In a 2025 by 2025 grid, every cell initially contains a `1'. Every minute, we simultaneously replace the number in each cell with the sum of numbers in the cells that share an edge with it. (For example, after the first minute, the number 2 is written in each of the four corner cells.) After 2025 minutes, we colour the board in checkerboard fashion, such that the top left corner is black. Find the difference between the sum of numbers in black cells and the sum of numbers in white cells. Proposed by chorn

Let $d(n)$ denote the number of positive divisors of $n$. For any given integer $a \geq 3$, define a sequence $\{a_i\}_{i=0}^\infty$ satisfying $a_{0}=a$, and $a_{n+1}=a_{n}+(-1)^{n} d(a_{n})$ for each integer $n \geq 0$. For example, if $a=275$, the sequence would be \[275, \overline{281,279,285,277,279,273}.\] Prove that for each positive integer $k$ there exists a positive integer $N$ such that if such a sequence has period $2k$ and all terms of the sequence are greater than $N$ then all terms of the sequence have the same parity. Proposed by Navid

Let $N$ be a positive integer. Let $R$ denote the smallest positive number that is the sum of $m$ terms $\sum^m_{i=1}{\pm \sqrt{a_i}}$, where each $a_i, i=1,\cdots, m$ is an integer not larger than $N$. Prove that \[R\le C\cdot N^{-m+\frac{3}{2}}\]for some positive real number $C$. Proposed by Navid (Clarification: note that the constant is allowed to depend on $m$ but should be independent of $N$, i.e. the equation $R(m,N)\le C(m)\cdot N^{-m+\frac{3}{2}}$ should hold for all positive integers $N$)

Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying \[f_k(x)=x+k\varphi(x)\]where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with $a_1=c$, and $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$ Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if (a) $k=2025$ ? (b) $k=2065$ ? Proposed by chorn

Let $ABC$ be a triangle and $H, O$ be its orthocenter and circumcenter, respectively. Construct a triangle by points $D_1, E_1, F_1,$ where $D_1$ lies on lines $BO$ and $AH$, $E_1$ lies on lines $CO$ and $BH$, and $F_1$ lies on lines $AO$ and $CH$. On the other hand, construct the other triangle $D_2E_2F_2$ that $D_2$ lies on $CO$ and $AH$, $E_2$ lies on $AO$ and $BH$, and $F_2$ lies on lines $BO$ and $CH$. Prove that triangles $D_1E_1F_1$ and $D_2E_2F_2$ are similar. Proposed by Saintan Wu

Let $\alpha, \beta$ be two rational numbers strictly between 0 and 1. Alice and Bob play a game. At the start of the game, Alice chooses a positive integer $n$. Knowing that, Bob then chooses a positive integer $T$. They then do the following for $T$ rounds: at the $i$th round, Bob chooses a set $X_i$ of $n$ positive integers that form a complete residue system modulo $n$. Then Alice chooses a subset $Y_i$ of $X_i$ such that the sum of elements in $Y_i$ is at most $\alpha$ times the sum of elements in $X_i$. After the $T$ rounds, Alice wins if it is possible to pick an integer $s$ between 0 and $n-1$ such that there are at least $\beta T$ positive integers among the elements in $Y_1, Y_2, . . . , Y_T$ (counted with multiplicities) that are equal to $s \pmod n$, and Bob wins otherwise. Find all pairs $(\alpha, \beta)$ of rational numbers strictly between 0 and 1 such that Alice has a winning strategy. Proposed by Hans