AMO-Bench 2025 Problem 29

Let $a$ be a prime greater than $100$ with $a \equiv 3 \pmod{5}$. For complex numbers $z_n$ $(1 \le n \le a)$, both the real and imaginary parts are nonzero. For each $n$, let $d_n$ be the distance from the point $(n,\lvert z_n\rvert)$ to the line $$y=\frac{202}{5}x+2025.$$ Choose one complex number at random from $\{z_n\}_{n=1}^a$; let $p$ be the probability that its modulus is a positive integer. Given $$\bigg(2a-\sum_{i=1}^{a} i^{d_i}\bigg)^{ 2} +\bigg(3+\sum_{k=2}^{a}(2k+2)-\sum_{i=1}^{a} i^{2d_i}\bigg)^{ 2} +\bigg(\frac{7}{2}+\sum_{j=2}^{a}\!\big(3j^3+3j+\tfrac{1}{2}\big)-\sum_{i=1}^{2022} i^{3d_i}\bigg)^{ 2}=0,$$ find the maximum value of $p$.