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Omni-MATH / Given a positive integer n≥2. Find all n-tuples of positive integers (a₁,a₂,…,a_(n)), such that 1<a₁≤a₂≤a₃≤⋯≤a_(n), a₁ is odd, and (1) M=(1)/(2^(n))(a₁−1)a₂a₃⋯a_(n) is a positive i…

Problem

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problem

Given a positive integer n≥2n \ge 2. Find all nn-tuples of positive integers (a1,a2,…,an)(a_1,a_2,\ldots,a_n), such that 1<a1≤a2≤a3≤⋯≤an1<a_1 \le a_2 \le a_3 \le \cdots \le a_n, a1a_1 is odd, and (1) M=12n(a1−1)a2a3⋯anM=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n is a positive integer; (2) One can pick nn-tuples of integers (ki,1,ki,2,…,ki,n)(k_{i,1},k_{i,2},\ldots,k_{i,n}) for i=1,2,…,Mi=1,2,\ldots,M such that for any 1≤i1<i2≤M1 \le i_1 <i_2 \le M, there exists j∈{1,2,…,n}j \in \{1,2,\ldots,n\} such that $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$.
Plain-text mathematical notation (without MathML)
Given a positive integer n≥2. Find all n-tuples of positive integers (a₁,a₂,…,a_(n)), such that 1<a₁≤a₂≤a₃≤⋯≤a_(n), a₁ is odd, and
(1) M=(1)/(2^(n))(a₁−1)a₂a₃⋯a_(n) is a positive integer;
(2) One can pick n-tuples of integers (k_(i,1),k_(i,2),…,k_(i,n)) for i=1,2,…,M such that for any 1≤i₁<i₂≤M, there exists j∈{1,2,…,n} such that $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$.
Original LaTeX notation
Given a positive integer $n \ge 2$. Find all $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$, such that $1<a_1 \le a_2 \le a_3 \le \cdots \le a_n$, $a_1$ is odd, and
(1) $M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n$ is a positive integer;
(2) One can pick $n$-tuples of integers $(k_{i,1},k_{i,2},\ldots,k_{i,n})$ for $i=1,2,\ldots,M$ such that for any $1 \le i_1 <i_2 \le M$, there exists $j \in \{1,2,\ldots,n\}$ such that $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$.

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Source and history

task_id
8462f740-3082-5871-820e-f8c3178b9625
task_key
test--8462f740-3082-5871-820e-f8c3178b9625
task_revision_id
5
config / split
/ test

Official source

initial import