| {"year": "2019", "tier": "T1", "problem_label": "4", "problem_type": null, "exam": "RMM", "problem": "Prove that for every positive integer $n$ there exists a (not necessarily convex) polygon with no three collinear vertices, which admits exactly $n$ different triangulations.\n(A triangulation is a dissection of the polygon into triangles by interior diagonals which have no common interior points with each other nor with the sides of the polygon.)", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2019-RMM2019-Day2-English.jsonl", "problem_match": "\nProblem 4.", "solution_match": ""}} |
| {"year": "2019", "tier": "T1", "problem_label": "5", "problem_type": null, "exam": "RMM", "problem": "Determine all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfying\n\n$$\nf(x+y f(x))+f(x y)=f(x)+f(2019 y)\n$$\n\nfor all real numbers $x$ and $y$.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2019-RMM2019-Day2-English.jsonl", "problem_match": "\nProblem 5.", "solution_match": ""}} |
| {"year": "2019", "tier": "T1", "problem_label": "6", "problem_type": null, "exam": "RMM", "problem": "Find all pairs of integers $(c, d)$, both greater than 1 , such that the following holds:\n\nFor any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p>c(2 c+1)$, there exists a set $S$ of at most $\\left(\\frac{2 c-1}{2 c+1}\\right) p$ integers, such that\n\n$$\n\\bigcup_{s \\in S}\\{s, Q(s), Q(Q(s)), Q(Q(Q(s))), \\ldots\\}\n$$\n\nis a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$ ).\n\nEach of the three problems is worth 7 points.\nTime allowed $4 \\frac{1}{2}$ hours.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2019-RMM2019-Day2-English.jsonl", "problem_match": "\nProblem 6.", "solution_match": ""}} |
|
|
