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Omni-MATH / A permutation of the set of positive integers [n]={1,2,…,n} is a sequence (a₁,a₂,…,a_(n)) such that each element of [n] appears precisely one time as a term of the sequence. For ex…

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A permutation of the set of positive integers [n]={1,2,…,n}[n] = \{1, 2, \ldots, n\} is a sequence (a1,a2,…,an)(a_1, a_2, \ldots, a_n) such that each element of [n][n] appears precisely one time as a term of the sequence. For example, (3,5,1,2,4)(3, 5, 1, 2, 4) is a permutation of [5][5] . Let P(n)P(n) be the number of permutations of [n][n] for which kakka_k is a perfect square for all 1≤k≤n1\leq k\leq n . Find with proof the smallest nn such that P(n)P(n) is a multiple of 20102010 .
Plain-text mathematical notation (without MathML)
A permutation of the set of positive integers [n]={1,2,…,n} is a sequence (a₁,a₂,…,a_(n)) such that each element of [n] appears precisely one time as a term of the sequence. For example, (3,5,1,2,4) is a permutation of [5] . Let P(n) be the number of permutations of [n] for which ka_(k) is a perfect square for all 1≤k≤n . Find with proof the smallest n such that P(n) is a multiple of 2010 .
Original LaTeX notation
A permutation of the set of positive integers $[n] = \{1, 2, \ldots, n\}$ is a sequence $(a_1, a_2, \ldots, a_n)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$ . Let $P(n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1\leq k\leq n$ . Find with proof the smallest $n$ such that $P(n)$ is a multiple of $2010$ .

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task_id
c2638a8e-dd82-5258-8897-f376db93d670
task_key
test--c2638a8e-dd82-5258-8897-f376db93d670
task_revision_id
1
config / split
/ test

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