add RMM 2025

RMM/md/en-2025-RMM2025-Day1-English.md

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+# The \(16^{\mathrm{th}}\) Romanian Master of Mathematics Competition  
+
+Day 1: 12 February, 2025, Bucharest  
+
+Language: English  
+
+Problem 1. Let \(n > 10\) be an integer, and let \(A_{1},A_{2},\ldots ,A_{n}\) be distinct points in the plane such that the distances between the points are pairwise different. Define \(f_{10}(j,k)\) to be the \(10^{\mathrm{th}}\) smallest of the distances from \(A_{j}\) to \(A_{1},A_{2},\ldots ,A_{k}\) , excluding \(A_{j}\) if \(k\geq j\) . Suppose that for all \(j\) and \(k\) satisfying \(11\leq j\leq k\leq n\) , we have \(f_{10}(j,j - 1)\geq f_{10}(k,j - 1)\) . Prove that \(f_{10}(j,n)\geq {\frac{1}{2}}f_{10}(n,n)\) for all \(j\) in the range \(1\leq j\leq n - 1\) .  
+
+Problem 2. Consider an infinite sequence of positive integers \(a_{1},a_{2},a_{3},\ldots\) such that \(a_{1} > 1\) and \((2^{a_{n}} - 1)a_{n + 1}\) is a square for all positive integers \(n\) . Is it possible for two terms of such a sequence to be equal?  
+
+Problem 3. Fix an integer \(n\geq 3\) . Determine the smallest positive integer \(k\) satisfying the following condition:  
+
+For any tree \(T\) with vertices \(v_{1},v_{2},\ldots ,v_{n}\) and any pairwise distinct complex numbers \(z_{1},z_{2},\ldots ,z_{n}\) , there is a polynomial \(P(X,Y)\) with complex coefficients of total degree at most \(k\) such that for all \(i\neq j\) satisfying \(1\leq i,j\leq n\) , we have \(P(z_{i},z_{j}) = 0\) if and only if there is an edge in \(T\) joining \(v_{i}\) to \(v_{j}\) .  
+
+Note, for example, that the total degree of the polynomial  
+
+\[9X^{3}Y^{4} + XY^{5} + X^{6} - 2\]  
+
+is 7 because \(7 = 3 + 4\) .  
+
+Each problem is worth 7 marks. Time allowed: \(4\frac{1}{2}\) hours.
+
+

RMM/md/en-2025-RMM2025-Day2-English.md

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+
+# The \(16^{\mathrm{th}}\) Romanian Master of Mathematics Competition  
+
+Day 2: 13 February, 2025, Bucharest  
+
+Language: English  
+
+Problem 4. Let \(\mathbb{Z}\) denote the set of integers, and let \(S \subset \mathbb{Z}\) be the set of integers that are at least \(10^{100}\) . Fix a positive integer \(c\) . Determine all functions \(f: S \to \mathbb{Z}\) satisfying \(f(xy + c) = f(x) + f(y)\) for all \(x, y \in S\) .  
+
+Problem 5. Let \(A B C\) be an acute triangle with \(A B< A C\) , and let \(H\) and \(o\) be its orthocentre and circumcentre, respectively. Let \(\Gamma\) be the circumcircle of triangle \(B O C\) . Circle \(\Gamma\) intersects line \(A O\) at points \(o\) and \(A^{\prime}\) , and \(\Gamma\) intersects the circle of radius \(A O\) with centre \(A\) at points \(o\) and \(F\) . Prove that the circle which has diameter \(A A^{\prime}\) , the circumcircle of triangle \(A F H\) and \(\Gamma\) pass through a common point.  
+
+Problem 6. Let \(k\) and \(m\) be integers greater than 1. Consider \(k\) pairwise disjoint sets \(S_{1}, S_{2}, \ldots , S_{k}\) , each of which has exactly \(m + 1\) elements: one red and \(m\) blue. Let \(\mathcal{F}\) be the family of all subsets \(T\) of \(S_{1} \cup S_{2} \cup \dots \cup S_{k}\) such that, for every \(i\) , the intersection \(T \cap S_{i}\) is monochromatic. Determine the largest possible number of sets in a subfamily \(\mathcal{G} \subseteq \mathcal{F}\) such that no two sets in \(\mathcal{G}\) are disjoint.  
+
+A set is monochromatic if all of its elements have the same colour. In particular, the empty set is monochromatic.  
+
+Each problem is worth 7 marks. Time allowed: \(4\frac{1}{2}\) hours.
+
+

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RMM/segmented/en-2025-RMM2025-Day1-English.jsonl

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+{"year": "2025", "tier": "T1", "problem_label": "1", "problem_type": null, "exam": "RMM", "problem": "Let \\(n > 10\\) be an integer, and let \\(A_{1},A_{2},\\ldots ,A_{n}\\) be distinct points in the plane such that the distances between the points are pairwise different. Define \\(f_{10}(j,k)\\) to be the \\(10^{\\mathrm{th}}\\) smallest of the distances from \\(A_{j}\\) to \\(A_{1},A_{2},\\ldots ,A_{k}\\) , excluding \\(A_{j}\\) if \\(k\\geq j\\) . Suppose that for all \\(j\\) and \\(k\\) satisfying \\(11\\leq j\\leq k\\leq n\\) , we have \\(f_{10}(j,j - 1)\\geq f_{10}(k,j - 1)\\) . Prove that \\(f_{10}(j,n)\\geq {\\frac{1}{2}}f_{10}(n,n)\\) for all \\(j\\) in the range \\(1\\leq j\\leq n - 1\\) .", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2025-RMM2025-Day1-English.jsonl", "problem_match": "\nProblem 1.", "solution_match": ""}}
+{"year": "2025", "tier": "T1", "problem_label": "2", "problem_type": null, "exam": "RMM", "problem": "Consider an infinite sequence of positive integers \\(a_{1},a_{2},a_{3},\\ldots\\) such that \\(a_{1} > 1\\) and \\((2^{a_{n}} - 1)a_{n + 1}\\) is a square for all positive integers \\(n\\) . Is it possible for two terms of such a sequence to be equal?", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2025-RMM2025-Day1-English.jsonl", "problem_match": "\nProblem 2.", "solution_match": ""}}
+{"year": "2025", "tier": "T1", "problem_label": "3", "problem_type": null, "exam": "RMM", "problem": "Fix an integer \\(n\\geq 3\\) . Determine the smallest positive integer \\(k\\) satisfying the following condition:  \n\nFor any tree \\(T\\) with vertices \\(v_{1},v_{2},\\ldots ,v_{n}\\) and any pairwise distinct complex numbers \\(z_{1},z_{2},\\ldots ,z_{n}\\) , there is a polynomial \\(P(X,Y)\\) with complex coefficients of total degree at most \\(k\\) such that for all \\(i\\neq j\\) satisfying \\(1\\leq i,j\\leq n\\) , we have \\(P(z_{i},z_{j}) = 0\\) if and only if there is an edge in \\(T\\) joining \\(v_{i}\\) to \\(v_{j}\\) .  \n\nNote, for example, that the total degree of the polynomial  \n\n\\[9X^{3}Y^{4} + XY^{5} + X^{6} - 2\\]  \n\nis 7 because \\(7 = 3 + 4\\) .  \n\nEach problem is worth 7 marks. Time allowed: \\(4\\frac{1}{2}\\) hours.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2025-RMM2025-Day1-English.jsonl", "problem_match": "\nProblem 3.", "solution_match": ""}}

RMM/segmented/en-2025-RMM2025-Day2-English.jsonl

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+{"year": "2025", "tier": "T1", "problem_label": "4", "problem_type": null, "exam": "RMM", "problem": "Let \\(\\mathbb{Z}\\) denote the set of integers, and let \\(S \\subset \\mathbb{Z}\\) be the set of integers that are at least \\(10^{100}\\) . Fix a positive integer \\(c\\) . Determine all functions \\(f: S \\to \\mathbb{Z}\\) satisfying \\(f(xy + c) = f(x) + f(y)\\) for all \\(x, y \\in S\\) .", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2025-RMM2025-Day2-English.jsonl", "problem_match": "\nProblem 4.", "solution_match": ""}}
+{"year": "2025", "tier": "T1", "problem_label": "5", "problem_type": null, "exam": "RMM", "problem": "Let \\(A B C\\) be an acute triangle with \\(A B< A C\\) , and let \\(H\\) and \\(o\\) be its orthocentre and circumcentre, respectively. Let \\(\\Gamma\\) be the circumcircle of triangle \\(B O C\\) . Circle \\(\\Gamma\\) intersects line \\(A O\\) at points \\(o\\) and \\(A^{\\prime}\\) , and \\(\\Gamma\\) intersects the circle of radius \\(A O\\) with centre \\(A\\) at points \\(o\\) and \\(F\\) . Prove that the circle which has diameter \\(A A^{\\prime}\\) , the circumcircle of triangle \\(A F H\\) and \\(\\Gamma\\) pass through a common point.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2025-RMM2025-Day2-English.jsonl", "problem_match": "\nProblem 5.", "solution_match": ""}}
+{"year": "2025", "tier": "T1", "problem_label": "6", "problem_type": null, "exam": "RMM", "problem": "Let \\(k\\) and \\(m\\) be integers greater than 1. Consider \\(k\\) pairwise disjoint sets \\(S_{1}, S_{2}, \\ldots , S_{k}\\) , each of which has exactly \\(m + 1\\) elements: one red and \\(m\\) blue. Let \\(\\mathcal{F}\\) be the family of all subsets \\(T\\) of \\(S_{1} \\cup S_{2} \\cup \\dots \\cup S_{k}\\) such that, for every \\(i\\) , the intersection \\(T \\cap S_{i}\\) is monochromatic. Determine the largest possible number of sets in a subfamily \\(\\mathcal{G} \\subseteq \\mathcal{F}\\) such that no two sets in \\(\\mathcal{G}\\) are disjoint.  \n\nA set is monochromatic if all of its elements have the same colour. In particular, the empty set is monochromatic.  \n\nEach problem is worth 7 marks. Time allowed: \\(4\\frac{1}{2}\\) hours.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2025-RMM2025-Day2-English.jsonl", "problem_match": "\nProblem 6.", "solution_match": ""}}