Duplicate from lmms-lab/imo-2025

Co-authored-by: Bo Li <luodian@users.noreply.huggingface.co>

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.gitignore

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+.venv/

README.md

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+---
+dataset_info:
+  features:
+  - name: id
+    dtype: string
+  - name: problem
+    dtype: string
+  - name: solution
+    dtype: string
+  splits:
+  - name: train
+    num_examples: 6
+license: mit
+language:
+- en
+tags:
+- mathematics
+- olympiad
+- problem-solving
+- competition-math
+---
+
+# IMO 2025 Problems Dataset
+
+This dataset contains the 6 problems from the 2025 International Mathematical Olympiad (IMO). The problems are formatted with proper LaTeX notation for mathematical expressions.
+
+## Dataset Structure
+
+Each example contains:
+- `id`: Problem identifier (e.g., "2025-imo-p1")
+- `problem`: The problem statement with LaTeX mathematical notation
+- `solution`: The solution (currently set to null)
+
+## Problem Types
+
+The dataset includes problems covering various mathematical areas:
+
+1. **Problem 1**: Combinatorial geometry (sunny lines)
+2. **Problem 2**: Euclidean geometry (circles and triangles)
+3. **Problem 3**: Number theory (bonza functions)
+4. **Problem 4**: Number theory (proper divisors and sequences)
+5. **Problem 5**: Game theory (inekoalaty game)
+6. **Problem 6**: Combinatorial geometry (grid tiling)
+
+## Mathematical Notation
+
+Mathematical expressions are formatted using LaTeX:
+- Variables and expressions: `$x$`, `$n \geq 3$`
+- Display equations: `$$f(a) \text{ divides } b^a - f(b)^{f(a)}$$`
+- Sets: `$\mathbb{N}$`, `$\mathbb{R}$`
+- Special formatting: *sunny*, *bonza*, *proper divisor*, *inekoalaty game*
+
+## Files
+
+- `imo_2025.json`: Full dataset in JSON format
+- `README.md`: This file
+
+## Usage
+
+```python
+from datasets import load_dataset
+
+# Load the dataset
+dataset = load_dataset("lmms-lab/imo-2025")
+
+# Access individual problems
+for problem in dataset['train']:
+    print(f"Problem: {problem['id']}")
+    print(f"Statement: {problem['problem']}")
+    print()
+```
+
+Or load directly from JSON:
+
+```python
+import json
+
+# Load from JSON
+with open("imo_2025.json", "r") as f:
+    problems = json.load(f)
+
+# Access problems
+for problem in problems:
+    print(f"ID: {problem['id']}")
+    print(f"Problem: {problem['problem']}")
+    print(f"Solution: {problem['solution']}")
+    print()
+```
+
+## Citation
+
+If you use this dataset in your research, please cite:
+
+```bibtex
+@dataset{imo2025,
+  title={IMO 2025 Problems Dataset},
+  author={LMMS Lab},
+  year={2025},
+  url={https://huggingface.co/datasets/lmms-lab/imo-2025}
+}
+```
+
+## Source
+
+Problems are from the 2025 International Mathematical Olympiad.
+Original source: https://www.imo-official.org/problems.aspx
+
+## License
+
+This dataset is released under the MIT License.

convert.py

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+from pathlib import Path
+from pydantic import BaseModel
+import json
+
+MARKDOWN_DIR = Path("markdown")
+
+class Problem(BaseModel):
+    year: str
+    problem_id: str
+    problem: str
+    solution: str | None
+    
+    def to_dict(self):
+        return {
+            "id": f"{self.year}-imo-{self.problem_id}",
+            "problem": self.problem,
+            "solution": self.solution
+        }
+
+def get_problem(year: str, problem_id: str, problem_dir: Path):
+    problem_file = problem_dir / "problem.md"
+    solution_file = problem_dir / "solution.md"
+    with open(problem_file, "r", encoding="utf-8") as f:
+        problem = f.read()
+    if solution_file.exists():
+        with open(solution_file, "r", encoding="utf-8") as f:
+            solution = f.read()
+    else:
+        solution = None
+    return Problem(year=year, problem_id=problem_id, problem=problem, solution=solution)
+
+def convert_problems():
+    result = []
+    for year_dir in MARKDOWN_DIR.iterdir():
+        if not year_dir.is_dir():
+            continue
+        year = year_dir.name
+        for problem_dir in year_dir.iterdir():
+            if not problem_dir.is_dir():
+                continue
+            problem_number = problem_dir.name
+            result.append(get_problem(year, problem_number, problem_dir).to_dict())
+    result.sort(key=lambda x: x["id"])
+    return result
+
+if __name__ == "__main__":
+    result = convert_problems()
+    with open("imo_2025.json", "w", encoding="utf-8") as f:
+        json.dump(result, f)

imo_2025.json

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+[{"id": "2025-imo-p1", "problem": "A line in the plane is called *sunny* if it is **not** parallel to any of the $x$-axis, the $y$-axis, and the line $x + y = 0$.\n\nLet $n \\geq 3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:\n\n* for all positive integers $a$ and $b$ with $a + b \\leq n + 1$, the point $(a, b)$ is on at least one of the lines; and\n* exactly $k$ of the $n$ lines are sunny.\n", "solution": null}, {"id": "2025-imo-p2", "problem": "Let $\\Omega$ and $\\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\\Omega$ is less than the radius of $\\Gamma$. Suppose circles $\\Omega$ and $\\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\\Omega$ at $C$ and $\\Gamma$ at $D$, such that points $C, M, N$ and $D$ lie on the line in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ intersects $\\Omega$ again at $E \\neq A$. Line $AP$ intersects $\\Gamma$ again at $F \\neq A$. Let $H$ be the orthocentre of triangle $PMN$.\n\nProve that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.\n\n(The *orthocentre* of a triangle is the point of intersection of its altitudes.)\n", "solution": null}, {"id": "2025-imo-p3", "problem": "Let $\\mathbb{N}$ denote the set of positive integers. A function $f: \\mathbb{N} \\to \\mathbb{N}$ is said to be *bonza* if\n\n$$f(a) \\text{ divides } b^a - f(b)^{f(a)}$$\n\nfor all positive integers $a$ and $b$.\n\nDetermine the smallest real constant $c$ such that $f(n) \\leq cn$ for all bonza functions $f$ and all positive integers $n$.\n", "solution": null}, {"id": "2025-imo-p4", "problem": "A *proper divisor* of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.\n\nThe infinite sequence $a_1, a_2, \\ldots$ consists of positive integers, each of which has at least three proper divisors.\n\nFor each $n \\geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.\n\nDetermine all possible values of $a_1$.\n", "solution": null}, {"id": "2025-imo-p5", "problem": "Alice and Bazza are playing the *inekoalaty game*, a two-player game whose rules depend on a positive real number $\\lambda$ which is known to both players. On the $n^{\\text{th}}$ turn of the game (starting with $n = 1$) the following happens:\n\n- If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that\n\n$$x_1 + x_2 + \\cdots + x_n \\leq \\lambda n.$$\n\n- If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that\n\n$$x_1^2 + x_2^2 + \\cdots + x_n^2 \\leq n.$$\n\nIf a player cannot choose a suitable number $x_n$, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.\n\nDetermine all values of $\\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.\n", "solution": null}, {"id": "2025-imo-p6", "problem": "Consider a $2025 \\times 2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.\n\nDetermine the minimum number of tiles Matilda needs to place to satisfy these conditions.", "solution": null}]

markdown/2025/p1/problem.md

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+A line in the plane is called *sunny* if it is **not** parallel to any of the $x$-axis, the $y$-axis, and the line $x + y = 0$.
+
+Let $n \geq 3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:
+
+* for all positive integers $a$ and $b$ with $a + b \leq n + 1$, the point $(a, b)$ is on at least one of the lines; and
+* exactly $k$ of the $n$ lines are sunny.

markdown/2025/p2/problem.md

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+Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose circles $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, such that points $C, M, N$ and $D$ lie on the line in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ intersects $\Omega$ again at $E \neq A$. Line $AP$ intersects $\Gamma$ again at $F \neq A$. Let $H$ be the orthocentre of triangle $PMN$.
+
+Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.
+
+(The *orthocentre* of a triangle is the point of intersection of its altitudes.)

markdown/2025/p3/problem.md

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+Let $\mathbb{N}$ denote the set of positive integers. A function $f: \mathbb{N} \to \mathbb{N}$ is said to be *bonza* if
+
+$$f(a) \text{ divides } b^a - f(b)^{f(a)}$$
+
+for all positive integers $a$ and $b$.
+
+Determine the smallest real constant $c$ such that $f(n) \leq cn$ for all bonza functions $f$ and all positive integers $n$.

markdown/2025/p4/problem.md

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+A *proper divisor* of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.
+
+The infinite sequence $a_1, a_2, \ldots$ consists of positive integers, each of which has at least three proper divisors.
+
+For each $n \geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.
+
+Determine all possible values of $a_1$.

markdown/2025/p5/problem.md

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+Alice and Bazza are playing the *inekoalaty game*, a two-player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n^{\text{th}}$ turn of the game (starting with $n = 1$) the following happens:
+
+- If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that
+
+$$x_1 + x_2 + \cdots + x_n \leq \lambda n.$$
+
+- If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that
+
+$$x_1^2 + x_2^2 + \cdots + x_n^2 \leq n.$$
+
+If a player cannot choose a suitable number $x_n$, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.
+
+Determine all values of $\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

markdown/2025/p6/problem.md

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+Consider a $2025 \times 2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
+
+Determine the minimum number of tiles Matilda needs to place to satisfy these conditions.

requirements.txt

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+pydantic>=2.11.7