Let $\alpha$ and $\beta$ be positive real numbers. Emerald makes a trip in the coordinate plane, starting off from the origin $(0,0)$. Each minute she moves one unit up or one unit to the right, restricting herself to the region $|x-y|\lt 2025$, in the coordinate plane. By the time she visits a point ( $x, y$ ) she writes down the integer $\lfloor x \alpha+y \beta\rfloor$ on it. It turns out that Emerald wrote each non-negative integer exactly once. Find all the possible pairs $(\alpha, \beta)$ for which such a trip would be possible.
Let $P(x)$ be a non-constant polynomial with integer coefficients such that $P(0) \neq 0$. Let $a_{1}, a_{2}, a_{3}, \ldots$ be an infinite sequence of integers such that $P(i-j)$ divides $a_{i}-a_{j}$ for all distinct positive integers $i, j$. Prove that the sequence $a_{1}, a_{2}, a_{3}, \ldots$ must be constant, that is, $a_{n}$ equals a constant $c$ for all $n$ positive integer.
Let $n \geq 3$ be an integer. There are $n$ cells on a circle, and each cell is assigned either $0$ or $1$. There is a rooster on one of these cells, and it repeats the following operations:
- If the rooster is on a cell assigned $0$, it changes the assigned number to $1$ and moves to the next cell counterclockwise.
- If the rooster is on a cell assigned $1$, it changes the assigned number to $0$ and moves to the cell after the next cell counterclockwise.
Prove that the following statement holds true after sufficiently many operations:
If the rooster is on a cell $C$, then the rooster would go around the circle exactly three times, stopping again at $C$. Moreover, every cell would be assigned the same number as it was assigned right before the rooster went around the circle $3$ times.
Consider an infinite sequence $a_{1}, a_{2}, \ldots$ of positive integers such that
$$
100!\left(a_{m}+a_{m+1}+\cdots+a_{n}\right) \quad \text{ is a multiple of } a_{n-m+1} a_{n+m}
$$
for all positive integers $m, n$ such that $m \leq n$.
Prove that the sequence is either bounded or linear.
Observation: A sequence of positive integers is bounded if there exists a constant $N$ such that $a_{n}\lt N$ for all $n \in \mathbb{Z}_{\gt 0}$. A sequence is linear if $a_{n}=n \cdot a_{1}$ for all $n \in \mathbb{Z}_{\gt 0}$.
Let $k$ be a positive integer and let $n_1, n_2, \dots, n_k \gt 1$ be $k$ distinct integers such that
$$
n_1! + \dots + n_k! = a^{n_1}
$$
for some positive integer $a$. Prove that at least one of $n_1, n_2, \dots, n_k$ is prime.
A positive integer $n \gt 1$ is called good if there exists some permutation of the numbers $1, 2, 3, \dots, n$, denoted by $(a_1, a_2, a_3, \dots, a_n)$ such that $a_i$ and $a_{i+1}$ have different parities for every $1 \le i \le n-1$; and for every $1 \le k \le n$, the sum $a_1 + a_2 + \dots + a_k$ is a quadratic residue modulo $n$. Prove that there exist infinitely many good numbers, as well as infinitely many numbers which are not good.
*Remark: Here an integer $x$ is considered a quadratic residue modulo $n$ if there exists an integer $y$ such that $x \equiv y^2 \pmod{n}$.*
Given a positive integer $k$, a pair of positive integers $(a, b)$ is said to be $k$-nice if $a \mid b^k + 1$ and $b \mid a^k + 1$. For which positive integers $k$ are there infinitely many $k$-nice pairs?
Let $k \gt 2$ be a given positive integer. Find all positive integers $d$, for which there exists a polynomial $P(x)$ with integer coefficients such that $\deg P(x) = d$ and $11^k \mid 2025^n + P(n)$ for all positive integers $n \gt k$.
Find all polynomials $P(x)$ with integer coefficients such that there exists a positive integer $N$ such that for all positive integers $n \gt N$ we have $P(n) \gt 0$ and
$$
n + P(n) \mid n^{P(n)} + P(n)^n.
$$
Let $f: \mathbb{R}^+ \to \mathbb{R}^+$ be a non-constant function. Prove that there exist $a, b \in \mathbb{R}^+$ such that
$$
f(a) + f(b) \gt 2f(\sqrt{ab}).
$$
For all $a$, $b$, $c$ positive real numbers with $a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 1$, show that
$$
\frac{ab + bc + ca}{1 + 2abc} \le \frac{(a + b + c)^2}{4}.
$$
Consider a table with $m$ rows and 22 columns. Each cell is filled with a number from the set $A = \{1, 2, 3, \dots, 2025\}$ (numbers may be repeated), such that for every pair of distinct numbers in $A$, there exists a row that contains exactly one of these two numbers. Find the minimum value of $m$.
A social network has $2025$ users. Two different users are either friends or not friends. A user is considered *lonely* if they have no friends. Initially, there are no lonely users, and two users Alice and Bob are not friends. A user may swap all their friends, meaning if they were friends with someone before the swap, they are no longer friends, and vice versa. Must there exist a way to arrange the $2025$ users in a sequence so that if, one-by-one in that sequence, each user swaps their friends, no user is ever lonely at any point during the process?
Let $n$ and $m$ be positive integers with $n \ge 2$. There are $n$ piles having $a_1, \dots, a_n$ stones such that for each $1 \le i \le n$ we have $m \le a_i \le m \cdot i$ (so $a_1 = m$). Aida and Bob play the following game: on each round, Bob picks two non-empty piles (if possible) and he removes a number of stones from the pile with fewer stones (in the case of equality, he chooses randomly). Then Aida decides whether she removes the same number of stones from the other pile or she moves the stones Bob just removed to the other pile. If there is at most one non-empty pile, the game ends.
Find the smallest positive integer $k$ such that Aida can guarantee that at some point there will be at most $k$ stones, regardless of the sequence $a_1, \dots, a_n$ and of how Bob plays.
Let $r$ be a positive rational number such that the numbers $r$ and $\sqrt{r+1}$ have the same fractional part. Show that $r$ is an integer.
Andrei Bâra