MathDB

Let

A

be the set of all permutations

a = (a_1, a_2, \ldots, a_{2003})

of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset

S

of the set

\{1, 2, \ldots, 2003\}

such that

\{a_k | k \in S\} = S.

For each

a = (a_1, a_2, \ldots, a_{2003}) \in A

, let

d(a) = \sum^{2003}_{k=1} \left(a_k - k \right)^2.

I. Find the least value of

d(a)

. Denote this least value by

d_0

. II. Find all permutations

a \in A

such that

d(a) = d_0

Problem Statement