Question 98
For an integer $n \ge 3$, a sequence of integers $a_1, a_2, \dots, a_n$ is called a **beautiful sequence** if the following conditions are all satisfied.
* $0 = a_1 \lt a_2 \lt \dots \lt a_n$.
* There exists an integer $i$ such that $1 \le i \le n$ and $a_i = 2025$.
* For all integers $i, j, k$ with $1 \le i \lt j \lt k \le n$, the inequality $\frac{a_i + a_k}{2} \le a_j$ holds.
Let $N$ be the maximum possible length of a beautiful sequence. For beautiful sequences $a_1, a_2, \dots, a_N$ of length $N$, determine the minimum possible value of $a_N$.
Here, the length of a sequence of integers $x_1, x_2, \dots, x_l$ is $l$.
* $0 = a_1 \lt a_2 \lt \dots \lt a_n$.
* There exists an integer $i$ such that $1 \le i \le n$ and $a_i = 2025$.
* For all integers $i, j, k$ with $1 \le i \lt j \lt k \le n$, the inequality $\frac{a_i + a_k}{2} \le a_j$ holds.
Let $N$ be the maximum possible length of a beautiful sequence. For beautiful sequences $a_1, a_2, \dots, a_N$ of length $N$, determine the minimum possible value of $a_N$.
Here, the length of a sequence of integers $x_1, x_2, \dots, x_l$ is $l$.