USAMO 2003 Problem 3

Let $n \neq 0$ . For every sequence of integers $$A = a_0,a_1,a_2,\dots, a_n$$ satisfying $0 \le a_i \le i$ , for $i=0,\dots,n$ , define another sequence $$t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n)$$ by setting $t(a_i)$ to be the number of terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$ . Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $B$ such that $t(B) = B$ .