Question 87
Let $n$ be a positive integer. Given $2n$ positive real numbers $a_1, a_2, a_3, \dots, a_n$ and $b_1, b_2, b_3, \dots, b_n$ satisfying the following condition:
$$
a_1 \gt a_2 \gt a_3 \gt \dots \gt a_n \text{ and } b_1 \gt b_2 \gt b_3 \gt \dots \gt b_n.
$$
A partition of these $2n$ numbers into $n$ disjoint pairs $(a_i, b_j), 1 \le i \le n, 1 \le j \le n$ is called regular if the difference between two numbers of each pair is less than $2013$.
a. Suppose that there exists a regular partition for these numbers. Show that the partition $(a_1, b_1), (a_2, b_2), \dots, (a_n, b_n)$ is also regular.
b. Suppose that there exists a regular partition in which $i \neq j$ for each pair $(a_i, b_j)$.
Compute the least number of regular partitions in this case.
$$
a_1 \gt a_2 \gt a_3 \gt \dots \gt a_n \text{ and } b_1 \gt b_2 \gt b_3 \gt \dots \gt b_n.
$$
A partition of these $2n$ numbers into $n$ disjoint pairs $(a_i, b_j), 1 \le i \le n, 1 \le j \le n$ is called regular if the difference between two numbers of each pair is less than $2013$.
a. Suppose that there exists a regular partition for these numbers. Show that the partition $(a_1, b_1), (a_2, b_2), \dots, (a_n, b_n)$ is also regular.
b. Suppose that there exists a regular partition in which $i \neq j$ for each pair $(a_i, b_j)$.
Compute the least number of regular partitions in this case.