residue of $101a+103b+107c+109d+m$ modulo $1000$.","answer":709,"answer_str":"709","tier":"AIME-Graduate"}
|
| 7 |
+
{"id":"aimepp-graduate-0007","problem":"Let $M$ be the rational homology $3$-sphere obtained by integral surgery on a framed $5$-component link whose linking matrix is $Q=\\begin{pmatrix}-41&17&-25&-50&-25\\\\17&-42&16&50&25\\\\-25&16&-43&-59&-25\\\\-50&50&-59&-109&-50\\\\-25&25&-25&-50&-25\\end{pmatrix}$. Identify $H_1(M;\\mathbb Z)$ with $\\mathbb Z^5/Q\\mathbb Z^5$ and define its linking pairing by $\\ell([x],[y])=x^{\\mathsf T}Q^{-1}y\\pmod{\\mathbb Z}$. Let $\\tau=|H_1(M;\\mathbb Z)|$, and let $N$ be the number of elements $u\\in H_1(M;\\mathbb Z)$ having exact order $60$ and satisfying $\\ell(u,u)=0$. Find the least nonnegative residue of $\\tau+37N$ modulo $1000$.","answer":336,"answer_str":"336","tier":"AIME-Graduate"}
|
| 8 |
+
{"id":"aimepp-graduate-0008","problem":"Let $G=\\operatorname{PSL}_2(\\mathbb F_7)$. For positive integers $a,b,c,d$, let $N_{a,b,c,d}$ be the number of $\\operatorname{Aut}(G)$-orbits, under componentwise action, of ordered generating quadruples $(x_1,x_2,x_3,x_4)\\in G^4$ such that $x_1x_2x_3x_4=1$ and the exact orders of $x_1,x_2,x_3,x_4$ are $a,b,c,d$, respectively. Find the least nonnegative residue of $N_{2,3,4,7}+17N_{3,4,4,7}$ modulo $1000$.","answer":470,"answer_str":"470","tier":"AIME-Graduate"}
|
| 9 |
+
{"id":"aimepp-graduate-0009","problem":"Over $\\mathbb F_2$, let $C_{23}=\\{a(x)g(x)\\bmod(x^{23}-1):\\deg a<12\\}$, where $g(x)=x^{11}+x^9+x^7+x^6+x^5+x+1$, and identify residues with coefficient vectors of length $23$. Extend every word by one parity coordinate so that its total Hamming weight is even, obtaining a binary code $C\\subset\\mathbb F_2^{24}$. For a set $E$ of $14$ coordinates, put $L(E)=\\{c\\in C:\\operatorname{Supp}(c)\\subseteq E\\}$. Let $N_j$ be the number of such sets $E$ for which $|L(E)|=2^j$. Find the least nonnegative residue of $N_2+7N_3$ modulo $1000$.","answer":736,"answer_str":"736","tier":"AIME-Graduate"}
|
| 10 |
+
{"id":"aimepp-graduate-0010","problem":"For $m\\ge2$, let $L^{2n+1}(m)=S^{2n+1}/\\mathbb Z_m$, where a generator acts by scalar multiplication by a primitive $m$th root of unity on every complex coordinate. Put $X=L^{11}(8)\\times L^{13}(12)\\times L^9(18)$. For $i=0,1$, let $\\tau_i$ be the order of the torsion subgroup of the complex topological $K$-group $K^i(X)$. Find the least nonnegative residue of $\\tau_0+7\\tau_1$ modulo $1000$.","answer":872,"answer_str":"872","tier":"AIME-Graduate"}
|
| 11 |
+
{"id":"aimepp-graduate-0011","problem":"Let $M$ be the countable homogeneous structure in the language $\\{E,F,<,R,S\\}$ with the following properties. The relations $E$ and $F$ are equivalence relations, $F$ refines $E$, there are infinitely many $E$-classes, every $E$-class contains infinitely many $F$-classes, and every $F$-class is infinite. The relation $<$ is constant on $E$-classes and induces a dense linear order without endpoints on $M/E$. The relation $R$ is symmetric and irreflexive, implies $F$, and induces the countable random graph on every $F$-class. The relation $S$ implies $E\\wedge\\neg F$, is constant on ordered pairs of $F$-classes, and induces the countable random tournament on the set of $F$-classes inside every $E$-class. Let $T=\\operatorname{Th}(M)$, and let $S_{12}(T)$ be the set of complete $12$-types over the empty set in variables $x_1,\\ldots,x_{12}$. Find the least nonnegative residue of $|S_{12}(T)|$ modulo $1000$.","answer":591,"answer_str":"591","tier":"AIME-Graduate"}
|
| 12 |
+
{"id":"aimepp-graduate-0012","problem":"In the $SU(2)$ Wess--Zumino--Witten fusion ring at level $42$, the simple classes $[V_0],\\ldots,[V_{42}]$ satisfy $[V_a][V_b]=\\sum[V_c]$, where the sum is over integers $c$ such that $|a-b|\\le c\\le\\min(a+b,84-a-b)$ and $c\\equiv a+b\\pmod2$. Put $\\Omega=\\sum_{a=0}^{42}[V_a]^2$, and let $D$ be the coefficient of $[V_0]$ in $[V_8][V_{11}][V_{13}][V_{17}][V_{22}][V_{25}][V_{31}][V_{35}]\\Omega^8$. Find the least nonnegative residue of $D$ modulo $1000$.","answer":808,"answer_str":"808","tier":"AIME-Graduate"}
|
| 13 |
+
{"id":"aimepp-graduate-0013","problem":"Let $X=\\operatorname{Gr}(4,\\mathbb F_3^8)$, equipped with the injection metric $d(U,V)=4-\\dim(U\\cap V)$. For $s\\in\\{2,5\\}$, define $\\mathcal M_s(X)=\\dfrac{|X|}{\\sum_{V\\in X}s^{-d(U,V)}}$, where $U\\in X$ is arbitrary, and write $\\mathcal M_s(X)=p_s/q_s$ in lowest terms with $q_s>0$. Let $D$ be the number of unordered pairs $\\{U,V\\}\\subset X$ with $d(U,V)=4$. Find the least nonnegative residue of $p_2+q_2+11(p_5+q_5)+D$ modulo $1000$.","answer":456,"answer_str":"456","tier":"AIME-Graduate"}
|
| 14 |
+
{"id":"aimepp-graduate-0014","problem":"Let $C/\\mathbb F_{101}$ be the smooth projective genus-$2$ curve with affine equation $y^2=x^5+7x^4+13x^3+17x^2+19x+23$. Let $\\alpha_1,\\ldots,\\alpha_4$ be its Frobenius eigenvalues, characterized by $\\#C(\\mathbb F_{101^r})=101^r+1-\\sum_{i=1}^4\\alpha_i^r$ for every $r\\ge1$, and put $J_r=\\prod_{i=1}^4(1-\\alpha_i^r)=\\#\\operatorname{Jac}(C)(\\mathbb F_{101^r})$. For $N=10^{18}+10^{12}+137$, find the least nonnegative residue of $J_N$ modulo $1000$.","answer":881,"answer_str":"881","tier":"AIME-Graduate"}
|
| 15 |
+
{"id":"aimepp-graduate-0015","problem":"Let $J\\in M_{34}(\\mathbb Q)$ be the nilpotent Jordan block with $J_{i,i+1}=1$. Define $P_8(z)=\\displaystyle\\sum_{k=0}^{8}\\frac{(16-k)!\\,8!}{16!\\,k!\\,(8-k)!}z^k$ and the diagonal Pad\\'e approximant $r_8(z)=P_8(z)/P_8(-z)$. Put $R=(r_8(J/32))^{32}$ and $E=e^J=\\sum_{k=0}^{33}J^k/k!$. Write $29!(R-E)_{1,30}+7\\cdot31!(R-E)_{1,32}+11\\cdot33!(R-E)_{1,34}=p/q$ in lowest terms with $q>0$. Find the least nonnegative residue of $p+q$ modulo $1000$.","answer":641,"answer_str":"641","tier":"AIME-Graduate"}
|
| 16 |
+
{"id":"aimepp-graduate-0016","problem":"Let $\\Gamma_E$ be the rooted $E_8$ graph on vertices $0,1,\\ldots,7$ with edges $\\{0,1\\},\\{1,2\\},\\{2,3\\},\\{3,4\\},\\{4,5\\},\\{5,6\\},\\{4,7\\}$, rooted at $0$. Let $\\Gamma_D$ be the rooted $D_{32}$ graph on vertices $0,1,\\ldots,31$ with edges $\\{i,i+1\\}$ for $0\\le i\\le28$ and edges $\\{29,30\\},\\{29,31\\}$, rooted at $0$. Let $a_n$ and $b_n$ be the numbers of length-$2n$ loops at the roots of $\\Gamma_E$ and $\\Gamma_D$, respectively; equivalently, these are the relative-commutant dimensions for the corresponding finite-depth subfactor standard invariants. Put $c_n=a_nb_n$ and $N=10^{18}+10^9+73$. Find the least nonnegative residue modulo $1000$ of $\\det\\bigl(c_{N+i+j}\\bigr)_{0\\le i,j\\le3}+7c_{2N+5}$.","answer":465,"answer_str":"465","tier":"AIME-Graduate"}
|
| 17 |
+
{"id":"aimepp-graduate-0017","problem":"Let $P$ and $S$ be the stabilizing symmetric solutions of $A^{\\mathsf T}P+PA-P^2+Q=0$ and $AS+SA^{\\mathsf T}-S^2+W=0$, where $A=\\begin{pmatrix}-12&2&-1&0\\\\-3&-14&2&1\\\\1&-2&-16&3\\\\0&1&-4&-18\\end{pmatrix}$, $Q=\\begin{pmatrix}52&5&-2&0\\\\5&93&2&-8\\\\-2&2&144&8\\\\0&-8&8&205\\end{pmatrix}$, and $W=\\begin{pmatrix}145&7&-2&0\\\\7&128&2&-6\\\\-2&2&105&6\\\\0&-6&6&76\\end{pmatrix}$. Let $X,Y,Z$ be the unique solutions of $(A-P)X+X(A-P)^{\\mathsf T}+I_4=0$, $(A-S)Y+Y(A-S)^{\\mathsf T}+I_4=0$, and $(A-P)Z+Z(A-S)^{\\mathsf T}+I_4=0$, respectively. If $\\det X+7\\det Y+11\\det Z=p/q$ in lowest terms with $q>0$, find the least nonnegative residue of $p+q$ modulo $1000$.","answer":857,"answer_str":"857","tier":"AIME-Graduate"}
|
| 18 |
+
{"id":"aimepp-graduate-0018","problem":"Let $X\\sim N_6(0,\\Sigma)$, where $\\Sigma=\\dfrac13\\begin{pmatrix}4&1&0&0&0&0\\\\1&5&1&0&0&0\\\\0&1&6&1&0&0\\\\0&0&1&7&1&0\\\\0&0&0&1&8&1\\\\0&0&0&0&1&9\\end{pmatrix}$. Put $Q=X^{\\mathsf T}AX$ and $R=X^{\\mathsf T}BX$, where $A=\\begin{pmatrix}5&1&0&1&0&0\\\\1&6&1&0&1&0\\\\0&1&7&1&0&1\\\\1&0&1&8&1&0\\\\0&1&0&1&9&1\\\\0&0&1&0&1&10\\end{pmatrix}$ and $B=\\begin{pmatrix}7&0&1&0&1&1\\\\0&8&1&1&0&1\\\\1&1&6&0&1&0\\\\0&1&0&9&1&1\\\\1&0&1&1&5&0\\\\1&1&0&1&0&10\\end{pmatrix}$. Write $\\mathbb E[(Q-\\mathbb E Q)^6(R-\\mathbb E R)^5]=p/q$ in lowest terms with $q>0$. Find the least nonnegative residue of $p+q$ modulo $1000$.","answer":249,"answer_str":"249","tier":"AIME-Graduate"}
|
| 19 |
+
{"id":"aimepp-graduate-0019","problem":"Let $\\zeta$ be a primitive fifth root of unity, and let $u_\\zeta^+(\\mathfrak e_8)$ be the positive part of the small quantum group, graded by root height. Thus, if $a_n$ is the dimension of its degree-$n$ component, then $\\displaystyle\\sum_{n\\ge0}a_nt^n=\\prod_{\\alpha\\in\\Phi^+(E_8)}(1+t^{\\operatorname{ht}(\\alpha)}+t^{2\\operatorname{ht}(\\alpha)}+t^{3\\operatorname{ht}(\\alpha)}+t^{4\\operatorname{ht}(\\alpha)})$. Find the least nonnegative residue of $a_{731}+13a_{1111}$ modulo $1000$.","answer":318,"answer_str":"318","tier":"AIME-Graduate"}
|
| 20 |
+
{"id":"aimepp-graduate-0020","problem":"Let $V=\\mathbb C^{14}$. For a partition $\\lambda\\vdash24$, let $m_\\lambda=\\dim\\operatorname{Hom}_{\\operatorname{GL}(V)}\\!\\left(\\mathbf S_\\lambda(V),\\operatorname{Sym}^6(\\bigwedge^4V)\\right)$. Put $\\lambda_1=(4,4,3,3,2,2,1,1,1,1,1,1)$, $\\lambda_2=(4,4,3,3,2,2,2,2,1,1)$, and $\\lambda_3=(4,4,3,3,2,2,2,1,1,1,1)$. Find the least nonnegative residue of $97m_{\\lambda_1}+89m_{\\lambda_2}+83m_{\\lambda_3}$ modulo $1000$.","answer":621,"answer_str":"621","tier":"AIME-Graduate"}
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| 1 |
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{"id":"aimepp-hard-0001","problem":"Define $K_N=\\displaystyle\\prod_{j=1}^N(j!)^j$. Write $K_{236}=19^E U$, where $19\\nmid U$. Let $Z$ be the least nonnegative residue of $E+U$ modulo $997$. Define $P(x)=x^{5} - 9 x^{4} + 349 x^{3} - 3 x^{2} + 7 x$ and $Q(x)=x^{4} + 692 x^{2} + 8 x - 63$. Find the least nonnegative residue modulo $1000$ of the resultant $\\operatorname{Res}_x(P,Q)$.","answer":929,"answer_str":"929","tier":"AIME Hard"}
|
| 2 |
+
{"id":"aimepp-hard-0002","problem":"Let $Z$ be the least nonnegative residue modulo $1000$ of the number of reduced fractions $a/b$ that satisfy $\\frac{1}{5}<a/b<\\frac{3}{5}$, $1\\le b\\le 554$, $\\omega(b)=2$, and $a+b\\equiv 1\\pmod{7}$. Here $\\omega(b)$ is the number of distinct prime divisors of $b$. Let $p=1361$, let $A$ and $B$ be the least nonnegative residues modulo $p$ of $128Z+288$ and $1345Z+304$, respectively, and consider $E: y^2=x^3+Ax+B$ over $\\mathbb F_p$. Including the point at infinity, let $C=\\#E(\\mathbb F_p)$, and let $X$ be the sum, as ordinary integers, of the $x$-coordinates of all affine points of $E$. Find the least nonnegative residue of $C+3X$ modulo $1000$.","answer":835,"answer_str":"835","tier":"AIME Hard"}
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| 3 |
+
{"id":"aimepp-hard-0003","problem":"A partition of $\\{1,2,\\ldots,9\\}$ has a crossing quadruple when $a<b<c<d$, the numbers $a,c$ lie in one block, and $b,d$ lie in a different block. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of set partitions that have exactly $4$ blocks, exactly $0$ singleton blocks, and a number of crossing quadruples congruent to $2$ modulo $5$. Put $L=Z+43$. Let $S(n,k)$ be a Stirling number of the second kind, and let $A(n,j)$ be the number of permutations of $\\{1,\\ldots,n\\}$ with exactly $j$ descents. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum_{j=0}^{5}(j+1)S(L+j,6)A(L,j)2^j$.","answer":284,"answer_str":"284","tier":"AIME Hard"}
|
| 4 |
+
{"id":"aimepp-hard-0004","problem":"Let $C_{253}=\\frac1{254}\\binom{506}{253}$ be a Catalan number. Write $C_{253}=5^e u$ with $5\\nmid u$. Let $Z$ be the least nonnegative residue of $u+84e$ modulo $1000$. Write $Z=100d_2+10d_1+d_0$ with $0\\le d_0,d_1,d_2\\le9$. Let $\\lambda=(12+d_2+d_1+d_0,8+d_1+d_0,4+d_0,4,2)$. Find the least nonnegative residue modulo $1000$ of the number of standard Young tableaux of shape $\\lambda$.","answer":192,"answer_str":"192","tier":"AIME Hard"}
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| 5 |
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{"id":"aimepp-hard-0005","problem":"Consider standard Young tableaux of skew shape $(4,2,2,2,2)/(0,0,0,0,0)$. A descent is an integer $i$ for which the entry $i+1$ lies in a lower row than the entry $i$. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of such tableaux that have exactly $5$ descents. Put $L=18+(Z\\bmod 13)$ and, for $1\\le j\\le L$, put $a_j=1+((Zj^2+12j+3)\\bmod 14)$. If $[a_1;a_2,\\ldots,a_L]=p_L/q_L$ and $p_{L-1}/q_{L-1}=[a_1;\\ldots,a_{L-1}]$ are in lowest terms, find the least nonnegative residue modulo $1000$ of $p_L+q_L+5p_{L-1}+7q_{L-1}$.","answer":24,"answer_str":"024","tier":"AIME Hard"}
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| 6 |
+
{"id":"aimepp-hard-0006","problem":"A tournament is formed by orienting every edge of the complete graph on $7$ labeled vertices. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of tournaments that have sorted outdegree sequence $(1, 1, 1, 4, 4, 5, 5)$ and exactly $3$ directed $3$-cycles. A Markov chain has transition matrix $\\frac1{11}\\begin{pmatrix}3 & 3 & 4 & 1\\\\2 & 7 & 1 & 1\\\\4 & 3 & 3 & 1\\\\8 & 1 & 1 & 1\\end{pmatrix}$. It starts in state $2$. After $Z+55$ steps, the probability that it is in one of the states $\\{3,4\\}$ is $p/q$ in lowest terms with $q>0$. Find the least nonnegative residue of $p+q$ modulo $1000$.","answer":771,"answer_str":"771","tier":"AIME Hard"}
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| 7 |
+
{"id":"aimepp-hard-0007","problem":"Let $\\theta_k=\\frac{2\\pi k}{19}$ for $1\\le k\\le 18$. If $\\displaystyle\\sum_{k=1}^{18}\\left(\\frac1{(8-2\\cos\\theta_k)^2}+\\frac{2}{8-2\\cos\\theta_k}\\right)=\\frac pq$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $m=8+(Z\\bmod 9)$. Form a wheel with hub $0$ and rim vertices $1,2,\\ldots,m$ in cyclic order. Give spoke $0j$ weight $1+((Z+j^2+9j)\\bmod5)$ and rim edge $j(j+1)$ weight $1+((2Z+4j+j^2)\\bmod5)$, with $m+1$ interpreted as $1$. The weight of a spanning tree is the product of its edge weights. Find the total weight of all spanning trees modulo $1000$.","answer":525,"answer_str":"525","tier":"AIME Hard"}
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| 8 |
+
{"id":"aimepp-hard-0008","problem":"Let $Z$ be the least nonnegative residue modulo $1000$ of the number of $5\\times5$ zero-one matrices that have every row sum equal to $2$ or $3$, exactly $1$ rows of sum $3$, no zero column, rank $4$ over $\\mathbb F_2$, trace $3$, an anti-diagonal sum congruent to $0$ modulo $3$, and exactly $5$ columns of odd sum. Find the least nonnegative residue modulo $1000$ of the number of words of length $Z+53$ over $\\{0,1,2\\}$ that contain none of the blocks $012$, $121$, $210$, have digit sum congruent to $3$ modulo $5$, and contain a number of $2$'s congruent to $2$ modulo $4$.","answer":702,"answer_str":"702","tier":"AIME Hard"}
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| 9 |
+
{"id":"aimepp-hard-0009","problem":"Define $K_N=\\displaystyle\\prod_{j=1}^N(j!)^j$. Write $K_{195}=11^E U$, where $11\\nmid U$. Let $Z$ be the least nonnegative residue of $E+U$ modulo $997$. Put $L=Z+30$. Define $G(n,0)=1$, $G(0,k)=0$ for $k>0$, and $G(n,k)=G(n-1,k)+7^{n-k}G(n-1,k-1)$. Find the least nonnegative residue modulo $1000$ of $G(L,11)+3G(L,10)$.","answer":208,"answer_str":"208","tier":"AIME Hard"}
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| 10 |
+
{"id":"aimepp-hard-0010","problem":"On an $8\\times 8$ board, row $i$ contains exactly the first $b_i$ allowed squares, where $(b_1,\\ldots,b_8)=(1, 2, 3, 7, 8, 8, 8, 8)$. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ways to place $8$ nonattacking rooks, one in each row and column, so that exactly $4$ rooks lie on the main diagonal, the induced column permutation has inversion parity $1$, and $\\sum_{i=1}^8 i\\pi(i)\\equiv 5\\pmod{8}$. Let $n=7+(Z\\bmod 4)$ and let $M=(m_{ij})_{0\\le i,j<n}$, where $m_{ij}$ is the least nonnegative residue modulo $17$ of $i^2+10ij+j^2+5Z(i+j+1)+6Z+8\\,[i-j\\equiv1\\pmod n]$. Here the bracket equals $1$ when its condition holds and $0$ otherwise. Find the least nonnegative residue of $\\det M$ modulo $1000$.","answer":447,"answer_str":"447","tier":"AIME Hard"}
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| 11 |
+
{"id":"aimepp-hard-0011","problem":"Let $Y(x)$ be the unique formal power series with constant term $0$ satisfying $Y=x(1+Y)^{2}(1+Y^2)^{3}(1-Y)^{-1}$. Let $Z$ be the least nonnegative residue modulo $997$ of the coefficient of $x^{36}$ in $Y(x)^{10}$. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $19^{Z^2+3Z+86}+31^{Z+86}+4510209742$, and write $R$ as a block of exactly ten decimal digits, allowing leading zeros. For $0\\le j<10$, let $R_j$ be the integer obtained by cyclically shifting this block left by $j$ places. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum (j+1)\\gcd(R_j,10^{10}-1)$, where the sum is over the integers $j$ with $0\\le j<10$ and $R_j\\equiv 6\\pmod{19}$.","answer":135,"answer_str":"135","tier":"AIME Hard"}
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| 12 |
+
{"id":"aimepp-hard-0012","problem":"Let $N=6452641$. For all coprime positive integer pairs $(x,y)$ with $x>y$ and $x^2-xy+y^2=N$, consider those satisfying $2x-y\\equiv 11\\pmod{13}$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of $3x+y$ over all such pairs. Define $u_0=81$ and $u_1=458$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 11u_{k+1}+9u_k+12(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+9Z+87}+13u_{2Z+43}$.","answer":545,"answer_str":"545","tier":"AIME Hard"}
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| 13 |
+
{"id":"aimepp-hard-0013","problem":"Let $Y(x)$ be the unique formal power series with constant term $0$ satisfying $Y=x(1+Y)^{2}(1+Y^2)^{3}(1-Y)^{-1}$. Let $Z$ be the least nonnegative residue modulo $991$ of the coefficient of $x^{43}$ in $Y(x)^{10}$. Let $p=2129$, let $A$ and $B$ be the least nonnegative residues modulo $p$ of $611Z+422$ and $1387Z+870$, respectively, and consider $E: y^2=x^3+Ax+B$ over $\\mathbb F_p$. Including the point at infinity, let $C=\\#E(\\mathbb F_p)$, and let $X$ be the sum, as ordinary integers, of the $x$-coordinates of all affine points of $E$. Find the least nonnegative residue of $C+5X$ modulo $1000$.","answer":723,"answer_str":"723","tier":"AIME Hard"}
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| 14 |
+
{"id":"aimepp-hard-0014","problem":"Over the ring $(\\mathbb Z/1000\\mathbb Z)[x]$, let $c_0+c_1x+\\cdots+c_6x^6$ be the remainder when $(x^2+x^4+3)^{154655378506}$ is divided by the monic polynomial $P(x)=x^{7} - 5 x^{6} - 6 x^{5} - 5 x^{4} - 2 x^{3} - 5 x - 5$. Let $Z$ be the least nonnegative residue of $9c_0+1c_1+2c_2+7c_3+5c_4+2c_5+9c_6$ modulo $1000$. Set $T=11^{Z+1}+\\displaystyle\\sum_{k=0}^Z(-1)^k\\binom Zk13^k23^{Z-k}(k^3+5k^2+7k+2)$. Find the least nonnegative residue of $T$ modulo $1000$.","answer":611,"answer_str":"611","tier":"AIME Hard"}
|
| 15 |
+
{"id":"aimepp-hard-0015","problem":"Let $Z$ be the least nonnegative residue modulo $1000$ of the number of reduced fractions $a/b$ that satisfy $\\frac{1}{5}<a/b<\\frac{3}{5}$, $1\\le b\\le 466$, $\\omega(b)=1$, and $a+b\\equiv 6\\pmod{13}$. Here $\\omega(b)$ is the number of distinct prime divisors of $b$. Let $n=7+(Z\\bmod 4)$ and let $M=(m_{ij})_{0\\le i,j<n}$, where $m_{ij}$ is the least nonnegative residue modulo $23$ of $i^2+15ij+j^2+15Z(i+j+1)+11Z+14\\,[i-j\\equiv1\\pmod n]$. Here the bracket equals $1$ when its condition holds and $0$ otherwise. Find the least nonnegative residue of $\\det M$ modulo $1000$.","answer":171,"answer_str":"171","tier":"AIME Hard"}
|
| 16 |
+
{"id":"aimepp-hard-0016","problem":"Let $Z$ be the least nonnegative residue modulo $1000$ of the number of $5\\times5$ zero-one matrices that have every row sum equal to $2$ or $3$, exactly $5$ rows of sum $3$, no zero column, rank $4$ over $\\mathbb F_2$, trace $2$, an anti-diagonal sum congruent to $0$ modulo $3$, and exactly $5$ columns of odd sum. Put $L=Z+48$. Let $S(n,k)$ be a Stirling number of the second kind, and let $A(n,j)$ be the number of permutations of $\\{1,\\ldots,n\\}$ with exactly $j$ descents. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum_{j=0}^{5}(j+1)S(L+j,6)A(L,j)11^j$.","answer":69,"answer_str":"069","tier":"AIME Hard"}
|
| 17 |
+
{"id":"aimepp-hard-0017","problem":"An urn contains $11$ red, $11$ blue, and $10$ green balls. Balls are drawn uniformly without replacement until the number drawn of some color first reaches its threshold: $5$ for red, $4$ for blue, and $2$ for green. The probability that the stopping color is green and the total number of draws is congruent to $3$ modulo $4$ is $p/q$ in lowest terms. Let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. In the quotient ring $(\\mathbb Z/1000\\mathbb Z)[x]/(x^{5}-3x^{4}-6x^{2}-4x+6)$, write $(-x^{4}-4x^{3}+5x^{2}+3x+4)^{Z^2+11Z+41}=r_0+r_1x+r_2x^2+r_3x^3+r_4x^4$. Find the least nonnegative residue modulo $1000$ of $4r_0+2r_1+6r_2+4r_3+9r_4$.","answer":205,"answer_str":"205","tier":"AIME Hard"}
|
| 18 |
+
{"id":"aimepp-hard-0018","problem":"Triangle $ABC$ has side lengths $BC=21$, $CA=15$, and $AB=16$. An interior point $P$ has barycentric coordinates $(8:7:3)$, and $Q$ is the isotomic conjugate of $P$. The perpendiculars from $Q$ to $BC,CA,AB$ meet those lines at $D,E,F$, respectively. If $([DEF]/[ABC])^2=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $p=1657$, set $x_0=Z$, and for $k\\ge0$ let $x_{k+1}$ be the least nonnegative residue modulo $p$ of $x_k^2+1399x_k+351$. Find the least nonnegative residue modulo $1000$ of $x_{10^{12}+13Z+71}+7\\displaystyle\\sum_{j=0}^{Z+44}x_j$.","answer":376,"answer_str":"376","tier":"AIME Hard"}
|
| 19 |
+
{"id":"aimepp-hard-0019","problem":"For a positive integer $t$, let $J_2(t)=t^2\\prod_{p\\mid t}(1-p^{-2})$. Let $N=780$. Let $Z$ be the least nonnegative residue modulo $997$ of $\\displaystyle\\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\\equiv 1\\pmod{11}$ and $\\Omega(d)\\equiv 1\\pmod2$. A Markov chain has transition matrix $\\frac1{14}\\begin{pmatrix}4 & 3 & 6 & 1\\\\5 & 1 & 7 & 1\\\\1 & 1 & 4 & 8\\\\1 & 6 & 3 & 4\\end{pmatrix}$. It starts in state $2$. After $Z+41$ steps, the probability that it is in one of the states $\\{1,3\\}$ is $p/q$ in lowest terms with $q>0$. Find the least nonnegative residue of $p+q$ modulo $1000$.","answer":115,"answer_str":"115","tier":"AIME Hard"}
|
| 20 |
+
{"id":"aimepp-hard-0020","problem":"The sequence $(a_n)$ is defined by $(a_0,a_1,\\ldots,a_5)=(17, 16, -12, -18, 13, 10)$ and $a_{n+6}=8a_{n+5}-2a_{n+4}+7a_{n+3}-6a_{n+2}-5a_{n+1}+8a_n$ for $n\\ge0$. Let $M$ be the $5\\times5$ matrix whose $(i,j)$ entry is $a_{8798202908543+i+j}-5a_{8798202908543+i+j+1}$, where $0\\le i,j\\le4$. Let $Z$ be the least nonnegative residue of $\\det(M)$ modulo $997$. Put $L=Z+58$. Define $G(n,0)=1$, $G(0,k)=0$ for $k>0$, and $G(n,k)=G(n-1,k)+7^{n-k}G(n-1,k-1)$. Find the least nonnegative residue modulo $1000$ of $G(L,9)+5G(L,8)$.","answer":475,"answer_str":"475","tier":"AIME Hard"}
|
| 21 |
+
{"id":"aimepp-hard-0021","problem":"The sequence $(a_n)$ is defined by $(a_0,a_1,\\ldots,a_5)=(-18, -13, 14, 11, -13, -9)$ and $a_{n+6}=5a_{n+5}+7a_{n+4}+8a_{n+3}-8a_{n+2}+7a_{n+1}+3a_n$ for $n\\ge0$. Let $M$ be the $5\\times5$ matrix whose $(i,j)$ entry is $a_{2431472563692+i+j}-1a_{2431472563692+i+j+1}$, where $0\\le i,j\\le4$. Let $Z$ be the least nonnegative residue of $\\det(M)$ modulo $997$. Put $L=18+(Z\\bmod 13)$ and, for $1\\le j\\le L$, put $a_j=1+((Zj^2+8j+4)\\bmod 11)$. If $[a_1;a_2,\\ldots,a_L]=p_L/q_L$ and $p_{L-1}/q_{L-1}=[a_1;\\ldots,a_{L-1}]$ are in lowest terms, find the least nonnegative residue modulo $1000$ of $p_L+q_L+3p_{L-1}+3q_{L-1}$.","answer":967,"answer_str":"967","tier":"AIME Hard"}
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| 22 |
+
{"id":"aimepp-hard-0022","problem":"Let $N=38909$. Among all pairs of positive integers $(x,y)$ satisfying $x<y$, $x^2+y^2=N$, and $\\gcd(x,y)=1$, retain only those for which $x+2y\\equiv 1\\pmod{13}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum(2x+3y)$ over the retained pairs. Put $L=Z+50$. Let $S(n,k)$ be a Stirling number of the second kind, and let $A(n,j)$ be the number of permutations of $\\{1,\\ldots,n\\}$ with exactly $j$ descents. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum_{j=0}^{9}(j+1)S(L+j,10)A(L,j)11^j$.","answer":65,"answer_str":"065","tier":"AIME Hard"}
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| 23 |
+
{"id":"aimepp-hard-0023","problem":"For $0\\le i,j\\le5$, define $m_{ij}=\\binom{29+i+j}{23+2i-j}+3\\binom{29+i+j}{23-i+2j}+3\\binom{29+i+j}{23+i-j-2}$, where a binomial coefficient is $0$ when its lower index is outside its usual range. If $M=(m_{ij})_{0\\le i,j\\le5}$, let $Z$ be the least nonnegative residue of $\\det M$ modulo $991$. A Markov chain has transition matrix $\\frac1{13}\\begin{pmatrix}1 & 1 & 1 & 10\\\\1 & 4 & 4 & 4\\\\2 & 1 & 9 & 1\\\\2 & 3 & 6 & 2\\end{pmatrix}$. It starts in state $2$. After $Z+78$ steps, the probability that it is in one of the states $\\{3,4\\}$ is $p/q$ in lowest terms with $q>0$. Find the least nonnegative residue of $p+q$ modulo $1000$.","answer":149,"answer_str":"149","tier":"AIME Hard"}
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| 24 |
+
{"id":"aimepp-hard-0024","problem":"On an $8\\times 8$ board, row $i$ contains exactly the first $b_i$ allowed squares, where $(b_1,\\ldots,b_8)=(2, 3, 6, 8, 8, 8, 8, 8)$. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ways to place $8$ nonattacking rooks, one in each row and column, so that exactly $0$ rooks lie on the main diagonal, the induced column permutation has inversion parity $1$, and $\\sum_{i=1}^8 i\\pi(i)\\equiv 2\\pmod{5}$. Put $H=Z+107$. Find the least nonnegative residue modulo $1000$ of the number of ordered pairs $(a,b)$ that satisfy $1\\le a<b\\le H$, $\\gcd(a,b)=1$, $a+b\\equiv 5\\pmod{11}$, and $\\left\\lfloor Ha/b\\right\\rfloor\\equiv 4\\pmod{5}$.","answer":145,"answer_str":"145","tier":"AIME Hard"}
|
| 25 |
+
{"id":"aimepp-hard-0025","problem":"Let $C_{267}=\\frac1{268}\\binom{534}{267}$ be a Catalan number. Write $C_{267}=5^e u$ with $5\\nmid u$. Let $Z$ be the least nonnegative residue of $u+28e$ modulo $1000$. Put $L=Z+70$. Let $S(n,k)$ be a Stirling number of the second kind, and let $A(n,j)$ be the number of permutations of $\\{1,\\ldots,n\\}$ with exactly $j$ descents. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum_{j=0}^{7}(j+1)S(L+j,8)A(L,j)5^j$.","answer":900,"answer_str":"900","tier":"AIME Hard"}
|
| 26 |
+
{"id":"aimepp-hard-0026","problem":"For $1\\le n\\le 72793$, set $a_n=\\lfloor n\\sqrt{12}\\rfloor$ and $b_n=\\lfloor n/\\sqrt{12}\\rfloor$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all $n$ such that $a_n\\equiv 2\\pmod{11}$, $b_n\\equiv 4\\pmod{6}$, and $n\\equiv 1\\pmod2$. Let $L=Z+52$ and $P=\\displaystyle\\prod_{j=1}^L(8j+7)$. Write $P=5^eU$ with $5\\nmid U$, and let $u$ be the least nonnegative residue of $U$ modulo $5^5=3125$. Find the least nonnegative residue modulo $1000$ of $u+7e+2(e\\bmod 5)^2$.","answer":546,"answer_str":"546","tier":"AIME Hard"}
|
| 27 |
+
{"id":"aimepp-hard-0027","problem":"Let $p=3079$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$11$ digits is congruent to $5$ modulo $11$, and the sum of the base-$11$ digits of its least positive inverse modulo $p$ is congruent to $9$ modulo $10$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all admissible $g$. Let $n=7+(Z\\bmod 4)$ and let $M=(m_{ij})_{0\\le i,j<n}$, where $m_{ij}$ is the least nonnegative residue modulo $17$ of $i^2+14ij+j^2+3Z(i+j+1)+4Z+2\\,[i-j\\equiv1\\pmod n]$. Here the bracket equals $1$ when its condition holds and $0$ otherwise. Find the least nonnegative residue of $\\det M$ modulo $1000$.","answer":460,"answer_str":"460","tier":"AIME Hard"}
|
| 28 |
+
{"id":"aimepp-hard-0028","problem":"Two walkers move simultaneously on the lattice, each taking one unit north or east step per second. The lower walker goes from $(0,0)$ to $(9,11)$, and the upper walker goes from $(0,2)$ to $(9,13)$. After every equal number of steps, their current vertices must be distinct. A turn is a change of direction by either walker. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ordered pairs of walks of such paths that have exactly $20$ turns in total and have the lower path visit exactly $2$ vertices on the line $y=x$, counting its initial vertex. Put $L=Z+64$. For an integer $q\\ge2$, let $P_q(L)$ be the number of rotation classes of aperiodic length-$L$ words over a $q$-letter alphabet. Find the least nonnegative residue modulo $1000$ of $P_6(L)+2P_5(L)+\\varphi(L)$, where here $q=6$.","answer":55,"answer_str":"055","tier":"AIME Hard"}
|
| 29 |
+
{"id":"aimepp-hard-0029","problem":"Let $N=30240$. Consider nondecreasing $4$-tuples of integers $2\\le a_1\\le\\cdots\\le a_4$ whose product is $N$. Among those having exactly $2$ even entries, greatest common divisor $1$, and $a_1+\\cdots+a_4\\equiv 4\\pmod{6}$, let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all possible values of $a_4$. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $11^{Z^2+6Z+69}+29^{Z+69}+2519666299$, and write $R$ as a block of exactly ten decimal digits, allowing leading zeros. For $0\\le j<10$, let $R_j$ be the integer obtained by cyclically shifting this block left by $j$ places. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum (j+1)\\gcd(R_j,10^{10}-1)$, where the sum is over the integers $j$ with $0\\le j<10$ and $R_j\\equiv 10\\pmod{19}$.","answer":17,"answer_str":"017","tier":"AIME Hard"}
|
| 30 |
+
{"id":"aimepp-hard-0030","problem":"A sphere $\\Sigma$ is orthogonal to four spheres having, respectively, center $(-8,5,-7)$ and squared radius $\\frac{221}{2}$; center $(-5,-1,4)$ and squared radius $\\frac{133}{2}$; center $(-6,8,-6)$ and squared radius $\\frac{177}{2}$; and center $(1,3,-5)$ and squared radius $\\frac{11}{2}$. If the center of $\\Sigma$ is $(h,k,\\ell)$ and its squared radius is $\\rho$, and $h^2+k^2+\\ell^2+\\rho=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Put $L=Z+31$. Define $G(n,0)=1$, $G(0,k)=0$ for $k>0$, and $G(n,k)=G(n-1,k)+7^{n-k}G(n-1,k-1)$. Find the least nonnegative residue modulo $1000$ of $G(L,9)+3G(L,8)$.","answer":413,"answer_str":"413","tier":"AIME Hard"}
|
| 31 |
+
{"id":"aimepp-hard-0031","problem":"In triangle $ABC$, the side lengths opposite $A,B,C$ are $26,32,12$, respectively. An interior point $P$ has barycentric coordinates $(5:2:2)$, and $Q$ is the isogonal conjugate of $P$. If $PQ^2=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. In the quotient ring $(\\mathbb Z/1000\\mathbb Z)[x]/(x^{5}+5x^{4}+3x^{3}+6x^{2}+5x-3)$, write $(-2x^{4}+3x^{3}-4x^{2}-x-2)^{Z^2+10Z+101}=r_0+r_1x+r_2x^2+r_3x^3+r_4x^4$. Find the least nonnegative residue modulo $1000$ of $-3r_0-5r_1-5r_2+9r_3+4r_4$.","answer":9,"answer_str":"009","tier":"AIME Hard"}
|
| 32 |
+
{"id":"aimepp-hard-0032","problem":"Let $r_1,r_2,r_3,r_4$ be the roots of $P(x)=x^{4} + 7 x^{3} - 3 x^{2} - 3 x + 7$. For $i<j$ define $u_{ij}=r_ir_j+4$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum_{i<j}\\left(u_{ij}^{1282}-7u_{ij}^{1281}\\right)$. Let $L=Z+47$ and $P=\\displaystyle\\prod_{j=1}^L(16j+9)$. Write $P=5^eU$ with $5\\nmid U$, and let $u$ be the least nonnegative residue of $U$ modulo $5^4=625$. Find the least nonnegative residue modulo $1000$ of $u+13e+2(e\\bmod 4)^2$.","answer":945,"answer_str":"945","tier":"AIME Hard"}
|
| 33 |
+
{"id":"aimepp-hard-0033","problem":"Write the base-$7$ expansion of $1/83$ as $0.\\overline{d_1d_2\\cdots d_h}_{7}$ with the shortest possible period, and read subscripts cyclically modulo $h$. For each $j$, let $B_j$ be the base-$7$ integer with digits $d_j,d_{j+1},\\ldots,d_{j+6}$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all $j\\in\\{1,\\ldots,h\\}$ for which $B_j\\equiv 3\\pmod{15}$ and $d_{j+7}+2d_{j+8}+3d_{j+9}+4d_{j+10}\\equiv 6\\pmod{9}$. Put $L=Z+66$. Define $G(n,0)=1$, $G(0,k)=0$ for $k>0$, and $G(n,k)=G(n-1,k)+2^{n-k}G(n-1,k-1)$. Find the least nonnegative residue modulo $1000$ of $G(L,8)+3G(L,7)$.","answer":828,"answer_str":"828","tier":"AIME Hard"}
|
| 34 |
+
{"id":"aimepp-hard-0034","problem":"Consider ordered pairs $(x,y)$ of least positive residues modulo $1296$ satisfying $xy\\equiv 701\\pmod{1296}$, $x+y\\equiv 8\\pmod{14}$, and $x+2y\\equiv 4\\pmod{5}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum(x+3y)$ over all such pairs. Find the least nonnegative residue modulo $1000$ of the number of words of length $Z+60$ over $\\{0,1,2\\}$ that contain none of the blocks $102$, $11$, $201$, have digit sum congruent to $5$ modulo $9$, and contain a number of $2$'s congruent to $1$ modulo $4$.","answer":124,"answer_str":"124","tier":"AIME Hard"}
|
| 35 |
+
{"id":"aimepp-hard-0035","problem":"Let $N=329509$. For all coprime positive integer pairs $(x,y)$ with $x>y$ and $x^2-xy+y^2=N$, consider those satisfying $2x-y\\equiv 7\\pmod{9}$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of $3x+y$ over all such pairs. A Markov chain has transition matrix $\\frac1{10}\\begin{pmatrix}4 & 2 & 2 & 2\\\\2 & 3 & 2 & 3\\\\7 & 1 & 1 & 1\\\\3 & 1 & 3 & 3\\end{pmatrix}$. It starts in state $1$. After $Z+33$ steps, the probability that it is in one of the states $\\{2,4\\}$ is $p/q$ in lowest terms with $q>0$. Find the least nonnegative residue of $p+q$ modulo $1000$.","answer":887,"answer_str":"887","tier":"AIME Hard"}
|
| 36 |
+
{"id":"aimepp-hard-0036","problem":"Let $C_{315}=\\frac1{316}\\binom{630}{315}$ be a Catalan number. Write $C_{315}=3^e u$ with $3\\nmid u$. Let $Z$ be the least nonnegative residue of $u+81e$ modulo $1000$. Define $u_0=402$ and $u_1=422$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 19u_{k+1}-17u_k+7(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+11Z+60}+7u_{2Z+12}$.","answer":192,"answer_str":"192","tier":"AIME Hard"}
|
| 37 |
+
{"id":"aimepp-hard-0037","problem":"Let $p=2017$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$10$ digits is congruent to $7$ modulo $9$, and the sum of the base-$10$ digits of its least positive inverse modulo $p$ is congruent to $3$ modulo $9$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all admissible $g$. Set $T=7^{Z+1}+\\displaystyle\\sum_{k=0}^Z(-1)^k\\binom Zk11^k5^{Z-k}(k^3+7k^2+8k-9)$. Find the least nonnegative residue of $T$ modulo $1000$.","answer":759,"answer_str":"759","tier":"AIME Hard"}
|
| 38 |
+
{"id":"aimepp-hard-0038","problem":"A sphere $\\Sigma$ is orthogonal to four spheres having, respectively, center $(7,5,4)$ and squared radius $\\frac{521}{4}$; center $(6,3,7)$ and squared radius $\\frac{617}{4}$; center $(-6,-3,-6)$ and squared radius $\\frac{133}{4}$; and center $(-6,4,7)$ and squared radius $\\frac{441}{4}$. If the center of $\\Sigma$ is $(h,k,\\ell)$ and its squared radius is $\\rho$, and $h^2+k^2+\\ell^2+\\rho=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Write $Z=100d_2+10d_1+d_0$ with $0\\le d_0,d_1,d_2\\le9$. Let $\\lambda=(13+d_2+d_1+d_0,9+d_1+d_0,5+d_0,4,2)$. Find the least nonnegative residue modulo $1000$ of the number of standard Young tableaux of shape $\\lambda$.","answer":800,"answer_str":"800","tier":"AIME Hard"}
|
| 39 |
+
{"id":"aimepp-hard-0039","problem":"Over the ring $(\\mathbb Z/1000\\mathbb Z)[x]$, let $c_0+c_1x+\\cdots+c_6x^6$ be the remainder when $(x^2+x^3-2)^{110460296634}$ is divided by the monic polynomial $P(x)=x^{7} + x^{6} + 4 x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 2$. Let $Z$ be the least nonnegative residue of $1c_0+7c_1+4c_2+4c_3+9c_4+2c_5+7c_6$ modulo $1000$. Let $m=8+(Z\\bmod 9)$. Form a wheel with hub $0$ and rim vertices $1,2,\\ldots,m$ in cyclic order. Give spoke $0j$ weight $1+((Z+j^2+2j)\\bmod5)$ and rim edge $j(j+1)$ weight $1+((2Z+j+j^2)\\bmod5)$, with $m+1$ interpreted as $1$. The weight of a spanning tree is the product of its edge weights. Find the total weight of all spanning trees modulo $1000$.","answer":864,"answer_str":"864","tier":"AIME Hard"}
|
| 40 |
+
{"id":"aimepp-hard-0040","problem":"Over the ring $(\\mathbb Z/1000\\mathbb Z)[x]$, let $c_0+c_1x+\\cdots+c_6x^6$ be the remainder when $(x^1+x^5+2)^{267703890501}$ is divided by the monic polynomial $P(x)=x^{7} + 5 x^{6} + 6 x^{5} - 6 x^{4} + 2 x^{3} - 5 x^{2} + x + 4$. Let $Z$ be the least nonnegative residue of $9c_0+2c_1+2c_2+4c_3+7c_4+1c_5+4c_6$ modulo $1000$. Write $Z=100d_2+10d_1+d_0$ with $0\\le d_0,d_1,d_2\\le9$. Let $\\lambda=(12+d_2+d_1+d_0,8+d_1+d_0,4+d_0,4,2)$. Find the least nonnegative residue modulo $1000$ of the number of standard Young tableaux of shape $\\lambda$.","answer":720,"answer_str":"720","tier":"AIME Hard"}
|
| 41 |
+
{"id":"aimepp-hard-0041","problem":"Let $\\theta_k=\\frac{2\\pi k}{9}$ for $1\\le k\\le 8$. If $\\displaystyle\\sum_{k=1}^{8}\\left(\\frac1{(8-2\\cos\\theta_k)^2}+\\frac{4}{8-2\\cos\\theta_k}\\right)=\\frac pq$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $p=1889$, set $x_0=Z$, and for $k\\ge0$ let $x_{k+1}$ be the least nonnegative residue modulo $p$ of $x_k^2+268x_k+1184$. Find the least nonnegative residue modulo $1000$ of $x_{10^{12}+7Z+59}+5\\displaystyle\\sum_{j=0}^{Z+54}x_j$.","answer":923,"answer_str":"923","tier":"AIME Hard"}
|
| 42 |
+
{"id":"aimepp-hard-0042","problem":"A plane partition in an $3\\times 5\\times 4$ box is a $3\\times 5$ array $(p_{ij})$ of integers from $0$ through $4$ that is weakly decreasing across every row and down every column. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of such arrays that have total sum congruent to $2$ modulo $5$, diagonal sum $\\sum_{i=1}^{\\min(3,5)}p_{ii}\\equiv 3\\pmod4$, and exactly $0$ entries equal to $4$. Define $u_0=468$ and $u_1=581$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 17u_{k+1}-11u_k+12(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+3Z+85}+7u_{2Z+36}$.","answer":36,"answer_str":"036","tier":"AIME Hard"}
|
| 43 |
+
{"id":"aimepp-hard-0043","problem":"A simple graph on $6$ labeled vertices is chosen uniformly from all connected such graphs. The probability that it has exactly $2$ vertices of odd degree and exactly $4$ triangles is $p/q$ in lowest terms. Let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Find the least nonnegative residue modulo $1000$ of the number of words of length $Z+34$ over $\\{0,1,2\\}$ that contain none of the blocks $010$, $021$, $120$, have digit sum congruent to $3$ modulo $9$, and contain a number of $2$'s congruent to $4$ modulo $5$.","answer":13,"answer_str":"013","tier":"AIME Hard"}
|
| 44 |
+
{"id":"aimepp-hard-0044","problem":"The eight vertices of a cube are colored red, green, or blue, using the colors exactly $2, 5, 1$ times, respectively. Two colorings are considered the same when a rotation of the cube carries one to the other. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of equivalence classes that have exactly $6$ edges whose endpoints have the same color. Let $p=1571$, set $x_0=Z$, and for $k\\ge0$ let $x_{k+1}$ be the least nonnegative residue modulo $p$ of $x_k^2+789x_k+403$. Find the least nonnegative residue modulo $1000$ of $x_{10^{12}+10Z+81}+7\\displaystyle\\sum_{j=0}^{Z+28}x_j$.","answer":276,"answer_str":"276","tier":"AIME Hard"}
|
| 45 |
+
{"id":"aimepp-hard-0045","problem":"Let $Z$ be the least nonnegative residue modulo $1000$ of the number of permutations $\\pi$ of $\\{1,2,\\ldots,9\\}$ that have exactly $3$ cycles in their disjoint-cycle decomposition, exactly $2$ excedances, exactly $1$ fixed points, and major index congruent to $6$ modulo $7$. Put $L=18+(Z\\bmod 13)$ and, for $1\\le j\\le L$, put $a_j=1+((Zj^2+15j+0)\\bmod 15)$. If $[a_1;a_2,\\ldots,a_L]=p_L/q_L$ and $p_{L-1}/q_{L-1}=[a_1;\\ldots,a_{L-1}]$ are in lowest terms, find the least nonnegative residue modulo $1000$ of $p_L+q_L+5p_{L-1}+11q_{L-1}$.","answer":743,"answer_str":"743","tier":"AIME Hard"}
|
| 46 |
+
{"id":"aimepp-hard-0046","problem":"Triangle $ABC$ has side lengths $BC=17$, $CA=12$, and $AB=26$. An interior point $P$ has barycentric coordinates $(6:8:7)$, and $Q$ is the isotomic conjugate of $P$. The perpendiculars from $Q$ to $BC,CA,AB$ meet those lines at $D,E,F$, respectively. If $([DEF]/[ABC])^2=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $p=1297$, let $A$ and $B$ be the least nonnegative residues modulo $p$ of $354Z+733$ and $440Z+973$, respectively, and consider $E: y^2=x^3+Ax+B$ over $\\mathbb F_p$. Including the point at infinity, let $C=\\#E(\\mathbb F_p)$, and let $X$ be the sum, as ordinary integers, of the $x$-coordinates of all affine points of $E$. Find the least nonnegative residue of $C+5X$ modulo $1000$.","answer":687,"answer_str":"687","tier":"AIME Hard"}
|
| 47 |
+
{"id":"aimepp-hard-0047","problem":"A partition of $\\{1,2,\\ldots,9\\}$ has a crossing quadruple when $a<b<c<d$, the numbers $a,c$ lie in one block, and $b,d$ lie in a different block. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of set partitions that have exactly $6$ blocks, exactly $5$ singleton blocks, and a number of crossing quadruples congruent to $0$ modulo $5$. Let $n=7+(Z\\bmod 4)$ and let $M=(m_{ij})_{0\\le i,j<n}$, where $m_{ij}$ is the least nonnegative residue modulo $19$ of $i^2+5ij+j^2+13Z(i+j+1)+14Z+6\\,[i-j\\equiv1\\pmod n]$. Here the bracket equals $1$ when its condition holds and $0$ otherwise. Find the least nonnegative residue of $\\det M$ modulo $1000$.","answer":508,"answer_str":"508","tier":"AIME Hard"}
|
| 48 |
+
{"id":"aimepp-hard-0048","problem":"In triangle $ABC$, the side lengths opposite $A,B,C$ are $33,33,13$, respectively. An interior point $P$ has barycentric coordinates $(5:4:3)$, and $Q$ is the isogonal conjugate of $P$. If $PQ^2=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $A=\\begin{pmatrix}3 & 4 & 1 & 4\\\\2 & 3 & 2 & 2\\\\2 & 2 & 0 & 2\\\\3 & 3 & 2 & 3\\end{pmatrix}$. Interpreting powers over the integers and reducing only the final result, find the least nonnegative residue modulo $1000$ of $\\operatorname{tr}(A^{Z+49})+11(A^{Z^2+38})_{1,4}$.","answer":295,"answer_str":"295","tier":"AIME Hard"}
|
| 49 |
+
{"id":"aimepp-hard-0049","problem":"For a positive integer $t$, let $J_2(t)=t^2\\prod_{p\\mid t}(1-p^{-2})$. Let $N=2012472$. Let $Z$ be the least nonnegative residue modulo $997$ of $\\displaystyle\\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\\equiv 6\\pmod{13}$ and $\\Omega(d)\\equiv 0\\pmod2$. Let $A=\\begin{pmatrix}0 & 0 & 4 & 4\\\\0 & 1 & 2 & 1\\\\2 & 1 & 1 & 1\\\\2 & 3 & 3 & 0\\end{pmatrix}$. Interpreting powers over the integers and reducing only the final result, find the least nonnegative residue modulo $1000$ of $\\operatorname{tr}(A^{Z+33})+5(A^{Z^2+41})_{1,4}$.","answer":334,"answer_str":"334","tier":"AIME Hard"}
|
| 50 |
+
{"id":"aimepp-hard-0050","problem":"Let $a_n$ be defined by $\\displaystyle\\sum_{n\\ge0}a_nx^n=\\frac{(1+x^8)^{5}(1+x^7+x^{14})^{5}}{(1-x^2)^{3}(1-x^5)^{2}}$. Let $Z$ be the least nonnegative residue modulo $983$ of $\\displaystyle\\sum_{\\substack{0\\le n\\le 116\\n\\equiv 6\\pmod{9}}}a_n$. Let $n=7+(Z\\bmod 4)$ and let $M=(m_{ij})_{0\\le i,j<n}$, where $m_{ij}$ is the least nonnegative residue modulo $29$ of $i^2+14ij+j^2+14Z(i+j+1)+4Z+5\\,[i-j\\equiv1\\pmod n]$. Here the bracket equals $1$ when its condition holds and $0$ otherwise. Find the least nonnegative residue of $\\det M$ modulo $1000$.","answer":720,"answer_str":"720","tier":"AIME Hard"}
|
| 51 |
+
{"id":"aimepp-hard-0051","problem":"The graph $C_5\\square P_15$ has $15$ cyclic rows of $5$ vertices; adjacent vertices in each row are joined, including the wraparound pair, and corresponding vertices in consecutive rows are joined. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of independent sets that contain exactly $5$ vertices and leave a number of rows empty that is congruent to $0$ modulo $3$. Define $u_0=248$ and $u_1=999$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 23u_{k+1}-7u_k-5(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+5Z+28}+3u_{2Z+32}$.","answer":36,"answer_str":"036","tier":"AIME Hard"}
|
| 52 |
+
{"id":"aimepp-hard-0052","problem":"A tetrahedron $ABCD$ has squared edge lengths $AB^2=36$, $AC^2=128$, $AD^2=54$, $BC^2=68$, $BD^2=102$, and $CD^2=230$. Let $V$ be its volume and $R$ its circumradius. Write $R^2=p/q$ in lowest terms. Let $Z$ be the least nonnegative residue modulo $1000$ of $288V^2+p+q$. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $7^{Z^2+6Z+79}+31^{Z+79}+8937579493$, and write $R$ as a block of exactly ten decimal digits, allowing leading zeros. For $0\\le j<10$, let $R_j$ be the integer obtained by cyclically shifting this block left by $j$ places. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum (j+1)\\gcd(R_j,10^{10}-1)$, where the sum is over the integers $j$ with $0\\le j<10$ and $R_j\\equiv 1\\pmod{17}$.","answer":30,"answer_str":"030","tier":"AIME Hard"}
|
| 53 |
+
{"id":"aimepp-hard-0053","problem":"Let $p=5843$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$5$ digits is congruent to $4$ modulo $10$, and the sum of the base-$5$ digits of its least positive inverse modulo $p$ is congruent to $5$ modulo $7$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all admissible $g$. Let $m=8+(Z\\bmod 9)$. Form a wheel with hub $0$ and rim vertices $1,2,\\ldots,m$ in cyclic order. Give spoke $0j$ weight $1+((Z+j^2+j)\\bmod5)$ and rim edge $j(j+1)$ weight $1+((2Z+2j+j^2)\\bmod5)$, with $m+1$ interpreted as $1$. The weight of a spanning tree is the product of its edge weights. Find the total weight of all spanning trees modulo $1000$.","answer":320,"answer_str":"320","tier":"AIME Hard"}
|
| 54 |
+
{"id":"aimepp-hard-0054","problem":"A tetrahedron $ABCD$ has squared edge lengths $AB^2=49$, $AC^2=208$, $AD^2=116$, $BC^2=145$, $BD^2=109$, and $CD^2=68$. Let $V$ be its volume and $R$ its circumradius. Write $R^2=p/q$ in lowest terms. Let $Z$ be the least nonnegative residue modulo $1000$ of $288V^2+p+q$. Define $u_0=616$ and $u_1=471$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 17u_{k+1}+9u_k+7(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+5Z+69}+13u_{2Z+27}$.","answer":536,"answer_str":"536","tier":"AIME Hard"}
|
| 55 |
+
{"id":"aimepp-hard-0055","problem":"Let $r_1,r_2,r_3,r_4$ be the complex roots, counted with multiplicity, of $P(x)=x^{4} + x^{3} + x^{2} - 3 x + 3$. For the six numbers $s_{ij}=r_i+r_j$ with $1\\le i<j\\le4$, set $T=\\displaystyle\\sum_{i<j}\\left(\\frac{1}{(11-s_{ij})^2}+4\\frac{1}{11-s_{ij}}\\right)$. If $T=\\frac pq$ in lowest terms with $q>0$, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Define $P(x)=x^{5} - 4 x^{4} + 457 x^{3} + 6 x^{2} - x - 3$ and $Q(x)=x^{4} + 2 x^{3} + 917 x^{2} + 2 x - 8$. Find the least nonnegative residue modulo $1000$ of the resultant $\\operatorname{Res}_x(P,Q)$.","answer":234,"answer_str":"234","tier":"AIME Hard"}
|
| 56 |
+
{"id":"aimepp-hard-0056","problem":"For $0\\le i,j\\le5$, define $m_{ij}=\\binom{37+i+j}{20+2i-j}+4\\binom{37+i+j}{20-i+2j}+2\\binom{37+i+j}{20+i-j-2}$, where a binomial coefficient is $0$ when its lower index is outside its usual range. If $M=(m_{ij})_{0\\le i,j\\le5}$, let $Z$ be the least nonnegative residue of $\\det M$ modulo $991$. Write $Z=100d_2+10d_1+d_0$ with $0\\le d_0,d_1,d_2\\le9$. Let $\\lambda=(16+d_2+d_1+d_0,12+d_1+d_0,8+d_0,4,2)$. Find the least nonnegative residue modulo $1000$ of the number of standard Young tableaux of shape $\\lambda$.","answer":0,"answer_str":"000","tier":"AIME Hard"}
|
| 57 |
+
{"id":"aimepp-hard-0057","problem":"Let $p=1579$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$11$ digits is congruent to $6$ modulo $11$, and the sum of the base-$11$ digits of its least positive inverse modulo $p$ is congruent to $3$ modulo $6$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all admissible $g$. Write $Z=100d_2+10d_1+d_0$ with $0\\le d_0,d_1,d_2\\le9$. Let $\\lambda=(15+d_2+d_1+d_0,11+d_1+d_0,7+d_0,4,2)$. Find the least nonnegative residue modulo $1000$ of the number of standard Young tableaux of shape $\\lambda$.","answer":0,"answer_str":"000","tier":"AIME Hard"}
|
| 58 |
+
{"id":"aimepp-hard-0058","problem":"The sequence $(a_n)$ is defined by $(a_0,a_1,\\ldots,a_5)=(-3, 14, -17, -17, 3, 13)$ and $a_{n+6}=8a_{n+5}-5a_{n+4}+2a_{n+2}+7a_{n+1}+a_n$ for $n\\ge0$. Let $M$ be the $5\\times5$ matrix whose $(i,j)$ entry is $a_{3800044329232+i+j}-3a_{3800044329232+i+j+1}$, where $0\\le i,j\\le4$. Let $Z$ be the least nonnegative residue of $\\det(M)$ modulo $997$. Define $P(x)=x^{5} + 5 x^{4} + 785 x^{3} + 3 x^{2} + x + 2$ and $Q(x)=x^{4} - 8 x^{3} + 1586 x^{2} - 3 x + 21$. Find the least nonnegative residue modulo $1000$ of the resultant $\\operatorname{Res}_x(P,Q)$.","answer":573,"answer_str":"573","tier":"AIME Hard"}
|
| 59 |
+
{"id":"aimepp-hard-0059","problem":"Let $Z$ be the least nonnegative residue modulo $1000$ of the number of length-$42$ words $a_1a_2\\cdots a_42$ over $\\{0,1,2,3\\}$ that contain none of the blocks $00$, $123$, or $232$, contain exactly $38$ occurrences of $3$, satisfy $\\sum_{i=1}^42 i a_i\\equiv 3\\pmod7$, and have a number of indices with $a_{i+1}>a_i$ congruent to $1$ modulo $3$. Let $m=8+(Z\\bmod 9)$. Form a wheel with hub $0$ and rim vertices $1,2,\\ldots,m$ in cyclic order. Give spoke $0j$ weight $1+((Z+j^2+j)\\bmod5)$ and rim edge $j(j+1)$ weight $1+((2Z+7j+j^2)\\bmod5)$, with $m+1$ interpreted as $1$. The weight of a spanning tree is the product of its edge weights. Find the total weight of all spanning trees modulo $1000$.","answer":133,"answer_str":"133","tier":"AIME Hard"}
|
| 60 |
+
{"id":"aimepp-hard-0060","problem":"Consider ordered pairs $(x,y)$ of least positive residues modulo $12960$ satisfying $xy\\equiv 5729\\pmod{12960}$, $x+y\\equiv 6\\pmod{8}$, and $x+2y\\equiv 2\\pmod{9}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum(x+3y)$ over all such pairs. Put $L=18+(Z\\bmod 13)$ and, for $1\\le j\\le L$, put $a_j=1+((Zj^2+4j+9)\\bmod 11)$. If $[a_1;a_2,\\ldots,a_L]=p_L/q_L$ and $p_{L-1}/q_{L-1}=[a_1;\\ldots,a_{L-1}]$ are in lowest terms, find the least nonnegative residue modulo $1000$ of $p_L+q_L+3p_{L-1}+11q_{L-1}$.","answer":534,"answer_str":"534","tier":"AIME Hard"}
|
| 61 |
+
{"id":"aimepp-hard-0061","problem":"The sequence $(a_n)$ is defined by $(a_0,a_1,\\ldots,a_5)=(16, -18, 18, -17, -16, 16)$ and $a_{n+6}=-8a_{n+5}+4a_{n+2}-7a_{n+1}-a_n$ for $n\\ge0$. Let $M$ be the $5\\times5$ matrix whose $(i,j)$ entry is $a_{917841055891+i+j}-4a_{917841055891+i+j+1}$, where $0\\le i,j\\le4$. Let $Z$ be the least nonnegative residue of $\\det(M)$ modulo $997$. Put $L=Z+39$. For an integer $q\\ge2$, let $P_q(L)$ be the number of rotation classes of aperiodic length-$L$ words over a $q$-letter alphabet. Find the least nonnegative residue modulo $1000$ of $P_6(L)+3P_5(L)+\\varphi(L)$, where here $q=6$.","answer":910,"answer_str":"910","tier":"AIME Hard"}
|
| 62 |
+
{"id":"aimepp-hard-0062","problem":"Let $N=1440$. Consider nondecreasing $4$-tuples of integers $2\\le a_1\\le\\cdots\\le a_4$ whose product is $N$. Among those having exactly $2$ even entries, greatest common divisor $1$, and $a_1+\\cdots+a_4\\equiv 7\\pmod{9}$, let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all possible values of $a_4$. In the quotient ring $(\\mathbb Z/1000\\mathbb Z)[x]/(x^{5}+5x^{4}-2x^{3}-6x^{2}-4x+4)$, write $(x^{3}+3x^{2}+x+5)^{Z^2+8Z+63}=r_0+r_1x+r_2x^2+r_3x^3+r_4x^4$. Find the least nonnegative residue modulo $1000$ of $4r_0+2r_1-7r_2-5r_3-3r_4$.","answer":228,"answer_str":"228","tier":"AIME Hard"}
|
| 63 |
+
{"id":"aimepp-hard-0063","problem":"Let $C_{243}=\\frac1{244}\\binom{486}{243}$ be a Catalan number. Write $C_{243}=11^e u$ with $11\\nmid u$. Let $Z$ be the least nonnegative residue of $u+67e$ modulo $1000$. Let $M=3080$ and let $R$ be the least nonnegative residue modulo $M$ of $(10Z+239)^2+1(10Z+239)$. Let $\\mathcal S$ be the set of residues $z\\in\\{0,1,\\ldots,M-1\\}$ satisfying $z^2+1z\\equiv R\\pmod M$, $z\\equiv 4\\pmod{12}$, and $s_7(z)\\equiv 4\\pmod{6}$, where $s_7$ is the base-7 digit sum. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum_{z\\in\\mathcal S}((z+1)^2+3z)$.","answer":236,"answer_str":"236","tier":"AIME Hard"}
|
| 64 |
+
{"id":"aimepp-hard-0064","problem":"Let $p=4597$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$8$ digits is congruent to $1$ modulo $5$, and the sum of the base-$8$ digits of its least positive inverse modulo $p$ is congruent to $0$ modulo $10$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all admissible $g$. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $11^{Z^2+5Z+50}+41^{Z+50}+2268065337$, and write $R$ as a block of exactly ten decimal digits, allowing leading zeros. For $0\\le j<10$, let $R_j$ be the integer obtained by cyclically shifting this block left by $j$ places. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum (j+1)\\gcd(R_j,10^{10}-1)$, where the sum is over the integers $j$ with $0\\le j<10$ and $R_j\\equiv 9\\pmod{11}$.","answer":30,"answer_str":"030","tier":"AIME Hard"}
|
| 65 |
+
{"id":"aimepp-hard-0065","problem":"In triangle $ABC$ with $BC=36$, $CA=36$, and $AB=31$, points $D,E,F$ lie on $BC,CA,AB$, respectively, and satisfy $BD:DC=2:4$, $CE:EA=1:5$, and $AF:FB=3:3$. The circles $(ADE)$, $(BEF)$, and $(CFD)$ have radical center $X$. If the power of $X$ with respect to the circumcircle of $ABC$ is $p_0/q_0$ in lowest terms with $q_0>0$, let $Z$ be the least nonnegative residue of $p_0+q_0$ modulo $1000$. Define $u_0=71$ and $u_1=384$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 13u_{k+1}-7u_k-13(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+8Z+21}+13u_{2Z+21}$.","answer":590,"answer_str":"590","tier":"AIME Hard"}
|
| 66 |
+
{"id":"aimepp-hard-0066","problem":"For a positive integer $t$, let $J_2(t)=t^2\\prod_{p\\mid t}(1-p^{-2})$. Let $N=26583700$. Let $Z$ be the least nonnegative residue modulo $997$ of $\\displaystyle\\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\\equiv 0\\pmod{10}$ and $\\Omega(d)\\equiv 1\\pmod2$. Write $Z=100d_2+10d_1+d_0$ with $0\\le d_0,d_1,d_2\\le9$. Let $\\lambda=(17+d_2+d_1+d_0,13+d_1+d_0,9+d_0,4,2)$. Find the least nonnegative residue modulo $1000$ of the number of standard Young tableaux of shape $\\lambda$.","answer":320,"answer_str":"320","tier":"AIME Hard"}
|
| 67 |
+
{"id":"aimepp-hard-0067","problem":"Let $C_{684}=\\frac1{685}\\binom{1368}{684}$ be a Catalan number. Write $C_{684}=7^e u$ with $7\\nmid u$. Let $Z$ be the least nonnegative residue of $u+70e$ modulo $1000$. Let $p=1657$, set $x_0=Z$, and for $k\\ge0$ let $x_{k+1}$ be the least nonnegative residue modulo $p$ of $x_k^2+939x_k+445$. Find the least nonnegative residue modulo $1000$ of $x_{10^{12}+8Z+50}+3\\displaystyle\\sum_{j=0}^{Z+50}x_j$.","answer":555,"answer_str":"555","tier":"AIME Hard"}
|
| 68 |
+
{"id":"aimepp-hard-0068","problem":"Two walkers move simultaneously on the lattice, each taking one unit north or east step per second. The lower walker goes from $(0,0)$ to $(9,12)$, and the upper walker goes from $(0,2)$ to $(9,14)$. After every equal number of steps, their current vertices must be distinct. A turn is a change of direction by either walker. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ordered pairs of walks of such paths that have exactly $24$ turns in total and have the lower path visit exactly $3$ vertices on the line $y=x$, counting its initial vertex. Put $L=Z+62$. Let $S(n,k)$ be a Stirling number of the second kind, and let $A(n,j)$ be the number of permutations of $\\{1,\\ldots,n\\}$ with exactly $j$ descents. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum_{j=0}^{7}(j+1)S(L+j,8)A(L,j)5^j$.","answer":920,"answer_str":"920","tier":"AIME Hard"}
|
| 69 |
+
{"id":"aimepp-hard-0069","problem":"Let $p=2393$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$7$ digits is congruent to $1$ modulo $9$, and the sum of the base-$7$ digits of its least positive inverse modulo $p$ is congruent to $4$ modulo $6$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all admissible $g$. Put $H=Z+69$. Find the least nonnegative residue modulo $1000$ of the number of ordered pairs $(a,b)$ that satisfy $1\\le a<b\\le H$, $\\gcd(a,b)=1$, $a+b\\equiv 7\\pmod{8}$, and $\\left\\lfloor Ha/b\\right\\rfloor\\equiv 2\\pmod{5}$.","answer":150,"answer_str":"150","tier":"AIME Hard"}
|
| 70 |
+
{"id":"aimepp-hard-0070","problem":"Let $C_{823}=\\frac1{824}\\binom{1646}{823}$ be a Catalan number. Write $C_{823}=3^e u$ with $3\\nmid u$. Let $Z$ be the least nonnegative residue of $u+66e$ modulo $1000$. Put $H=Z+100$. Find the least nonnegative residue modulo $1000$ of the number of ordered pairs $(a,b)$ that satisfy $1\\le a<b\\le H$, $\\gcd(a,b)=1$, $a+b\\equiv 8\\pmod{11}$, and $\\left\\lfloor Ha/b\\right\\rfloor\\equiv 4\\pmod{6}$.","answer":46,"answer_str":"046","tier":"AIME Hard"}
|
| 71 |
+
{"id":"aimepp-hard-0071","problem":"For a positive integer $t$, let $J_2(t)=t^2\\prod_{p\\mid t}(1-p^{-2})$. Let $N=242219250$. Let $Z$ be the least nonnegative residue modulo $997$ of $\\displaystyle\\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\\equiv 0\\pmod{13}$ and $\\Omega(d)\\equiv 0\\pmod2$. Put $L=Z+42$. For an integer $q\\ge2$, let $P_q(L)$ be the number of rotation classes of aperiodic length-$L$ words over a $q$-letter alphabet. Find the least nonnegative residue modulo $1000$ of $P_7(L)+5P_6(L)+\\varphi(L)$, where here $q=7$.","answer":254,"answer_str":"254","tier":"AIME Hard"}
|
| 72 |
+
{"id":"aimepp-hard-0072","problem":"Consider ordered pairs $(x,y)$ of least positive residues modulo $21600$ satisfying $xy\\equiv 20813\\pmod{21600}$, $x+y\\equiv 4\\pmod{10}$, and $x+2y\\equiv 7\\pmod{9}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum(x+3y)$ over all such pairs. In the quotient ring $(\\mathbb Z/1000\\mathbb Z)[x]/(x^{5}-5x^{4}-5x^{3}+3x^{2}-4x-2)$, write $(-x^{3}-2x^{2}-x+4)^{Z^2+5Z+72}=r_0+r_1x+r_2x^2+r_3x^3+r_4x^4$. Find the least nonnegative residue modulo $1000$ of $-7r_0-7r_1-5r_2-3r_3-3r_4$.","answer":728,"answer_str":"728","tier":"AIME Hard"}
|
| 73 |
+
{"id":"aimepp-hard-0073","problem":"For $1\\le k<1680$ with $\\gcd(k,1680)=1$ and $k\\equiv 5\\pmod{7}$, define $w_k=\\gcd(k^2+5k+12,1680)\\gcd(k^3+6,1680)$. Let $Z$ be the least nonnegative residue of $\\sum w_k$ modulo $1000$. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $23^{Z^2+9Z+56}+41^{Z+56}+5823509168$, and write $R$ as a block of exactly ten decimal digits, allowing leading zeros. For $0\\le j<10$, let $R_j$ be the integer obtained by cyclically shifting this block left by $j$ places. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum (j+1)\\gcd(R_j,10^{10}-1)$, where the sum is over the integers $j$ with $0\\le j<10$ and $R_j\\equiv 5\\pmod{13}$.","answer":1,"answer_str":"001","tier":"AIME Hard"}
|
| 74 |
+
{"id":"aimepp-hard-0074","problem":"Let $E$ be the exponent of the prime $17$ in the integer $\\displaystyle\\prod_{k=1}^{120725}(6^k-1)^{\\lfloor 120725/k\\rfloor+\\lfloor k/12\\rfloor}$. Let $Z$ be the least nonnegative residue of $E$ modulo $1000$. Define $u_0=996$ and $u_1=576$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 11u_{k+1}+14u_k-19(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+10Z+39}+13u_{2Z+45}$.","answer":243,"answer_str":"243","tier":"AIME Hard"}
|
| 75 |
+
{"id":"aimepp-hard-0075","problem":"Let $\\theta_k=\\frac{2\\pi k}{11}$ for $1\\le k\\le 10$. If $\\displaystyle\\sum_{k=1}^{10}\\left(\\frac1{(10-2\\cos\\theta_k)^2}+\\frac{6}{10-2\\cos\\theta_k}\\right)=\\frac pq$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $p=2179$, let $A$ and $B$ be the least nonnegative residues modulo $p$ of $1552Z+1051$ and $112Z+1558$, respectively, and consider $E: y^2=x^3+Ax+B$ over $\\mathbb F_p$. Including the point at infinity, let $C=\\#E(\\mathbb F_p)$, and let $X$ be the sum, as ordinary integers, of the $x$-coordinates of all affine points of $E$. Find the least nonnegative residue of $C+2X$ modulo $1000$.","answer":513,"answer_str":"513","tier":"AIME Hard"}
|
| 76 |
+
{"id":"aimepp-hard-0076","problem":"A partition of $\\{1,2,\\ldots,10\\}$ has a crossing quadruple when $a<b<c<d$, the numbers $a,c$ lie in one block, and $b,d$ lie in a different block. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of set partitions that have exactly $5$ blocks, exactly $2$ singleton blocks, and a number of crossing quadruples congruent to $0$ modulo $5$. Put $L=Z+59$. Define $G(n,0)=1$, $G(0,k)=0$ for $k>0$, and $G(n,k)=G(n-1,k)+5^{n-k}G(n-1,k-1)$. Find the least nonnegative residue modulo $1000$ of $G(L,7)+2G(L,6)$.","answer":793,"answer_str":"793","tier":"AIME Hard"}
|
| 77 |
+
{"id":"aimepp-hard-0077","problem":"Let $N=15211$. Consider triples of positive integers $x\\le y\\le z$ satisfying $x^2+y^2+z^2=N$ and $\\gcd(x,y,z)=1$. Among those with $x+y+z\\equiv 5\\pmod{8}$, let $Z$ be the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum(x+2y+3z)$. Set $T=7^{Z+1}+\\displaystyle\\sum_{k=0}^Z(-1)^k\\binom Zk7^k5^{Z-k}(k^3+7k^2-8k-17)$. Find the least nonnegative residue of $T$ modulo $1000$.","answer":711,"answer_str":"711","tier":"AIME Hard"}
|
| 78 |
+
{"id":"aimepp-hard-0078","problem":"A polynomial $F$ of degree at most $9$ has remainder $1958 + 9901(x-2) + 22240(x-2)^{2} + 28856(x-2)^{3}$ upon division by $(x-2)^4$, remainder $-4622 + 21037(x+2) - 42336(x+2)^{2}$ upon division by $(x+2)^3$, and remainder $-2365048 + 5287741(x+4) - 5245496(x+4)^{2}$ upon division by $(x+4)^3$. Let $Z$ be the least nonnegative residue of $F(-7)$ modulo $1000$. Let $L=Z+31$ and $P=\\displaystyle\\prod_{j=1}^L(4j+4)$. Write $P=5^eU$ with $5\\nmid U$, and let $u$ be the least nonnegative residue of $U$ modulo $5^4=625$. Find the least nonnegative residue modulo $1000$ of $u+13e+2(e\\bmod 4)^2$.","answer":398,"answer_str":"398","tier":"AIME Hard"}
|
| 79 |
+
{"id":"aimepp-hard-0079","problem":"Let $a_n$ be defined by $\\displaystyle\\sum_{n\\ge0}a_nx^n=\\frac{(1+x^3)^{3}(1+x^7+x^{14})^{2}}{(1-x^5)^{2}(1-x^4)^{4}}$. Let $Z$ be the least nonnegative residue modulo $997$ of $\\displaystyle\\sum_{\\substack{0\\le n\\le 116\\n\\equiv 4\\pmod{9}}}a_n$. Put $L=Z+74$. For an integer $q\\ge2$, let $P_q(L)$ be the number of rotation classes of aperiodic length-$L$ words over a $q$-letter alphabet. Find the least nonnegative residue modulo $1000$ of $P_8(L)+2P_7(L)+\\varphi(L)$, where here $q=8$.","answer":904,"answer_str":"904","tier":"AIME Hard"}
|
| 80 |
+
{"id":"aimepp-hard-0080","problem":"For a positive integer $t$, let $J_2(t)=t^2\\prod_{p\\mid t}(1-p^{-2})$. Let $N=50611275$. Let $Z$ be the least nonnegative residue modulo $997$ of $\\displaystyle\\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\\equiv 4\\pmod{7}$ and $\\Omega(d)\\equiv 1\\pmod2$. Define $u_0=473$ and $u_1=160$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 11u_{k+1}+5u_k+7(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+9Z+82}+7u_{2Z+34}$.","answer":636,"answer_str":"636","tier":"AIME Hard"}
|
| 81 |
+
{"id":"aimepp-hard-0081","problem":"Two walkers move simultaneously on the lattice, each taking one unit north or east step per second. The lower walker goes from $(0,0)$ to $(8,7)$, and the upper walker goes from $(0,2)$ to $(8,9)$. After every equal number of steps, their current vertices must be distinct. A turn is a change of direction by either walker. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ordered pairs of walks of such paths that have exactly $10$ turns in total and have the lower path visit exactly $5$ vertices on the line $y=x$, counting its initial vertex. Let $A=\\begin{pmatrix}4 & 0 & 1 & 3\\\\2 & 4 & 4 & 3\\\\0 & 3 & 3 & 1\\\\1 & 2 & 4 & 4\\end{pmatrix}$. Interpreting powers over the integers and reducing only the final result, find the least nonnegative residue modulo $1000$ of $\\operatorname{tr}(A^{Z+17})+2(A^{Z^2+70})_{1,4}$.","answer":130,"answer_str":"130","tier":"AIME Hard"}
|
| 82 |
+
{"id":"aimepp-hard-0082","problem":"The eight vertices of a cube are colored red, green, or blue, using the colors exactly $3, 2, 3$ times, respectively. Two colorings are considered the same when a rotation of the cube carries one to the other. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of equivalence classes that have exactly $5$ edges whose endpoints have the same color. Put $L=Z+23$. Let $S(n,k)$ be a Stirling number of the second kind, and let $A(n,j)$ be the number of permutations of $\\{1,\\ldots,n\\}$ with exactly $j$ descents. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum_{j=0}^{7}(j+1)S(L+j,8)A(L,j)7^j$.","answer":406,"answer_str":"406","tier":"AIME Hard"}
|
| 83 |
+
{"id":"aimepp-hard-0083","problem":"The eight vertices of a cube are colored red, green, or blue, using the colors exactly $4, 3, 1$ times, respectively. Two colorings are considered the same when a rotation of the cube carries one to the other. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of equivalence classes that have exactly $3$ edges whose endpoints have the same color. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $19^{Z^2+5Z+52}+29^{Z+52}+3143072797$, and write $R$ as a block of exactly ten decimal digits, allowing leading zeros. For $0\\le j<10$, let $R_j$ be the integer obtained by cyclically shifting this block left by $j$ places. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum (j+1)\\gcd(R_j,10^{10}-1)$, where the sum is over the integers $j$ with $0\\le j<10$ and $R_j\\equiv 2\\pmod{13}$.","answer":75,"answer_str":"075","tier":"AIME Hard"}
|
| 84 |
+
{"id":"aimepp-hard-0084","problem":"Let $\\theta_k=\\frac{2\\pi k}{23}$ for $1\\le k\\le 22$. If $\\displaystyle\\sum_{k=1}^{22}\\left(\\frac1{(3-2\\cos\\theta_k)^2}+\\frac{5}{3-2\\cos\\theta_k}\\right)=\\frac pq$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. In the quotient ring $(\\mathbb Z/1000\\mathbb Z)[x]/(x^{5}+6x^{4}-6x^{3}-3x^{2}+3x+2)$, write $(-5x^{4}-2x^{3}-3x^{2}+2x-2)^{Z^2+6Z+118}=r_0+r_1x+r_2x^2+r_3x^3+r_4x^4$. Find the least nonnegative residue modulo $1000$ of $-3r_0+4r_1+6r_2+9r_3+2r_4$.","answer":814,"answer_str":"814","tier":"AIME Hard"}
|
| 85 |
+
{"id":"aimepp-hard-0085","problem":"The sequence $(a_n)$ is defined by $(a_0,a_1,\\ldots,a_5)=(-10, -15, -7, -7, -6, -10)$ and $a_{n+6}=6a_{n+5}-3a_{n+4}-6a_{n+2}-6a_n$ for $n\\ge0$. Let $M$ be the $5\\times5$ matrix whose $(i,j)$ entry is $a_{1024512110523+i+j}-2a_{1024512110523+i+j+1}$, where $0\\le i,j\\le4$. Let $Z$ be the least nonnegative residue of $\\det(M)$ modulo $997$. Put $H=Z+105$. Find the least nonnegative residue modulo $1000$ of the number of ordered pairs $(a,b)$ that satisfy $1\\le a<b\\le H$, $\\gcd(a,b)=1$, $a+b\\equiv 1\\pmod{7}$, and $\\left\\lfloor Ha/b\\right\\rfloor\\equiv 1\\pmod{4}$.","answer":37,"answer_str":"037","tier":"AIME Hard"}
|
| 86 |
+
{"id":"aimepp-hard-0086","problem":"Triangle $ABC$ has side lengths $BC=12$, $CA=25$, and $AB=22$. An interior point $P$ has barycentric coordinates $(5:5:1)$, and $Q$ is the isotomic conjugate of $P$. The perpendiculars from $Q$ to $BC,CA,AB$ meet those lines at $D,E,F$, respectively. If $([DEF]/[ABC])^2=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $19^{Z^2+10Z+60}+29^{Z+60}+7821323505$, and write $R$ as a block of exactly ten decimal digits, allowing leading zeros. For $0\\le j<10$, let $R_j$ be the integer obtained by cyclically shifting this block left by $j$ places. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum (j+1)\\gcd(R_j,10^{10}-1)$, where the sum is over the integers $j$ with $0\\le j<10$ and $R_j\\equiv 1\\pmod{7}$.","answer":330,"answer_str":"330","tier":"AIME Hard"}
|
| 87 |
+
{"id":"aimepp-hard-0087","problem":"A simple graph on $6$ labeled vertices is chosen uniformly from all connected such graphs. The probability that it has exactly $2$ vertices of odd degree and exactly $8$ triangles is $p/q$ in lowest terms. Let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $A=\\begin{pmatrix}0 & 0 & 2 & 0\\\\2 & 3 & 2 & 3\\\\1 & 4 & 1 & 3\\\\4 & 2 & 1 & 1\\end{pmatrix}$. Interpreting powers over the integers and reducing only the final result, find the least nonnegative residue modulo $1000$ of $\\operatorname{tr}(A^{Z+20})+11(A^{Z^2+58})_{1,4}$.","answer":115,"answer_str":"115","tier":"AIME Hard"}
|
| 88 |
+
{"id":"aimepp-hard-0088","problem":"In triangle $ABC$ with $BC=33$, $CA=40$, and $AB=18$, points $D,E,F$ lie on $BC,CA,AB$, respectively, and satisfy $BD:DC=7:2$, $CE:EA=5:5$, and $AF:FB=2:2$. The circles $(ADE)$, $(BEF)$, and $(CFD)$ have radical center $X$. If the power of $X$ with respect to the circumcircle of $ABC$ is $p_0/q_0$ in lowest terms with $q_0>0$, let $Z$ be the least nonnegative residue of $p_0+q_0$ modulo $1000$. A Markov chain has transition matrix $\\frac1{12}\\begin{pmatrix}1 & 9 & 1 & 1\\\\5 & 2 & 4 & 1\\\\2 & 2 & 3 & 5\\\\3 & 6 & 1 & 2\\end{pmatrix}$. It starts in state $4$. After $Z+88$ steps, the probability that it is in one of the states $\\{3,4\\}$ is $p/q$ in lowest terms with $q>0$. Find the least nonnegative residue of $p+q$ modulo $1000$.","answer":385,"answer_str":"385","tier":"AIME Hard"}
|
| 89 |
+
{"id":"aimepp-hard-0089","problem":"A circular ladder has two disjoint $8$-cycles, with corresponding vertices joined by $8$ rungs. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of matchings that contain exactly $8$ edges, exactly $4$ rungs, and a number of edges from the first cycle congruent to $2$ modulo $3$. Put $L=18+(Z\\bmod 13)$ and, for $1\\le j\\le L$, put $a_j=1+((Zj^2+2j+10)\\bmod 15)$. If $[a_1;a_2,\\ldots,a_L]=p_L/q_L$ and $p_{L-1}/q_{L-1}=[a_1;\\ldots,a_{L-1}]$ are in lowest terms, find the least nonnegative residue modulo $1000$ of $p_L+q_L+5p_{L-1}+3q_{L-1}$.","answer":763,"answer_str":"763","tier":"AIME Hard"}
|
| 90 |
+
{"id":"aimepp-hard-0090","problem":"For $1\\le k<5400$ with $\\gcd(k,5400)=1$ and $k\\equiv 6\\pmod{7}$, define $w_k=\\gcd(k^2+7k+14,5400)\\gcd(k^3+5,5400)$. Let $Z$ be the least nonnegative residue of $\\sum w_k$ modulo $1000$. Let $p=1489$, set $x_0=Z$, and for $k\\ge0$ let $x_{k+1}$ be the least nonnegative residue modulo $p$ of $x_k^2+1400x_k+597$. Find the least nonnegative residue modulo $1000$ of $x_{10^{12}+9Z+78}+7\\displaystyle\\sum_{j=0}^{Z+35}x_j$.","answer":545,"answer_str":"545","tier":"AIME Hard"}
|
| 91 |
+
{"id":"aimepp-hard-0091","problem":"Let $N=378000$. Consider nondecreasing $4$-tuples of integers $2\\le a_1\\le\\cdots\\le a_4$ whose product is $N$. Among those having exactly $2$ even entries, greatest common divisor $1$, and $a_1+\\cdots+a_4\\equiv 4\\pmod{12}$, let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all possible values of $a_4$. A Markov chain has transition matrix $\\frac1{12}\\begin{pmatrix}1 & 1 & 7 & 3\\\\1 & 3 & 5 & 3\\\\3 & 2 & 6 & 1\\\\2 & 6 & 3 & 1\\end{pmatrix}$. It starts in state $4$. After $Z+88$ steps, the probability that it is in one of the states $\\{1,4\\}$ is $p/q$ in lowest terms with $q>0$. Find the least nonnegative residue of $p+q$ modulo $1000$.","answer":511,"answer_str":"511","tier":"AIME Hard"}
|
| 92 |
+
{"id":"aimepp-hard-0092","problem":"Let $F(x)=P(x^2+2x+2)$, where $P(x)=x^{4} + 4 x^{3} - 4 x^{2} + x + 3$. The discriminant of $F$ is an integer $D$. Let $Z$ be the least nonnegative residue of $D$ modulo $983$. Put $L=Z+57$. Let $S(n,k)$ be a Stirling number of the second kind, and let $A(n,j)$ be the number of permutations of $\\{1,\\ldots,n\\}$ with exactly $j$ descents. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum_{j=0}^{6}(j+1)S(L+j,7)A(L,j)3^j$.","answer":640,"answer_str":"640","tier":"AIME Hard"}
|
| 93 |
+
{"id":"aimepp-hard-0093","problem":"A sphere $\\Sigma$ is orthogonal to four spheres having, respectively, center $(6,4,2)$ and squared radius $\\frac{213}{2}$; center $(1,0,2)$ and squared radius $\\frac{71}{2}$; center $(5,7,-6)$ and squared radius $\\frac{239}{2}$; and center $(-3,-8,-6)$ and squared radius $\\frac{87}{2}$. If the center of $\\Sigma$ is $(h,k,\\ell)$ and its squared radius is $\\rho$, and $h^2+k^2+\\ell^2+\\rho=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Define $u_0=445$ and $u_1=369$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 19u_{k+1}+5u_k-19(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+5Z+33}+13u_{2Z+34}$.","answer":588,"answer_str":"588","tier":"AIME Hard"}
|
| 94 |
+
{"id":"aimepp-hard-0094","problem":"Let $F(x)=P(x^2+3x+0)$, where $P(x)=x^{4} - 4 x^{3} - 2 x^{2} + 4 x - 1$. The discriminant of $F$ is an integer $D$. Let $Z$ be the least nonnegative residue of $D$ modulo $991$. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $17^{Z^2+7Z+99}+41^{Z+99}+6210865758$, and write $R$ as a block of exactly ten decimal digits, allowing leading zeros. For $0\\le j<10$, let $R_j$ be the integer obtained by cyclically shifting this block left by $j$ places. Find the least nonnegative residue modulo $1000$ of $\\displaystyle\\sum (j+1)\\gcd(R_j,10^{10}-1)$, where the sum is over the integers $j$ with $0\\le j<10$ and $R_j\\equiv 10\\pmod{13}$.","answer":36,"answer_str":"036","tier":"AIME Hard"}
|
| 95 |
+
{"id":"aimepp-hard-0095","problem":"A simple graph on $6$ labeled vertices is chosen uniformly from all connected such graphs. The probability that it has exactly $6$ vertices of odd degree and exactly $0$ triangles is $p/q$ in lowest terms. Let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Put $L=Z+27$. Define $G(n,0)=1$, $G(0,k)=0$ for $k>0$, and $G(n,k)=G(n-1,k)+4^{n-k}G(n-1,k-1)$. Find the least nonnegative residue modulo $1000$ of $G(L,12)+7G(L,11)$.","answer":328,"answer_str":"328","tier":"AIME Hard"}
|
| 96 |
+
{"id":"aimepp-hard-0096","problem":"Two walkers move simultaneously on the lattice, each taking one unit north or east step per second. The lower walker goes from $(0,0)$ to $(10,7)$, and the upper walker goes from $(0,2)$ to $(10,9)$. After every equal number of steps, their current vertices must be distinct. A turn is a change of direction by either walker. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ordered pairs of walks of such paths that have exactly $12$ turns in total and have the lower path visit exactly $6$ vertices on the line $y=x$, counting its initial vertex. Put $H=Z+85$. Find the least nonnegative residue modulo $1000$ of the number of ordered pairs $(a,b)$ that satisfy $1\\le a<b\\le H$, $\\gcd(a,b)=1$, $a+b\\equiv 0\\pmod{10}$, and $\\left\\lfloor Ha/b\\right\\rfloor\\equiv 4\\pmod{6}$.","answer":268,"answer_str":"268","tier":"AIME Hard"}
|
| 97 |
+
{"id":"aimepp-hard-0097","problem":"For $1\\le k<8400$ with $\\gcd(k,8400)=1$ and $k\\equiv 7\\pmod{12}$, define $w_k=\\gcd(k^2+9k+8,8400)\\gcd(k^3+5,8400)$. Let $Z$ be the least nonnegative residue of $\\sum w_k$ modulo $1000$. Put $H=Z+89$. Find the least nonnegative residue modulo $1000$ of the number of ordered pairs $(a,b)$ that satisfy $1\\le a<b\\le H$, $\\gcd(a,b)=1$, $a+b\\equiv 0\\pmod{7}$, and $\\left\\lfloor Ha/b\\right\\rfloor\\equiv 1\\pmod{4}$.","answer":628,"answer_str":"628","tier":"AIME Hard"}
|
| 98 |
+
{"id":"aimepp-hard-0098","problem":"Let $r_1,r_2,r_3,r_4$ be the roots of $P(x)=x^{4} - 4 x^{3} - 3 x^{2} - 4 x - 5$. For each unordered pair $i<j$, put $t_{ij}=r_ir_j$. If $\\displaystyle\\sum_{i<j}\\frac{7+t_{ij}}{(13-t_{ij})^2}=\\frac pq$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Define $u_0=27$ and $u_1=888$, and for $k\\ge 0$ define $u_{k+2}$ by $u_{k+2}\\equiv 11u_{k+1}+21u_k-5(-1)^k\\pmod{1000}$, with every $u_k$ chosen from $\\{0,1,\\ldots,999\\}$. Find the least nonnegative residue modulo $1000$ of $u_{Z^2+3Z+49}+11u_{2Z+21}$.","answer":910,"answer_str":"910","tier":"AIME Hard"}
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data/aime-researcher.jsonl
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{"id":"aimepp-researcher-0001","problem":"Let $\\operatorname{RM}(2,5)\\subset\\mathbb F_2^{32}$ be the binary Reed--Muller code obtained by evaluating all squarefree polynomials in $x_1,\\ldots,x_5$ of total degree at most $2$ at the points of $\\mathbb F_2^5$. Let $S=\\{x\\in\\mathbb F_2^5:x_4=x_5=0\\}$, and let $C$ be the code obtained by shortening $\\operatorname{RM}(2,5)$ on the eight coordinates indexed by $S$: retain the codewords that vanish on every coordinate in $S$, then delete those coordinates. Put $D=C^\\perp\\subset\\mathbb F_2^{24}$. For a binary linear code $E$ of length $24$, let $A_w(E)$ be the number of its codewords of Hamming weight $w$, and define its undetected-error probability on the binary symmetric channel with crossover probability $\\varepsilon$ by $U_\\varepsilon(E)=\\displaystyle\\sum_{w=1}^{24}A_w(E)\\varepsilon^w(1-\\varepsilon)^{24-w}$. If $U_{3/11}(C)+U_{2/7}(D)=p/q$ in lowest terms with $q>0$, find the least nonnegative residue of $p+q$ modulo $1000$.","answer":534,"answer_str":"534","tier":"AIME-Researcher"}
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| 2 |
+
{"id":"aimepp-researcher-0002","problem":"Let $A=\\begin{pmatrix}0&0&0&0&0&-1\\\\1&0&0&0&0&-7\\\\0&1&0&0&0&5\\\\0&0&1&0&0&-4\\\\0&0&0&1&0&-6\\\\0&0&0&0&1&3\\end{pmatrix}\\in\\mathrm{SL}_6(\\mathbb Z)$, and let $f_A:\\mathbb T^6\\to\\mathbb T^6$ be the induced torus automorphism. Let $M_A=(\\mathbb T^6\\times[0,1])/((x,1)\\sim(f_A(x),0))$ be its mapping torus. Define $\\alpha_A:C(\\mathbb T^6)\\to C(\\mathbb T^6)$ by $\\alpha_A(g)=g\\circ f_A^{-1}$, and put $\\mathcal B_A=C(\\mathbb T^6)\\rtimes_{\\alpha_A}\\mathbb Z$. For a finitely generated abelian group $G$, write $\\tau(G)=|\\operatorname{Tor}G|$. Set $T_i=\\tau(K_i(\\mathcal B_A))$ for $i=0,1$, and $h_j=\\tau(H_j(M_A;\\mathbb Z))$ for $1\\le j\\le5$. Find the least nonnegative residue modulo $1000$ of $T_0+3T_1+7h_2h_3+11h_1h_5$.","answer":302,"answer_str":"302","tier":"AIME-Researcher"}
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| 3 |
+
{"id":"aimepp-researcher-0003","problem":"Let $V=\\mathbb F_2^6$ and $\\Omega=\\operatorname{Gr}(3,V)$. Matrices act on column vectors, and each $U\\in\\Omega$ has two successors $A(U)$ and $B(U)$, where $A=\\begin{pmatrix}1&0&1&1&1&1\\\\1&0&1&0&0&1\\\\1&0&0&0&1&0\\\\0&0&0&0&1&0\\\\1&1&0&1&1&0\\\\0&0&0&1&1&1\\end{pmatrix}$ and $B=\\begin{pmatrix}0&0&1&0&0&0\\\\0&0&1&0&0&1\\\\1&0&0&0&0&0\\\\1&0&0&0&1&1\\\\0&1&0&0&0&0\\\\1&1&1&1&0&0\\end{pmatrix}$ in $\\mathrm{GL}_6(\\mathbb F_2)$. Put $L_0=\\langle e_1,e_2,e_3\\rangle$, $L_1=\\langle e_3,e_4,e_5\\rangle$, and $L_2=\\langle e_2,e_5,e_6\\rangle$. For $U\\in\\Omega$, let $\\pi(U)$ be the residue modulo $3$ of $\\dim(U\\cap L_0)+2\\dim(U\\cap L_1)+\\dim(U\\cap L_2)$, and set $P_i=\\{U:\\pi(U)=i\\}$. Let $\\mathcal E=\\{U:\\dim(U\\cap L_0)+\\dim(U\\cap L_2)\\equiv0\\pmod2\\}$. For $S\\subseteq\\Omega$, define $\\operatorname{CPre}(S)=\\{U\\in\\mathcal E:A(U)\\in S\\text{ or }B(U)\\in S\\}\\cup\\{U\\notin\\mathcal E:A(U)\\in S\\text{ and }B(U)\\in S\\}$. In the complete lattice $2^\\Omega$, let $W$ be the subset denoted by the modal $\\mu$-calculus formula $\\nu X.\\,\\mu Y.\\,\\nu Z.\\,\\bigl((P_0\\cap\\operatorname{CPre}(X))\\cup(P_1\\cap\\operatorname{CPre}(Y))\\cup(P_2\\cap\\operatorname{CPre}(Z))\\bigr)$, where $\\mu$ and $\\nu$ denote least and greatest fixed points. Find the least nonnegative residue of $|W|$ modulo $1000$.","answer":715,"answer_str":"715","tier":"AIME-Researcher"}
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| 4 |
+
{"id":"aimepp-researcher-0004","problem":"In the $\\mathrm{SU}(2)$ Wess--Zumino--Witten model at level $23$, let $\\mathcal V_g(a_1,\\ldots,a_n)$ be the genus-$g$ conformal-block space with labels $a_i\\in\\{0,1,\\ldots,23\\}$, whose dimension is given by the Verlinde formula $\\displaystyle\\dim\\mathcal V_g(a_1,\\ldots,a_n)=\\sum_{s=0}^{23}(S_{0s})^{\\,2-2g-n}\\prod_{i=1}^n S_{a_i s}$, where $\\displaystyle S_{as}=\\sqrt{\\frac{2}{25}}\\sin\\!\\left(\\frac{(a+1)(s+1)\\pi}{25}\\right)$. Put $D=\\dim\\mathcal V_7(1,4,7,10,13,16,19,22)+3\\dim\\mathcal V_6(2,5,8,11,14,17,20,23)$. Write $D=2^\\alpha5^\\beta u$ with $\\gcd(u,10)=1$. Find the least nonnegative residue of $u+7\\alpha+11\\beta$ modulo $1000$.","answer":579,"answer_str":"579","tier":"AIME-Researcher"}
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| 5 |
+
{"id":"aimepp-researcher-0005","problem":"Let $V=\\mathbb F_2^7$ and $X=\\displaystyle\\bigcup_{r\\in\\{1,2,4,6\\}}\\operatorname{Gr}(r,V)$, equipped with the subspace metric $d(U,W)=\\dim U+\\dim W-2\\dim(U\\cap W)$. For $t\\in\\{1/3,1/5\\}$, let $w_t:X\\to\\mathbb Q$ be the unique $\\mathrm{GL}(V)$-invariant weighting satisfying $\\displaystyle\\sum_{W\\in X}t^{d(U,W)}w_t(W)=1$ for every $U\\in X$, and define the magnitude $\\operatorname{Mag}_t(X)=\\displaystyle\\sum_{U\\in X}w_t(U)$. If $\\operatorname{Mag}_{1/3}(X)/\\operatorname{Mag}_{1/5}(X)=p/q$ in lowest terms with $q>0$, find the least nonnegative residue of $p+q$ modulo $1000$.","answer":571,"answer_str":"571","tier":"AIME-Researcher"}
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data/aime.jsonl
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+
{"id":"aimepp-aime-0001","problem":"Let $X,Y,Z>1$ and $W>1$ satisfy $\\log_X W=54$, $\\log_Y W=24$, and $\\log_{X^{3}Y^{2}Z^{2}}W=6$. Determine $\\log_Z W$.","answer":72,"answer_str":"072","tier":"AIME"}
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| 2 |
+
{"id":"aimepp-aime-0002","problem":"For real $x$ with $-8\\le x\\le 25$, define $F(x)=|x+4|+2|x-10|+|x-12|+2|x-21|$. Find the minimum value of $F(x)$.","answer":38,"answer_str":"038","tier":"AIME"}
|
| 3 |
+
{"id":"aimepp-aime-0003","problem":"Find the product of all real roots of \\[x^2-5x+15=8\\sqrt{x^2-5x}\\].","answer":225,"answer_str":"225","tier":"AIME"}
|
| 4 |
+
{"id":"aimepp-aime-0004","problem":"A circle has radius $25$. Two parallel chords of lengths $48$ and $30$ lie on opposite sides of the center. Find the distance between the chords.","answer":27,"answer_str":"027","tier":"AIME"}
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| 5 |
+
{"id":"aimepp-aime-0005","problem":"Complex numbers $x$ and $y$ satisfy $x^2+y^2=91$ and $x^3+y^3=836$. Find the largest real value that $x+y$ can have.","answer":11,"answer_str":"011","tier":"AIME"}
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| 6 |
+
{"id":"aimepp-aime-0006","problem":"A sequence is defined by $a_1=8$, $a_2=27$, and $a_n=3a_{n-1}+6a_{n-2}$ for $n\\ge3$. Find the remainder when $a_{829}$ is divided by $1000$.","answer":293,"answer_str":"293","tier":"AIME"}
|
| 7 |
+
{"id":"aimepp-aime-0007","problem":"Of $31$ people seated around a circular table, three are chosen uniformly at random. The probability that at least two chosen people were adjacent is $\\frac{p}{q}$ in lowest terms. Find $p+q$.","answer":173,"answer_str":"173","tier":"AIME"}
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| 8 |
+
{"id":"aimepp-aime-0008","problem":"Find the largest two-digit prime divisor of $\\binom{164}{82}$.","answer":97,"answer_str":"097","tier":"AIME"}
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| 9 |
+
{"id":"aimepp-aime-0009","problem":"Find the minimum value of $\\dfrac{36x^2\\sin^2x+25}{x\\sin x}$ for $0<x<\\pi$.","answer":60,"answer_str":"060","tier":"AIME"}
|
| 10 |
+
{"id":"aimepp-aime-0010","problem":"Consider $5$-digit decimal integers beginning with $8$ that have exactly one pair of equal digits, with every other digit occurring once. Find the remainder when the number of such integers is divided by $1000$.","answer":40,"answer_str":"040","tier":"AIME"}
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| 11 |
+
{"id":"aimepp-aime-0011","problem":"A rectangular box has side lengths $23$, $25$, and $h$. The centers of the three faces meeting at one vertex form a triangle of area $\\frac{579}{8}$. Find $h$.","answer":2,"answer_str":"002","tier":"AIME"}
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| 12 |
+
{"id":"aimepp-aime-0012","problem":"In a circle, a diameter has the two-digit base-$10$ length $\\overline{ab}_{10}$, and a perpendicular chord has length $\\overline{ba}_{10}$. The diameter is longer, the two digits sum to $11$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$.","answer":65,"answer_str":"065","tier":"AIME"}
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| 13 |
+
{"id":"aimepp-aime-0013","problem":"For every nonempty subset of $\\{1,2,\\ldots,14\\}$, list its elements in decreasing order and alternately add and subtract, beginning with addition. Find the remainder when the sum of these alternating sums is divided by $1000$.","answer":688,"answer_str":"688","tier":"AIME"}
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| 14 |
+
{"id":"aimepp-aime-0014","problem":"In a circle, a diameter has the two-digit base-$9$ length $\\overline{ab}_{9}$, and a perpendicular chord has length $\\overline{ba}_{9}$. The diameter is longer, the two digits sum to $9$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$.","answer":65,"answer_str":"065","tier":"AIME"}
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| 15 |
+
{"id":"aimepp-aime-0015","problem":"In a circle, a diameter has the two-digit base-$8$ length $\\overline{ab}_{8}$, and a perpendicular chord has length $\\overline{ba}_{8}$. The diameter is longer, the two digits sum to $7$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$.","answer":35,"answer_str":"035","tier":"AIME"}
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| 16 |
+
{"id":"aimepp-aime-0016","problem":"The $8$ positive integer terms of an arithmetic sequence have total $420$, and the greatest term is $2$ times the least term. Find the greatest term.","answer":70,"answer_str":"070","tier":"AIME"}
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| 17 |
+
{"id":"aimepp-aime-0017","problem":"How many odd integers from $250$ through $999$ have no repeated decimal digit and have digit sum divisible by $5$?","answer":54,"answer_str":"054","tier":"AIME"}
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| 18 |
+
{"id":"aimepp-aime-0018","problem":"Find the area of a triangle with side lengths $10,13,13$.","answer":60,"answer_str":"060","tier":"AIME"}
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| 19 |
+
{"id":"aimepp-aime-0019","problem":"Positive integers $a\\le b$ satisfy $\\gcd(a,b)=3$ and $\\operatorname{lcm}(a,b)=228$. Find the sum of all possible values of $a+b$.","answer":300,"answer_str":"300","tier":"AIME"}
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| 20 |
+
{"id":"aimepp-aime-0020","problem":"Let $X,Y,Z>1$ and $W>1$ satisfy $\\log_X W=792$, $\\log_Y W=352$, and $\\log_{X^{3}YZ^{2}}W=96$. Determine $\\log_Z W$.","answer":528,"answer_str":"528","tier":"AIME"}
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| 21 |
+
{"id":"aimepp-aime-0021","problem":"For real $x$ with $-21\\le x\\le 26$, define $F(x)=2|x+17|+3|x-8|+|x-18|+3|x-22|$. Find the minimum value of $F(x)$.","answer":102,"answer_str":"102","tier":"AIME"}
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| 22 |
+
{"id":"aimepp-aime-0022","problem":"A polynomial $P$ has degree at most $4$ and satisfies $P(0)=1$, $P(1)=-1$, $P(2)=-47$, $P(3)=-257$, $P(4)=-823$. Find the remainder when $P(5)$ is divided by $1000$.","answer":991,"answer_str":"991","tier":"AIME"}
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| 23 |
+
{"id":"aimepp-aime-0023","problem":"The nonreal roots of $z^2-7z+24=0$ are $\\alpha$ and $\\beta$. Find the remainder when $\\alpha^{36}+\\beta^{36}$ is divided by $1000$.","answer":977,"answer_str":"977","tier":"AIME"}
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| 24 |
+
{"id":"aimepp-aime-0024","problem":"A rectangular box has side lengths $3$, $8$, and $h$. The centers of the three faces meeting at one vertex form a triangle of area $\\frac{61}{4}$. Find $h$.","answer":14,"answer_str":"014","tier":"AIME"}
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| 25 |
+
{"id":"aimepp-aime-0025","problem":"How many pairs of positive integers $(x,y)$ with $x\\le y$ satisfy $\\frac1x+\\frac1y=\\frac1{147}$?","answer":8,"answer_str":"008","tier":"AIME"}
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| 26 |
+
{"id":"aimepp-aime-0026","problem":"Three distinct vertices of a regular $10$-gon are chosen uniformly at random. The probability that they form an obtuse triangle is $\\frac{m}{q}$ in lowest terms. Find $m+q$.","answer":3,"answer_str":"003","tier":"AIME"}
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| 27 |
+
{"id":"aimepp-aime-0027","problem":"How many subsets $S$ of $\\{1,2,\\ldots,10\\}$ satisfy $\\sum_{s\\in S}s\\equiv 4\\pmod{7}$? The empty set is allowed.","answer":146,"answer_str":"146","tier":"AIME"}
|
| 28 |
+
{"id":"aimepp-aime-0028","problem":"Angles $u$ and $v$ satisfy $\\tan u+\\tan v=34$ and $\\cot u+\\cot v=68$. Find $\\tan(u+v)$.","answer":68,"answer_str":"068","tier":"AIME"}
|
| 29 |
+
{"id":"aimepp-aime-0029","problem":"Find the smallest positive integer whose cube has last three digits $237$.","answer":933,"answer_str":"933","tier":"AIME"}
|
| 30 |
+
{"id":"aimepp-aime-0030","problem":"A faculty has three departments: department 1 has 2 men and 3 women; department 2 has 3 men and 2 women; department 3 has 3 men and 3 women. A six-person committee must contain exactly two people from each department and exactly three men. How many committees are possible?","answer":558,"answer_str":"558","tier":"AIME"}
|
| 31 |
+
{"id":"aimepp-aime-0031","problem":"A $6\\times 5\\times 9$ rectangular block is painted on all six faces and cut into unit cubes. How many unit cubes have exactly two painted faces?","answer":56,"answer_str":"056","tier":"AIME"}
|
| 32 |
+
{"id":"aimepp-aime-0032","problem":"A faculty has three departments: department 1 has 2 men and 3 women; department 2 has 3 men and 2 women; department 3 has 4 men and 2 women. A six-person committee must contain exactly two people from each department and exactly three men. How many committees are possible?","answer":536,"answer_str":"536","tier":"AIME"}
|
| 33 |
+
{"id":"aimepp-aime-0033","problem":"How many rectangles have all four vertices among the vertices of a regular $32$-gon?","answer":120,"answer_str":"120","tier":"AIME"}
|
| 34 |
+
{"id":"aimepp-aime-0034","problem":"A polynomial $P$ has degree at most $4$ and satisfies $P(0)=-2$, $P(1)=8$, $P(2)=30$, $P(3)=40$, $P(4)=-10$. Find the remainder when $P(5)$ is divided by $1000$.","answer":808,"answer_str":"808","tier":"AIME"}
|
docs/COMMERCIAL_ACCESS.md
ADDED
|
@@ -0,0 +1,54 @@
|
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|
|
|
|
| 1 |
+
# Full-dataset and commercial access
|
| 2 |
+
|
| 3 |
+
AIME++ is built for model teams that need large volumes of difficult mathematical reasoning problems with a transparent, deterministic success condition. Each problem resolves to an integer from `0` through `999`, enabling exact-match evaluation and RLVR rewards without a learned judge.
|
| 4 |
+
|
| 5 |
+
This repository is a 157-problem inspection and integration sample released under the MIT License. Teams may use it to validate their data pipeline, run initial model experiments, test exact-match rewards, and inspect the progression from competition-style to research-level mathematics.
|
| 6 |
+
|
| 7 |
+
## Available scale
|
| 8 |
+
|
| 9 |
+
Ulam AI's broader database contains more than 100,000 problems. Available AIME++ collections include:
|
| 10 |
+
|
| 11 |
+
| Collection | Available volume | Description |
|
| 12 |
+
|---|---:|---|
|
| 13 |
+
| AIME family | 24,700+ problems | Standard AIME and AIME Hard problems; graduate and research-level problems are excluded from this count |
|
| 14 |
+
| AIME-Graduate | 1,000+ problems | Graduate-level problems with the same `0–999` answer contract |
|
| 15 |
+
| AIME-Researcher | 100+ problems | Research-level problems with the same `0–999` answer contract |
|
| 16 |
+
|
| 17 |
+
The public sample represents approximately 132 AIME-family examples, 20 graduate examples, and 5 researcher examples. It is large enough for technical diligence while preserving the overwhelming majority of the catalog for licensed delivery.
|
| 18 |
+
|
| 19 |
+
## Why teams buy the larger collection
|
| 20 |
+
|
| 21 |
+
- **Production scale:** tens of thousands of consistent prompt-and-answer pairs rather than a small benchmark.
|
| 22 |
+
- **Deterministic rewards:** every item supports inexpensive, reproducible exact-match scoring.
|
| 23 |
+
- **Difficulty coverage:** the same output contract spans AIME, AIME Hard, AIME-Graduate, and AIME-Researcher tasks.
|
| 24 |
+
- **Low integration cost:** JSONL, stable IDs, explicit tiers, a JSON Schema, validation tooling, and a reference scorer are already defined in the sample.
|
| 25 |
+
- **Internal authorship:** the problems were created internally by Ulam AI, which holds the dataset rights.
|
| 26 |
+
- **Flexible deployment:** collections can support supervised fine-tuning, RLVR, capability evaluation, regression suites, and custom tier mixes under the applicable commercial terms.
|
| 27 |
+
|
| 28 |
+
The dataset is intentionally answer-only. Each supplied final answer is the authoritative golden solution for its problem. Worked derivations and chain-of-thought traces are not part of the product format.
|
| 29 |
+
|
| 30 |
+
## Commercial delivery can be scoped by
|
| 31 |
+
|
| 32 |
+
- collection and number of problems;
|
| 33 |
+
- AIME versus AIME Hard composition within the AIME family;
|
| 34 |
+
- AIME-Graduate and AIME-Researcher volume;
|
| 35 |
+
- training, fine-tuning, distillation, RLVR, or internal-evaluation use;
|
| 36 |
+
- public-answer versus separated or sealed answer-key delivery;
|
| 37 |
+
- custom topic or difficulty targeting;
|
| 38 |
+
- JSONL, Parquet, or buyer-specific export format;
|
| 39 |
+
- versioning, validation reports, updates, and support; and
|
| 40 |
+
- private evaluation execution and scorecards.
|
| 41 |
+
|
| 42 |
+
## Information to include in an inquiry
|
| 43 |
+
|
| 44 |
+
To make scoping fast, prospective partners should provide:
|
| 45 |
+
|
| 46 |
+
- intended use;
|
| 47 |
+
- desired problem count and tier mix;
|
| 48 |
+
- whether the answer key should be delivered, separated, or retained by Ulam;
|
| 49 |
+
- model and tool-access constraints;
|
| 50 |
+
- required delivery format and security controls;
|
| 51 |
+
- desired evaluation or update cadence; and
|
| 52 |
+
- target date and license term.
|
| 53 |
+
|
| 54 |
+
Visit [ulam.ai](https://ulam.ai/) to contact Ulam AI. Final inventory, deliverables, usage rights, pricing, representations, and support are defined in the applicable written agreement.
|
schema/dataset.schema.json
ADDED
|
@@ -0,0 +1,37 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
{
|
| 2 |
+
"$schema": "https://json-schema.org/draft/2020-12/schema",
|
| 3 |
+
"$id": "https://ulam.ai/schemas/aime-plus-plus-sample-0.1.0.json",
|
| 4 |
+
"title": "AIME++ Sample Record",
|
| 5 |
+
"description": "One exact-answer mathematical reasoning problem from the AIME++ sample.",
|
| 6 |
+
"type": "object",
|
| 7 |
+
"additionalProperties": false,
|
| 8 |
+
"required": ["id", "problem", "answer", "answer_str", "tier"],
|
| 9 |
+
"properties": {
|
| 10 |
+
"id": {
|
| 11 |
+
"type": "string",
|
| 12 |
+
"pattern": "^aimepp-(aime|hard|graduate|researcher)-[0-9]{4}$",
|
| 13 |
+
"description": "Stable record identifier."
|
| 14 |
+
},
|
| 15 |
+
"problem": {
|
| 16 |
+
"type": "string",
|
| 17 |
+
"minLength": 1,
|
| 18 |
+
"description": "English mathematical problem statement with LaTeX markup."
|
| 19 |
+
},
|
| 20 |
+
"answer": {
|
| 21 |
+
"type": "integer",
|
| 22 |
+
"minimum": 0,
|
| 23 |
+
"maximum": 999,
|
| 24 |
+
"description": "Canonical numeric answer."
|
| 25 |
+
},
|
| 26 |
+
"answer_str": {
|
| 27 |
+
"type": "string",
|
| 28 |
+
"pattern": "^[0-9]{3}$",
|
| 29 |
+
"description": "Canonical zero-padded three-digit answer used for exact-match evaluation."
|
| 30 |
+
},
|
| 31 |
+
"tier": {
|
| 32 |
+
"type": "string",
|
| 33 |
+
"enum": ["AIME", "AIME Hard", "AIME-Graduate", "AIME-Researcher"],
|
| 34 |
+
"description": "Human-readable difficulty tier."
|
| 35 |
+
}
|
| 36 |
+
}
|
| 37 |
+
}
|
scripts/score.py
ADDED
|
@@ -0,0 +1,156 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
#!/usr/bin/env python3
|
| 2 |
+
"""Score AIME++ JSONL predictions with deterministic exact matching."""
|
| 3 |
+
|
| 4 |
+
from __future__ import annotations
|
| 5 |
+
|
| 6 |
+
import argparse
|
| 7 |
+
import json
|
| 8 |
+
import re
|
| 9 |
+
from collections import defaultdict
|
| 10 |
+
from pathlib import Path
|
| 11 |
+
|
| 12 |
+
|
| 13 |
+
ROOT = Path(__file__).resolve().parents[1]
|
| 14 |
+
CONFIG_FILES = {
|
| 15 |
+
"all": (
|
| 16 |
+
"aime.jsonl",
|
| 17 |
+
"aime-hard.jsonl",
|
| 18 |
+
"aime-graduate.jsonl",
|
| 19 |
+
"aime-researcher.jsonl",
|
| 20 |
+
),
|
| 21 |
+
"aime": ("aime.jsonl",),
|
| 22 |
+
"aime-hard": ("aime-hard.jsonl",),
|
| 23 |
+
"aime-graduate": ("aime-graduate.jsonl",),
|
| 24 |
+
"aime-researcher": ("aime-researcher.jsonl",),
|
| 25 |
+
}
|
| 26 |
+
STRICT_ANSWER = re.compile(r"\s*([0-9]{1,3})\s*")
|
| 27 |
+
BOXED_ANSWER = re.compile(r"\\boxed\{\s*([0-9]{1,3})\s*\}")
|
| 28 |
+
|
| 29 |
+
|
| 30 |
+
def parse_prediction(value: object, allow_boxed: bool) -> int | None:
|
| 31 |
+
if isinstance(value, bool):
|
| 32 |
+
return None
|
| 33 |
+
if isinstance(value, int):
|
| 34 |
+
return value if 0 <= value <= 999 else None
|
| 35 |
+
if not isinstance(value, str):
|
| 36 |
+
return None
|
| 37 |
+
|
| 38 |
+
strict = STRICT_ANSWER.fullmatch(value)
|
| 39 |
+
if strict:
|
| 40 |
+
return int(strict.group(1))
|
| 41 |
+
if allow_boxed:
|
| 42 |
+
boxed = BOXED_ANSWER.findall(value)
|
| 43 |
+
if boxed:
|
| 44 |
+
return int(boxed[-1])
|
| 45 |
+
return None
|
| 46 |
+
|
| 47 |
+
|
| 48 |
+
def load_gold(data_dir: Path, config: str) -> dict[str, dict[str, object]]:
|
| 49 |
+
gold: dict[str, dict[str, object]] = {}
|
| 50 |
+
for filename in CONFIG_FILES[config]:
|
| 51 |
+
path = data_dir / filename
|
| 52 |
+
with path.open(encoding="utf-8") as handle:
|
| 53 |
+
for line_number, line in enumerate(handle, start=1):
|
| 54 |
+
record = json.loads(line)
|
| 55 |
+
record_id = record["id"]
|
| 56 |
+
if record_id in gold:
|
| 57 |
+
raise ValueError(f"duplicate gold id {record_id!r} in {path}:{line_number}")
|
| 58 |
+
gold[record_id] = record
|
| 59 |
+
return gold
|
| 60 |
+
|
| 61 |
+
|
| 62 |
+
def load_predictions(path: Path) -> dict[str, object]:
|
| 63 |
+
predictions: dict[str, object] = {}
|
| 64 |
+
with path.open(encoding="utf-8") as handle:
|
| 65 |
+
for line_number, line in enumerate(handle, start=1):
|
| 66 |
+
if not line.strip():
|
| 67 |
+
continue
|
| 68 |
+
record = json.loads(line)
|
| 69 |
+
if not isinstance(record, dict) or "id" not in record or "prediction" not in record:
|
| 70 |
+
raise ValueError(f"{path}:{line_number}: expected fields 'id' and 'prediction'")
|
| 71 |
+
record_id = record["id"]
|
| 72 |
+
if not isinstance(record_id, str):
|
| 73 |
+
raise ValueError(f"{path}:{line_number}: id must be a string")
|
| 74 |
+
if record_id in predictions:
|
| 75 |
+
raise ValueError(f"{path}:{line_number}: duplicate prediction id {record_id!r}")
|
| 76 |
+
predictions[record_id] = record["prediction"]
|
| 77 |
+
return predictions
|
| 78 |
+
|
| 79 |
+
|
| 80 |
+
def main() -> int:
|
| 81 |
+
parser = argparse.ArgumentParser(description=__doc__)
|
| 82 |
+
parser.add_argument("predictions", type=Path, help="JSONL with id and prediction fields")
|
| 83 |
+
parser.add_argument("--data-dir", type=Path, default=ROOT / "data")
|
| 84 |
+
parser.add_argument(
|
| 85 |
+
"--config",
|
| 86 |
+
choices=tuple(CONFIG_FILES),
|
| 87 |
+
default="all",
|
| 88 |
+
help="gold configuration to score (default: all)",
|
| 89 |
+
)
|
| 90 |
+
parser.add_argument(
|
| 91 |
+
"--allow-boxed",
|
| 92 |
+
action="store_true",
|
| 93 |
+
help=r"also accept the last \boxed{N} found in a string; strict whole-string matching is the default",
|
| 94 |
+
)
|
| 95 |
+
parser.add_argument("--json", action="store_true", help="emit machine-readable JSON")
|
| 96 |
+
args = parser.parse_args()
|
| 97 |
+
|
| 98 |
+
gold = load_gold(args.data_dir, args.config)
|
| 99 |
+
predictions = load_predictions(args.predictions)
|
| 100 |
+
unknown_ids = sorted(set(predictions) - set(gold))
|
| 101 |
+
|
| 102 |
+
correct = 0
|
| 103 |
+
valid = 0
|
| 104 |
+
submitted = 0
|
| 105 |
+
by_tier: dict[str, dict[str, int]] = defaultdict(lambda: {"correct": 0, "total": 0})
|
| 106 |
+
for record_id, record in gold.items():
|
| 107 |
+
tier = str(record["tier"])
|
| 108 |
+
by_tier[tier]["total"] += 1
|
| 109 |
+
if record_id not in predictions:
|
| 110 |
+
continue
|
| 111 |
+
submitted += 1
|
| 112 |
+
parsed = parse_prediction(predictions[record_id], args.allow_boxed)
|
| 113 |
+
if parsed is None:
|
| 114 |
+
continue
|
| 115 |
+
valid += 1
|
| 116 |
+
if parsed == record["answer"]:
|
| 117 |
+
correct += 1
|
| 118 |
+
by_tier[tier]["correct"] += 1
|
| 119 |
+
|
| 120 |
+
total = len(gold)
|
| 121 |
+
report = {
|
| 122 |
+
"config": args.config,
|
| 123 |
+
"accuracy": correct / total if total else 0.0,
|
| 124 |
+
"correct": correct,
|
| 125 |
+
"total": total,
|
| 126 |
+
"submitted": submitted,
|
| 127 |
+
"valid": valid,
|
| 128 |
+
"invalid": submitted - valid,
|
| 129 |
+
"missing": total - submitted,
|
| 130 |
+
"unknown_ids": unknown_ids,
|
| 131 |
+
"tiers": {
|
| 132 |
+
tier: {
|
| 133 |
+
**counts,
|
| 134 |
+
"accuracy": counts["correct"] / counts["total"] if counts["total"] else 0.0,
|
| 135 |
+
}
|
| 136 |
+
for tier, counts in by_tier.items()
|
| 137 |
+
},
|
| 138 |
+
}
|
| 139 |
+
|
| 140 |
+
if args.json:
|
| 141 |
+
print(json.dumps(report, indent=2, sort_keys=True))
|
| 142 |
+
else:
|
| 143 |
+
print(f"overall: {correct}/{total} ({report['accuracy']:.2%})")
|
| 144 |
+
print(
|
| 145 |
+
f"coverage: submitted={submitted}, valid={valid}, "
|
| 146 |
+
f"invalid={submitted - valid}, missing={total - submitted}"
|
| 147 |
+
)
|
| 148 |
+
for tier, counts in report["tiers"].items():
|
| 149 |
+
print(f"- {tier}: {counts['correct']}/{counts['total']} ({counts['accuracy']:.2%})")
|
| 150 |
+
if unknown_ids:
|
| 151 |
+
print(f"unknown prediction ids: {', '.join(unknown_ids)}")
|
| 152 |
+
return 0
|
| 153 |
+
|
| 154 |
+
|
| 155 |
+
if __name__ == "__main__":
|
| 156 |
+
raise SystemExit(main())
|
scripts/validate_dataset.py
ADDED
|
@@ -0,0 +1,199 @@
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|
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|
|
|
|
|
|
|
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|
|
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|
|
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|
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|
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|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
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|
|
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|
|
|
|
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|
|
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|
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|
|
|
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|
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|
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|
|
|
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|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
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|
|
|
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|
|
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|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
#!/usr/bin/env python3
|
| 2 |
+
"""Validate the AIME++ sample using only the Python standard library."""
|
| 3 |
+
|
| 4 |
+
from __future__ import annotations
|
| 5 |
+
|
| 6 |
+
import argparse
|
| 7 |
+
import hashlib
|
| 8 |
+
import json
|
| 9 |
+
import re
|
| 10 |
+
import sys
|
| 11 |
+
from dataclasses import dataclass
|
| 12 |
+
from pathlib import Path
|
| 13 |
+
|
| 14 |
+
|
| 15 |
+
ROOT = Path(__file__).resolve().parents[1]
|
| 16 |
+
FIELDS = {"id", "problem", "answer", "answer_str", "tier"}
|
| 17 |
+
|
| 18 |
+
|
| 19 |
+
@dataclass(frozen=True)
|
| 20 |
+
class FileSpec:
|
| 21 |
+
path: str
|
| 22 |
+
tier: str
|
| 23 |
+
id_stem: str
|
| 24 |
+
expected_count: int
|
| 25 |
+
|
| 26 |
+
|
| 27 |
+
FILE_SPECS = (
|
| 28 |
+
FileSpec("data/aime.jsonl", "AIME", "aime", 34),
|
| 29 |
+
FileSpec("data/aime-hard.jsonl", "AIME Hard", "hard", 98),
|
| 30 |
+
FileSpec("data/aime-graduate.jsonl", "AIME-Graduate", "graduate", 20),
|
| 31 |
+
FileSpec("data/aime-researcher.jsonl", "AIME-Researcher", "researcher", 5),
|
| 32 |
+
)
|
| 33 |
+
|
| 34 |
+
|
| 35 |
+
def normalized_problem(problem: str) -> str:
|
| 36 |
+
return re.sub(r"\s+", " ", problem).strip().casefold()
|
| 37 |
+
|
| 38 |
+
|
| 39 |
+
def sha256(path: Path) -> str:
|
| 40 |
+
digest = hashlib.sha256()
|
| 41 |
+
with path.open("rb") as handle:
|
| 42 |
+
for chunk in iter(lambda: handle.read(1024 * 1024), b""):
|
| 43 |
+
digest.update(chunk)
|
| 44 |
+
return digest.hexdigest()
|
| 45 |
+
|
| 46 |
+
|
| 47 |
+
def validate(root: Path) -> tuple[list[dict[str, object]], list[str]]:
|
| 48 |
+
errors: list[str] = []
|
| 49 |
+
stats: list[dict[str, object]] = []
|
| 50 |
+
seen_ids: set[str] = set()
|
| 51 |
+
seen_problems: dict[str, str] = {}
|
| 52 |
+
|
| 53 |
+
for spec in FILE_SPECS:
|
| 54 |
+
path = root / spec.path
|
| 55 |
+
if not path.is_file():
|
| 56 |
+
errors.append(f"missing file: {spec.path}")
|
| 57 |
+
continue
|
| 58 |
+
|
| 59 |
+
records: list[dict[str, object]] = []
|
| 60 |
+
with path.open(encoding="utf-8") as handle:
|
| 61 |
+
for line_number, line in enumerate(handle, start=1):
|
| 62 |
+
if not line.strip():
|
| 63 |
+
errors.append(f"{spec.path}:{line_number}: blank lines are not allowed")
|
| 64 |
+
continue
|
| 65 |
+
try:
|
| 66 |
+
record = json.loads(line)
|
| 67 |
+
except json.JSONDecodeError as exc:
|
| 68 |
+
errors.append(f"{spec.path}:{line_number}: invalid JSON: {exc.msg}")
|
| 69 |
+
continue
|
| 70 |
+
if not isinstance(record, dict):
|
| 71 |
+
errors.append(f"{spec.path}:{line_number}: record must be an object")
|
| 72 |
+
continue
|
| 73 |
+
records.append(record)
|
| 74 |
+
|
| 75 |
+
if len(records) != spec.expected_count:
|
| 76 |
+
errors.append(
|
| 77 |
+
f"{spec.path}: expected {spec.expected_count} records, found {len(records)}"
|
| 78 |
+
)
|
| 79 |
+
|
| 80 |
+
lengths: list[int] = []
|
| 81 |
+
for index, record in enumerate(records, start=1):
|
| 82 |
+
where = f"{spec.path}:{index}"
|
| 83 |
+
if set(record) != FIELDS:
|
| 84 |
+
missing = sorted(FIELDS - set(record))
|
| 85 |
+
extra = sorted(set(record) - FIELDS)
|
| 86 |
+
errors.append(f"{where}: schema mismatch; missing={missing}, extra={extra}")
|
| 87 |
+
continue
|
| 88 |
+
|
| 89 |
+
record_id = record["id"]
|
| 90 |
+
expected_id = f"aimepp-{spec.id_stem}-{index:04d}"
|
| 91 |
+
if record_id != expected_id:
|
| 92 |
+
errors.append(f"{where}: expected id {expected_id!r}, found {record_id!r}")
|
| 93 |
+
if not isinstance(record_id, str):
|
| 94 |
+
errors.append(f"{where}: id must be a string")
|
| 95 |
+
elif record_id in seen_ids:
|
| 96 |
+
errors.append(f"{where}: duplicate id {record_id!r}")
|
| 97 |
+
else:
|
| 98 |
+
seen_ids.add(record_id)
|
| 99 |
+
|
| 100 |
+
if record["tier"] != spec.tier:
|
| 101 |
+
errors.append(f"{where}: expected tier {spec.tier!r}")
|
| 102 |
+
|
| 103 |
+
problem = record["problem"]
|
| 104 |
+
if not isinstance(problem, str) or not problem.strip():
|
| 105 |
+
errors.append(f"{where}: problem must be non-empty text")
|
| 106 |
+
else:
|
| 107 |
+
lengths.append(len(problem))
|
| 108 |
+
if problem != problem.strip():
|
| 109 |
+
errors.append(f"{where}: problem has leading or trailing whitespace")
|
| 110 |
+
if any(ord(character) < 32 for character in problem):
|
| 111 |
+
errors.append(f"{where}: problem contains an ASCII control character")
|
| 112 |
+
if problem.count("$") % 2:
|
| 113 |
+
errors.append(f"{where}: problem has unbalanced dollar-sign LaTeX delimiters")
|
| 114 |
+
normalized = normalized_problem(problem)
|
| 115 |
+
if normalized in seen_problems:
|
| 116 |
+
errors.append(
|
| 117 |
+
f"{where}: normalized duplicate of {seen_problems[normalized]}"
|
| 118 |
+
)
|
| 119 |
+
else:
|
| 120 |
+
seen_problems[normalized] = where
|
| 121 |
+
|
| 122 |
+
answer = record["answer"]
|
| 123 |
+
if isinstance(answer, bool) or not isinstance(answer, int) or not 0 <= answer <= 999:
|
| 124 |
+
errors.append(f"{where}: answer must be an integer in 0..999")
|
| 125 |
+
elif record["answer_str"] != f"{answer:03d}":
|
| 126 |
+
errors.append(f"{where}: answer_str must equal the zero-padded answer")
|
| 127 |
+
|
| 128 |
+
stats.append(
|
| 129 |
+
{
|
| 130 |
+
"path": spec.path,
|
| 131 |
+
"tier": spec.tier,
|
| 132 |
+
"count": len(records),
|
| 133 |
+
"min_chars": min(lengths) if lengths else 0,
|
| 134 |
+
"max_chars": max(lengths) if lengths else 0,
|
| 135 |
+
"mean_chars": round(sum(lengths) / len(lengths), 1) if lengths else 0,
|
| 136 |
+
"sha256": sha256(path),
|
| 137 |
+
}
|
| 138 |
+
)
|
| 139 |
+
|
| 140 |
+
checksums_path = root / "CHECKSUMS.sha256"
|
| 141 |
+
if not checksums_path.is_file():
|
| 142 |
+
errors.append("missing file: CHECKSUMS.sha256")
|
| 143 |
+
else:
|
| 144 |
+
declared: dict[str, str] = {}
|
| 145 |
+
for line_number, line in enumerate(
|
| 146 |
+
checksums_path.read_text(encoding="utf-8").splitlines(), start=1
|
| 147 |
+
):
|
| 148 |
+
match = re.fullmatch(r"([0-9a-f]{64}) (data/[^\s]+\.jsonl)", line)
|
| 149 |
+
if not match:
|
| 150 |
+
errors.append(f"CHECKSUMS.sha256:{line_number}: malformed checksum line")
|
| 151 |
+
continue
|
| 152 |
+
declared[match.group(2)] = match.group(1)
|
| 153 |
+
expected_paths = {spec.path for spec in FILE_SPECS}
|
| 154 |
+
if set(declared) != expected_paths:
|
| 155 |
+
errors.append("CHECKSUMS.sha256: file list does not match the release data files")
|
| 156 |
+
for row in stats:
|
| 157 |
+
path = str(row["path"])
|
| 158 |
+
if declared.get(path) != row["sha256"]:
|
| 159 |
+
errors.append(f"CHECKSUMS.sha256: digest mismatch for {path}")
|
| 160 |
+
|
| 161 |
+
return stats, errors
|
| 162 |
+
|
| 163 |
+
|
| 164 |
+
def print_markdown(stats: list[dict[str, object]]) -> None:
|
| 165 |
+
print("| Tier | Records | Min chars | Mean chars | Max chars |")
|
| 166 |
+
print("|---|---:|---:|---:|---:|")
|
| 167 |
+
for row in stats:
|
| 168 |
+
print(
|
| 169 |
+
f"| {row['tier']} | {row['count']} | {row['min_chars']} | "
|
| 170 |
+
f"{row['mean_chars']} | {row['max_chars']} |"
|
| 171 |
+
)
|
| 172 |
+
print(f"\n**Total:** {sum(int(row['count']) for row in stats)} records")
|
| 173 |
+
|
| 174 |
+
|
| 175 |
+
def main() -> int:
|
| 176 |
+
parser = argparse.ArgumentParser(description=__doc__)
|
| 177 |
+
parser.add_argument("--root", type=Path, default=ROOT, help="dataset repository root")
|
| 178 |
+
parser.add_argument("--markdown", action="store_true", help="print a Markdown statistics table")
|
| 179 |
+
args = parser.parse_args()
|
| 180 |
+
|
| 181 |
+
stats, errors = validate(args.root.resolve())
|
| 182 |
+
if errors:
|
| 183 |
+
print(f"FAILED: {len(errors)} validation error(s)", file=sys.stderr)
|
| 184 |
+
for error in errors:
|
| 185 |
+
print(f"- {error}", file=sys.stderr)
|
| 186 |
+
return 1
|
| 187 |
+
|
| 188 |
+
if args.markdown:
|
| 189 |
+
print_markdown(stats)
|
| 190 |
+
else:
|
| 191 |
+
total = sum(int(row["count"]) for row in stats)
|
| 192 |
+
print(f"PASS: {total} records across {len(stats)} tiers")
|
| 193 |
+
for row in stats:
|
| 194 |
+
print(f"- {row['path']}: {row['count']} records; sha256={row['sha256']}")
|
| 195 |
+
return 0
|
| 196 |
+
|
| 197 |
+
|
| 198 |
+
if __name__ == "__main__":
|
| 199 |
+
raise SystemExit(main())
|