add USA TSTST 2025

USA_TSTST/md/en-probs-TSTST-2025.md

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+
+# USA TST Selection Test for \(67^{\mathrm{th}}\) IMO and \(15^{\mathrm{th}}\) EGMO  
+
+Aurora, IL  
+
+Day I 12:30pm - 5:00pm  
+
+Tuesday, June 24, 2025  
+
+Time limit: 4.5 hours. If you need to add page headers after the time limit, you must do so under proctor supervision. Proctors may not answer clarification questions.  
+
+You may keep the problems, but you cannot discuss them publicly until they are posted by staff online.  
+
+Problem 1. In a finite group of people, some pairs are friends (friendship is mutual). Each person \(p\) has a list \(f_{1}(p), f_{2}(p), \ldots , f_{d(p)}(p)\) of their friends, where \(d(p)\) is the number of distinct friends \(p\) has. Additionally, any two people are connected by a series of friendships. Each person also has a water balloon. The following game is played until someone ends up with more than one water balloon: on round \(r\) , each person \(p\) throws the current water balloon they have to their friend \(f_{s}(p)\) such that \(d(p)\) divides \(r - s\) . Show that if the game never ends, then everyone has the same number of friends.  
+
+Problem 2. Find all sets \(S \subseteq \mathbb{Z}\) for which there exists a function \(f: \mathbb{R} \to \mathbb{Z}\) such that  
+
+- \(f(x - y) - 2f(x) + f(x + y) \geq -1\) for all \(x, y \in \mathbb{R}\) , and  
+
+- \(S = \{f(z) \mid z \in \mathbb{R}\}\) .  
+
+Problem 3. Let \(a_{1}, a_{2}, r\) , and \(s\) be positive integers with \(r\) and \(s\) odd. The sequence \(a_{1}, a_{2}, a_{3}, \ldots\) is defined by  
+
+\[a_{n + 2} = r a_{n + 1} + s a_{n}\]  
+
+for all \(n \geq 1\) . Determine the maximum possible number of integers \(1 \leq \ell \leq 2025\) such that \(a_{\ell}\) divides \(a_{\ell +1}\) , over all possible choices of \(a_{1}, a_{2}, r\) , and \(s\) .
+
+
+
+# USA TST Selection Test for \(67^{\mathrm{th}}\) IMO and \(15^{\mathrm{th}}\) EGMO  
+
+Aurora, IL  
+
+Day II 12:30pm - 5:00pm  
+
+Thursday, June 26, 2025  
+
+Time limit: 4.5 hours. If you need to add page headers after the time limit, you must do so under proctor supervision. Proctors may not answer clarification questions.  
+
+You may keep the problems, but you cannot discuss them publicly until they are posted by staff online.  
+
+Problem 4. Let \(n \geq 2\) be a positive integer. Let \(a_{1}, a_{2}, \ldots , a_{n}\) be a sequence of positive integers such that  
+
+\[\gcd (a_{1},a_{2}),\gcd (a_{2},a_{3}),\ldots ,\gcd (a_{n - 1},a_{n})\]  
+
+is a strictly increasing sequence. Find, in terms of \(n\) , the maximum possible value of  
+
+\[\frac{1}{a_{1}} +\frac{1}{a_{2}} +\dots +\frac{1}{a_{n}}\]  
+
+over all such sequences.  
+
+Problem 5. A tetrahedron \(ABCD\) is said to be angelic if it has nonzero volume and satisfies  
+
+\[\angle BAC + \angle CAD + \angle DAB = \angle ABC + \angle CBD + \angle DBA,\] \[\angle ACB + \angle BCD + \angle DCA = \angle ADB + \angle BDC + \angle CDA.\]  
+
+Across all angelic tetrahedrons, what is the maximum number of distinct lengths that could appear in the set \(\{AB, AC, AD, BC, BD, CD\}\) ?  
+
+Problem 6. Alice and Bob play a game on \(n\) vertices labelled \(1, 2, \ldots , n\) . They take turns adding edges \(\{i, j\}\) , with Alice going first. Neither player is allowed to make a move that creates a cycle, and the game ends after \(n - 1\) total turns.  
+
+Let the weight of the edge \(\{i, j\}\) be \(|i - j|\) , and let \(W\) be the total weight of all edges at the end of the game. Alice plays to maximize \(W\) and Bob plays to minimize \(W\) . If both play optimally, what will \(W\) be?
+
+
+
+# USA TST Selection Test for \(67^{\mathrm{th}}\) IMO and \(15^{\mathrm{th}}\) EGMO  
+
+Aurora, IL  
+
+Day III 12:30pm - 5:00pm  
+
+Saturday, June 28, 2025  
+
+Time limit: 4.5 hours. If you need to add page headers after the time limit, you must do so under proctor supervision. Proctors may not answer clarification questions.  
+
+You may keep the problems, but you cannot discuss them publicly until they are posted by staff online.  
+
+Problem 7. For a positive real number \(c\) , the sequence \(a_{1}, a_{2}, \ldots\) of real numbers is defined as follows. Let \(a_{1} = c\) , and for \(n \geq 2\) , let  
+
+\[a_{n} = \sum_{i = 1}^{n - 1}(a_{i})^{n - i + 1}.\]  
+
+Find all positive real numbers \(c\) such that \(a_{i} > a_{i + 1}\) for all positive integers \(i\) .  
+
+Problem 8. Find all polynomials \(f\) with integer coefficients such that for all positive integers \(n\) ,  
+
+\[n \text{ divides } \frac{f(f(\ldots(f(0)) \ldots) - 1}{n + 1 f^{\mathrm{s}}}\]  
+
+Problem 9. Let acute triangle \(ABC\) have orthocenter \(H\) . Let \(B_{1}\) , \(C_{1}\) , \(B_{2}\) , and \(C_{2}\) be collinear points which lie on lines \(AB\) , \(AC\) , \(BH\) , and \(CH\) , respectively. Let \(\omega_{B}\) and \(\omega_{C}\) be the circumcircles of triangles \(BB_{1}B_{2}\) and \(CC_{1}C_{2}\) , respectively. Prove that the radical axis of \(\omega_{B}\) and \(\omega_{C}\) intersects the line through their centers on the nine- point circle of triangle \(ABC\) .
+
+

USA_TSTST/raw/en-probs-TSTST-2025.pdf

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USA_TSTST/raw/en-sols-TSTST-2025.pdf

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USA_TSTST/segment_script/segment_2025.py

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+# -----------------------------------------------------------------------------
+# Author: Jiawei Liu
+# Date: 2025-10-29
+# -----------------------------------------------------------------------------
+import json
+import re
+from pathlib import Path
+
+from rapidfuzz import fuzz
+
+
+def clean_text(text: str):
+    text = re.sub(
+        r"^# USA TST Selection Test for.*?them publicly until they are posted by staff online\.",
+        "",
+        text,
+        flags=re.DOTALL | re.MULTILINE | re.IGNORECASE,
+    )
+    return text
+
+
+def extract_problems(markdown_text: str):
+    problems = {}
+
+    # Each problem starts with a number followed by a dot and space (e.g., 'Problem 1. ')
+    matchs = list(
+        re.finditer(
+            r"^Problem\s*(\d+)\.\s+", markdown_text, flags=re.DOTALL | re.MULTILINE
+        )
+    )
+
+    # Split the text into parts using the problem numbers as delimiters
+    for i, m in enumerate(matchs):
+        problem_label = m.group(1)
+        problem = markdown_text[
+            m.end() : matchs[i + 1].start()
+            if i + 1 < len(matchs)
+            else len(markdown_text)
+        ]
+
+        problems[problem_label] = (problem.strip(), m.group())
+
+    print(f"  - Extracted {len(problems)} problems.")
+
+    return problems
+
+
+def extract_solutions(markdown_text):
+    solutions = {}
+
+    # 1. Find the block of text containing the solutions
+    # Block starts with '## \(S 1\) Solutions to Day 1' and end of document
+    solutions_section = re.search(
+        r"^##.*?Solutions to Day 1\s*$(.*)",
+        markdown_text,
+        re.DOTALL | re.MULTILINE,
+    )
+
+
+    if not solutions_section:
+        print("  - Not found Solutions section.")
+        return solutions
+
+    solutions_block = solutions_section.group(1)
+
+    ## \(\S 1.1\) Solution to TSTST 1, by Milan Haiman  
+    # 2. Split the block into individual solutions
+    matchs = list(
+        re.finditer(r"^##.*?Solution.*?TSTST\s*(\d+).*?\n$", solutions_block, flags=re.MULTILINE)
+    )
+
+    # Split the text into parts using the solution numbers as delimiters
+    for i, m in enumerate(matchs):
+        problem_label = m.group(1)
+        solution = solutions_block[
+            m.end() : matchs[i + 1].start()
+            if i + 1 < len(matchs)
+            else len(solutions_block)
+        ]
+
+        solutions[problem_label] = (solution.strip(), m.group())
+
+    print(f"  - Extracted {len(solutions)} solutions.")
+
+    return solutions
+
+
+def join(problems: dict, solutions: dict):
+    pairs = []
+
+    for problem_label, (problem, p_match) in problems.items():
+        solution, s_match = solutions.get(problem_label)
+
+        # Clean solution by removing the part that overlaps with the problem statement
+        problem_align = fuzz.partial_ratio_alignment(solution, problem)
+        solution = solution.replace(
+            solution[problem_align.src_start : problem_align.src_end], ""
+        )
+        solution = re.sub(
+            r"##.*?Solutions to Day \d", "", solution, flags=re.IGNORECASE
+        ).strip()
+
+        if not solution:
+            print(f"  - Warning: No solution found for problem {problem_label}.")
+        pairs.append((problem, solution, problem_label, p_match, s_match))
+
+    return pairs
+
+
+def write_pairs(output_file: Path, pairs: list, year: str, project_root: Path):
+    output_jsonl_text = ""
+    for problem, solution, problem_label, p_match, s_match in pairs:
+        output_jsonl_text += (
+            json.dumps(
+                {
+                    "year": year,
+                    "tier": "T0",
+                    "problem_label": problem_label,
+                    "problem_type": None,
+                    "exam": "IMO",
+                    "problem": problem,
+                    "solution": solution,
+                    "metadata": {
+                        "resource_path": output_file.relative_to(
+                            project_root
+                        ).as_posix(),
+                        "problem_match": p_match,
+                        "solution_match": s_match,
+                    },
+                },
+                ensure_ascii=False,
+            )
+            + "\n"
+        )
+
+    output_file.write_text(output_jsonl_text, encoding="utf-8")
+
+
+if __name__ == "__main__":
+    compet_base_path = Path(__file__).resolve().parent.parent
+    compet_md_path = compet_base_path / "md"
+    seg_output_path = compet_base_path / "segmented"
+    project_root = compet_base_path.parent
+
+    num_problems = 0
+    num_solutions = 0
+
+    problems_md = list(compet_md_path.glob("**/*probs*2025.md"))[0]
+    solutions_md = list(compet_md_path.glob("**/*sols*2025.md"))[0]
+
+    output_file = seg_output_path / solutions_md.relative_to(
+        compet_md_path
+    ).with_suffix(".jsonl")
+    output_file.parent.mkdir(parents=True, exist_ok=True)
+
+    # Read the markdown file
+    problems_md_text = "\n" + clean_text(problems_md.read_text(encoding="utf-8"))
+    solutions_md_text = "\n" + clean_text(solutions_md.read_text(encoding="utf-8"))
+
+    problems = extract_problems(problems_md_text)
+    solutions = extract_solutions(solutions_md_text)
+
+    num_problems += len(problems)
+    num_solutions += len(solutions)
+
+    pairs = join(problems, solutions)
+
+    year = re.search(r"\d{4}", output_file.stem).group()
+
+    write_pairs(output_file, pairs, year, project_root)
+
+    print()
+
+    print(f"Total problems extracted: {num_problems}")
+    print(f"Total solutions extracted: {num_solutions}")