Let $n$ be a positive integer. Ana and Banana play a game. Banana thinks of a function $f\colon\mathbb{Z}\to\mathbb{Z}$ and a prime number $p$. He tells Ana that $f$ is nonconstant, $p<100$, and $f(x+p)=f(x)$ for all integers $x$. Ana's goal is to determine the value of $p$. She writes down $n$ integers $x_1,\dots,x_n$. After seeing this list, Banana writes down $f(x_1),\dots,f(x_n)$ in order. Ana wins if she can determine the value of $p$ from this information. Find the smallest value of $n$ for which Ana has a winning strategy. Anthony Wang

Let $a_1, a_2, \dots$ and $b_1, b_2, \dots$ be sequences of real numbers for which $a_1 > b_1$ and \begin{align*} a_{n+1} &= a_n^2 - 2b_n\\ b_{n+1} &= b_n^2 - 2a_n \end{align*}for all positive integers $n$. Prove that $a_1, a_2, \dots$ is eventually increasing (that is, there exists a positive integer $N$ for which $a_k < a_{k+1}$ for all $k > N$). Holden Mui

Let $A_1A_2\dotsm A_{2025}$ 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 $P^{j}_{i}$ and $Q^{j}_{i}$ for integers $i$ and positive integers $j$ as follows: For all $i$, $P^1_i = Q^1_i = A_i$. For all $i$ and $j$, $P^{j+1}_{i}$ and $Q^{j+1}_i$ are the circumcenters of $PP^j_iP^j_{i+1}$ and $QQ^j_iQ^{j}_{i+1}$, respectively. Let $\mathcal{P}$ and $\mathcal{Q}$ be the polygons $P^{2025}_{1}P^{2025}_{2}\dotsm P^{2025}_{2025}$ and $Q^{2025}_{1}Q^{2025}_{2}\dotsm Q^{2025}_{2025}$, respectively. Prove that $\mathcal{P}$ and $\mathcal{Q}$ are cyclic. 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 $O_PO_Q$ is parallel to $PQ$. Ruben Carpenter