Question 41
Find the smallest positive integer $n$ with the following property:
For any sequence $a_1, a_2, \dots, a_{2021}$ of real numbers satisfying
$$
0 \lt a_1, a_2, \dots, a_{2021} \lt 2 \quad \text{and} \quad a_1 + a_2 + \dots + a_{2021} = 2021,
$$
there is a sequence $b_1, b_2, \dots, b_n$ of real numbers satisfying
$$
0 \lt b_1, b_2, \dots, b_n \lt 2 \quad \text{and} \quad b_1 + b_2 + \dots + b_n = n
$$
and a permutation $c_1, c_2, \dots, c_{n+2021}$ of the sequence $a_1, a_2, \dots, a_{2021}, b_1, b_2, \dots, b_n$ such that
$$
\begin{cases} c_1 + c_2 + \dots + c_l \le l, & \text{for all } 1 \le l \le n + 2021 \text{ odd} \\ c_1 + c_2 + \dots + c_l \ge l, & \text{for all } 1 \le l \le n + 2021 \text{ even}. \end{cases}
$$
For any sequence $a_1, a_2, \dots, a_{2021}$ of real numbers satisfying
$$
0 \lt a_1, a_2, \dots, a_{2021} \lt 2 \quad \text{and} \quad a_1 + a_2 + \dots + a_{2021} = 2021,
$$
there is a sequence $b_1, b_2, \dots, b_n$ of real numbers satisfying
$$
0 \lt b_1, b_2, \dots, b_n \lt 2 \quad \text{and} \quad b_1 + b_2 + \dots + b_n = n
$$
and a permutation $c_1, c_2, \dots, c_{n+2021}$ of the sequence $a_1, a_2, \dots, a_{2021}, b_1, b_2, \dots, b_n$ such that
$$
\begin{cases} c_1 + c_2 + \dots + c_l \le l, & \text{for all } 1 \le l \le n + 2021 \text{ odd} \\ c_1 + c_2 + \dots + c_l \ge l, & \text{for all } 1 \le l \le n + 2021 \text{ even}. \end{cases}
$$