Question 5
Let $\mathcal A$ be the set of permutations of $1,2,\ldots,2003$ that fix no proper subset of $\{1,\ldots,2003\}$. For $a=(a_1,\ldots,a_{2003})\in\mathcal A$, let
$$d(a)=\sum_{i=1}^{2003}(a_i-i)^2.$$
(a) Find its minimum $d_0$. (b) Find all permutations attaining $d_0$.
Editorial clarification: “fix no proper subset” means leave no nonempty proper subset invariant as a set under $i\mapsto a_i$. The empty set is always invariant and must be excluded. Thus the permitted permutations are exactly single cycles of length $2003$.
$$d(a)=\sum_{i=1}^{2003}(a_i-i)^2.$$
(a) Find its minimum $d_0$. (b) Find all permutations attaining $d_0$.
Editorial clarification: “fix no proper subset” means leave no nonempty proper subset invariant as a set under $i\mapsto a_i$. The empty set is always invariant and must be excluded. Thus the permitted permutations are exactly single cycles of length $2003$.