KMO 2022

Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions:

• $a_i \leq a_j$ for every positive integers $i <j$ .
• For any positive integer $k \geq 3$ , the following inequality holds: $$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$

Prove that $\{a_n\}$ is constant.

$p$ is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers $(x,y)$ that satisfy the following equation. $$p^2 x^4-6px^2+1=y^2$$