{"year": "2024", "tier": "T1", "problem_label": "1", "problem_type": null, "exam": "RMM", "problem": "Let $n$ be a positive integer. Initially, a bishop is placed in each square of the top row of a $2^{n} \\times 2^{n}$ chessboard; those bishops are numbered from 1 to $2^{n}$, from left to right. A jump is a simultaneous move made by all bishops such that the following conditions are satisfied:\n\n- each bishop moves diagonally, in a straight line, some number of squares, and\n- at the end of the jump, the bishops all stand in different squares of the same row.\n\nFind the total number of permutations $\\sigma$ of the numbers $1,2, \\ldots, 2^{n}$ with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order $\\sigma(1), \\sigma(2), \\ldots, \\sigma\\left(2^{n}\\right)$, from left to right.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2024-RMM2024-Day1-English.jsonl", "problem_match": "\nProblem 1.", "solution_match": ""}}
{"year": "2024", "tier": "T1", "problem_label": "2", "problem_type": null, "exam": "RMM", "problem": "Consider an odd prime $p$ and a positive integer $N<50 p$. Let $a_{1}, a_{2}, \\ldots, a_{N}$ be a list of positive integers less than $p$ such that any specific value occurs at most $\\frac{51}{100} N$ times and $a_{1}+a_{2}+\\cdots+a_{N}$ is not divisible by $p$. Prove that there exists a permutation $b_{1}, b_{2}, \\ldots, b_{N}$ of the $a_{i}$ such that, for all $k=1,2, \\ldots, N$, the sum $b_{1}+b_{2}+\\cdots+b_{k}$ is not divisible by $p$.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2024-RMM2024-Day1-English.jsonl", "problem_match": "\nProblem 2.", "solution_match": ""}}
{"year": "2024", "tier": "T1", "problem_label": "3", "problem_type": null, "exam": "RMM", "problem": "Given a positive integer $n$, a set $\\mathcal{S}$ is $n$-admissible if\n\n- each element of $\\mathcal{S}$ is an unordered triple of integers in $\\{1,2, \\ldots, n\\}$,\n- $|\\mathcal{S}|=n-2$, and\n- for each $1 \\leq k \\leq n-2$ and each choice of $k$ distinct $A_{1}, A_{2}, \\ldots, A_{k} \\in \\mathcal{S}$,\n\n$$\n\\left|A_{1} \\cup A_{2} \\cup \\cdots \\cup A_{k}\\right| \\geq k+2\n$$\n\nIs it true that, for all $n>3$ and for each $n$-admissible set $\\mathcal{S}$, there exist pairwise distinct points $P_{1}, \\ldots, P_{n}$ in the plane such that the angles of the triangle $P_{i} P_{j} P_{k}$ are all less than $61^{\\circ}$ for any triple $\\{i, j, k\\}$ in $\\mathcal{S}$ ?\n\nEach problem is worth 7 marks.\nTime allowed: $4 \\frac{1}{2}$ hours.", "solution": "", "metadata": {"resource_path": "RMM/segmented/en-2024-RMM2024-Day1-English.jsonl", "problem_match": "\nProblem 3.", "solution_match": ""}}