| {"year": "2024", "tier": "T1", "problem_label": "4", "problem_type": null, "exam": "RMM", "problem": "Fix integers $a$ and $b$ greater than 1 . For any positive integer $n$, let $r_{n}$ be the (non-negative) remainder that $b^{n}$ leaves upon division by $a^{n}$. Assume there exists a positive integer $N$ such that $r_{n}<2^{n} / n$ for all integers $n \\geq N$. Prove that $a$ divides $b$.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2024-RMM2024-Day2-English.jsonl", "problem_match": "\nProblem 4.", "solution_match": ""}} |
| {"year": "2024", "tier": "T1", "problem_label": "5", "problem_type": null, "exam": "RMM", "problem": "Let $B C$ be a fixed segment in the plane, and let $A$ be a variable point in the plane not on the line $B C$. Distinct points $X$ and $Y$ are chosen on the rays $\\overrightarrow{C A}$ and $\\overrightarrow{B A}$, respectively, such that $\\angle C B X=\\angle Y C B=\\angle B A C$. Assume that the tangents to the circumcircle of $A B C$ at $B$ and $C$ meet line $X Y$ at $P$ and $Q$, respectively, such that the points $X, P, Y$, and $Q$ are pairwise distinct and lie on the same side of $B C$. Let $\\Omega_{1}$ be the circle through $X$ and $P$ centred on $B C$. Similarly, let $\\Omega_{2}$ be the circle through $Y$ and $Q$ centred on $B C$. Prove that $\\Omega_{1}$ and $\\Omega_{2}$ intersect at two fixed points as $A$ varies.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2024-RMM2024-Day2-English.jsonl", "problem_match": "\nProblem 5.", "solution_match": ""}} |
| {"year": "2024", "tier": "T1", "problem_label": "6", "problem_type": null, "exam": "RMM", "problem": "A polynomial $P$ with integer coefficients is square-free if it is not expressible in the form $P=Q^{2} R$, where $Q$ and $R$ are polynomials with integer coefficients and $Q$ is not constant. For a positive integer $n$, let $\\mathcal{P}_{n}$ be the set of polynomials of the form\n\n$$\n1+a_{1} x+a_{2} x^{2}+\\cdots+a_{n} x^{n}\n$$\n\nwith $a_{1}, a_{2}, \\ldots, a_{n} \\in\\{0,1\\}$. Prove that there exists an integer $N$ so that, for all integers $n \\geq N$, more than $99 \\%$ of the polynomials in $\\mathcal{P}_{n}$ are square-free.\n\nEach problem is worth 7 marks.\nTime allowed: $4 \\frac{1}{2}$ hours.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2024-RMM2024-Day2-English.jsonl", "problem_match": "\nProblem 6.", "solution_match": ""}} |
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