| {"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": ""}} |
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