Question 161
Let $A_1A_2\cdots A_{2025}$ be a convex 2025-gon, and let $A_i = A_{i+2025}$ for all integers $i$. Distinct points $P$ and $Q$ lie in its interior such that $\angle A_{i-1}A_iP = \angle QA_iA_{i+1}$ for all $i$. Define points $P_i^j$ and $Q_i^j$ for integers $i$ and positive integers $j$ as follows:
* For all $i$, $P_i^1 = Q_i^1 = A_i$.
* For all $i$ and $j$, $P_i^{j+1}$ and $Q_i^{j+1}$ are the circumcenters of $PP_i^jP_{i+1}^j$ and $QQ_i^jQ_{i+1}^j$, respectively.
Let $\mathcal{P}$ and $\mathcal{Q}$ be the polygons $P_1^{2025}P_2^{2025}\cdots P_{2025}^{2025}$ and $Q_1^{2025}Q_2^{2025}\cdots Q_{2025}^{2025}$, respectively.
(a) Prove that $\mathcal{P}$ and $\mathcal{Q}$ are cyclic.
(b) Let $O_P$ and $O_Q$ be the circumcenters of $\mathcal{P}$ and $\mathcal{Q}$, respectively. Assuming that $O_P \neq O_Q$, show that $O_P O_Q$ is parallel to $PQ$.
* For all $i$, $P_i^1 = Q_i^1 = A_i$.
* For all $i$ and $j$, $P_i^{j+1}$ and $Q_i^{j+1}$ are the circumcenters of $PP_i^jP_{i+1}^j$ and $QQ_i^jQ_{i+1}^j$, respectively.
Let $\mathcal{P}$ and $\mathcal{Q}$ be the polygons $P_1^{2025}P_2^{2025}\cdots P_{2025}^{2025}$ and $Q_1^{2025}Q_2^{2025}\cdots Q_{2025}^{2025}$, respectively.
(a) Prove that $\mathcal{P}$ and $\mathcal{Q}$ are cyclic.
(b) Let $O_P$ and $O_Q$ be the circumcenters of $\mathcal{P}$ and $\mathcal{Q}$, respectively. Assuming that $O_P \neq O_Q$, show that $O_P O_Q$ is parallel to $PQ$.