MathDB
Permutations Part 1: 2010 USAJMO #1

Source:

April 29, 2010
AMCUSAJMOPerfect SquarespermutationsUSA(J)MOnumber theory

Problem Statement

A permutation of the set of positive integers [n]={1,2,...,n}[n] = \{1, 2, . . . , 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.
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