| {"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": ""}} |
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