Question 13
A finite sequence of integers $a_1, a_2, \dots, a_n$ is called *regular* if there exists a real number $x$ satisfying
$$
\lfloor kx \rfloor = a_k \quad \text{for } 1 \le k \le n.
$$
Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say the term $a_k$ is *forced* if the following condition is satisfied: the sequence
$$
a_1, a_2, \dots, a_{k-1}, b
$$
is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with 1000 terms.
$$
\lfloor kx \rfloor = a_k \quad \text{for } 1 \le k \le n.
$$
Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say the term $a_k$ is *forced* if the following condition is satisfied: the sequence
$$
a_1, a_2, \dots, a_{k-1}, b
$$
is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with 1000 terms.