AMO-Bench 2025 Problem 16

Let $n$ be a positive integer such that $2 \leq n \leq 5$. For each such $n$, let $A_nB_nC_nD_n$ be a rectangle with vertex $A_n$ at $(-n, 1 + 6 \sqrt{2} - \sqrt{2}n)$, and vertex $B_n$ the reflection of $A_n$ through the origin. The line segments $A_nC_n$ and $B_nD_n$ are parallel to the $x$-axis, and the line segments $A_nD_n$ and $B_nC_n$ are parallel to the $y$-axis. Let $P_i (1 \leq i \leq 36)$ be 36 distinct points on the curve $(y - |x|)^2 + x^2 = 1$. For each point $P_i$, let the distances to the 16 sides of the four rectangles be denoted by $r_{i_1}, r_{i_2}, \ldots, r_{i_{16}}$. Let $d_i = \min \Big( r_{i_k} \Big) , (1 \leq k \leq 16)$, and suppose that these values satisfy the condition $3 \sum_{i=1}^{36}d_{i}^{3} -15\sum_{i=1}^{36} d_{i}^{2} = -1200$. Find the maximum value of $\sum_{i=1}^{36} d_i$.