Question 75
Let $A$ be the set of all permutations $a = (a_1, a_2, ..., a_{2003})$ of $2003$ first positive integers such that each permutation satisfies the condition: there is no nonvoid proper subset $S$ of the set $\{1, 2, ..., 2003\}$ such that $\{a_k \mid k \in S\} = S$.
For each $a = (a_1, a_2, ..., a_{2003}) \in A$, put $d(a) = \sum_{k=1}^{2003} (a_k - k)^2$.
a. Find the least value of $d(a)$. Denote this least value by $d_0$.
b. Find all permutations $a \in A$ such that $d(a) = d_0$.
For each $a = (a_1, a_2, ..., a_{2003}) \in A$, put $d(a) = \sum_{k=1}^{2003} (a_k - k)^2$.
a. Find the least value of $d(a)$. Denote this least value by $d_0$.
b. Find all permutations $a \in A$ such that $d(a) = d_0$.