Let $\triangle ABC$ be a triangle with $AB \lt AC$, and let $D$ be a point on the side $AC$ such that $AD = \frac{AC-AB}{2}$. Points $X$ and $Y$ are chosen on the line through $A$ parallel to $BC$ such that $BX = CY$ and line $AC$ is tangent to the circumcircle of $\triangle XDY$. Prove that the tangents to the circumcircle of $\triangle XDY$ at points $X$ and $Y$ meet on line $BC$.
Let $\triangle ABC$ be an acute-angled triangle with $AB \lt AC$, which is inscribed in the circle $\omega_1$ centered at $O$. Denote by $H$ the orthocenter of $\triangle ABC$ and by $M$ the midpoint of side $BC$. Let $\omega_2$ be the circumcircle of triangle $\triangle BHC$, let the line $HO$ meet $\omega_2$ at $K \neq H$, and let the line $MK$ meet $\omega_2$ at $P \neq K$. Prove that the tangent line to $\omega_1$ at $A$ and the tangent line to $\omega_2$ at $P$ meet on the circumcircle of triangle $\triangle APK$.
Let $a$, $b$ and $k$ be positive integers. Suppose that for any integer $n \ge 2025$ there exists a positive integer $x_n \gt n$ such that $x_n \mid n^2 + a$ and $x_n + k \mid n^2 + b$. Prove that $k = b - a$.
There are $n$ cities in a country, where $n \geq 100$ is a positive integer. Some pairs of cities are connected by (two-way) flights. For two cities $A$ and $B$, a path is a sequence of distinct cities $C_0, C_1, C_2, \ldots, C_k, C_{k+1}$, such that there are flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$, with $C_0 = A$ and $C_{k+1} = B$.
A long path between $A$ and $B$ is defined as a path such that no other path has more vertices. Similarly, a short path is defined as a path with the fewest vertices. In particular, if $A$ and $B$ have a direct flight, that is the shortest path.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). For a given $n$, find all possible numbers of flights in the country.
Let $a_{0}, a_{1}, \ldots, a_{10}$ be integers such that, for each $i \in \{0,1,\ldots,2047\}$, there exists a subset $S \subseteq \{0,1,\ldots,10\}$ with
$$
\sum_{j \in S} a_{j} \equiv i \pmod{2048}.
$$
Show that for each $i \in \{0,1,\ldots,10\}$, there is exactly one $j \in \{0,1,\ldots,10\}$ such that $a_{j}$ is divisible by $2^{i}$ but not by $2^{i+1}$.
Note: $\sum_{j \in S} a_{j}$ is the summation notation, for instance, $\sum_{j \in \{2,5\}} a_{j} = a_{2} + a_{5}$, while, for the empty set $\varnothing$, one defines $\sum_{j \in \varnothing} a_{j} = 0$.
The $n$ players of a hockey team gather to select their team captain. Initially, they stand in a circle, and each person votes for the person on their left.
The players will update their votes via a series of rounds. In one round, each player $a$ updates their vote, one at a time, according to the following procedure: At the time of the update, if $a$ is voting for $b$, and $b$ is voting for $c$, then $a$ updates their vote to $c$. (Note that $a$, $b$, and $c$ need not be distinct; if $b = c$, then $a$'s vote does not change for this update.) Every player updates their vote exactly once in each round, in an order determined by the players (possibly different across different rounds).
They repeat this updating procedure for $n$ rounds. Prove that at this time, all $n$ players will unanimously vote for the same person.
A polynomial $c_{d}x^{d} + c_{d - 1}x^{d - 1} + \dots + c_{1}x + c_{0}$ with degree $d$ is reflexive if there is an integer $n \geq d$ such that $c_{i} = c_{n - i}$ for every $0 \leq i \leq n$, where $c_{i} = 0$ for $i \gt d$. Let $\ell \geq 2$ be an integer and $p(x)$ be a polynomial with integer coefficients. Prove that there exist reflexive polynomials $q(x), r(x)$ with integer coefficients such that
$$(1 + x + x^{2} + \dots + x^{\ell -1})p(x) = q(x) + x^{\ell}r(x).$$
A rectangle $R$ is divided into a set $S$ of finitely many smaller rectangles with sides parallel to the sides of $R$ such that no three rectangles in $S$ share a common corner. An ant is initially located at the bottom-left corner of $R$. In one operation, we can choose a rectangle $r \in S$ such that the ant is currently located at one of the corners of $r$, say $c$, and move the ant to one of the two corners of $r$ adjacent to $c$.
Suppose that after a finite number of operations, the ant ends up at the top-right corner of $R$. Prove that some rectangle $r \in S$ was chosen in at least two operations.
Prove that the real-coefficient polynomial in $x, y, z$,
$$
x^{4}(x - y)(x - z) + y^{4}(y - z)(y - x) + z^{4}(z - x)(z - y),
$$
cannot be expressed as a finite sum of squares of real-coefficient polynomials in $x, y, z$.
Let $k$, $l$, $n$ be positive integers, and let $a_1, a_2, \dots, a_k \in \{1, 2, \dots, n\}$ satisfy the following three conditions:
(1) $n \ge 3$, $l \le n-2$, and $l-k \le \frac{n-3}{2}$;
(2) For each $t \in \{1, 2, \dots, l\}$, there exists a non-empty subset $I \subseteq \{1, 2, \dots, k\}$ such that
$$
\sum_{i \in I} a_i \equiv t \pmod{n};
$$
(3) For each $t \in \{l+1, l+2, \dots, n\}$, there does not exist a non-empty subset $I \subseteq \{1, 2, \dots, k\}$ such that
$$
\sum_{i \in I} a_i \equiv t \pmod{n}.
$$
Prove that $a_1 + a_2 + \dots + a_k = l$.
There are $2025$ people and $66$ given colors. Each person has $66$ balls, one of each color, with a total weight of $1$ for all $66$ balls.
Find the smallest real number $C$ such that, no matter how the balls are weighted, one can always select exactly one ball from each person so that for every color, the total weight of the selected balls of that color does not exceed $C$.
Given a positive integer $k$, let integers $a, b$ (not both zero) satisfy $0 \le a \lt k$, $0 \le b \lt k+1$. Define the sequence $\{T_n\}_{n \ge k}$ as follows:
$$
T_k = a, \quad T_{k+1} = b,
$$
and for $n \ge k + 2$,
$$
T_n \equiv T_{n-1} + T_{n-2} \pmod{n} \quad \text{with} \quad 0 \le T_n \lt n.
$$
Concatenating the decimal expansions of $T_k, T_{k+1}, \dots$ after the decimal point yields an infinite decimal $x = 0.T_kT_{k+1}\dots$. For example, if $k = 66$, $a = 5$, $b = 20$, then $T_{66} = 5$, $T_{67} = 20$, $T_{68} = 25$, $T_{69} = 45$, $T_{70} = 0$, $T_{71} = 45$, $T_{72} = 45$, $T_{73} = 17$, etc., so $x = 0.52025450454517\dots$.
Prove that $x$ is irrational.
Let $A_1A_2A_3A_4$ be a convex quadrilateral that is **not** cyclic and whose opposite sides are **not** parallel. For $1 \le i \le 4$, let $M_i$ be the midpoint of $A_{i-1}A_{i+1}$. Let $B_i$ be a point on the tangent to the circumcircle of triangle $A_{i-1}A_iA_{i+1}$ at $A_i$, such that the reflection of $M_i$ over the angle bisector of $\angle A_{i+1}A_{i+2}A_{i+3}$ lies on line $B_iA_{i+2}$. Let $C_i$ be the unique intersection point of lines $A_iA_{i+1}$ and $B_iB_{i+1}$. (All indices are modulo 4.)
Prove that $C_1, C_2, C_3, C_4$ are collinear.
A finite point set $P = \{(x_i, y_i) \mid i = 1, 2, \dots, n\}$ in the plane is called **sparse** if for any $1 \le i \lt j \le n$,
$$
|x_i - x_j| \cdot |y_i - y_j| \ge 1.
$$
(1) Let $\{Q_1, Q_2, Q_3\}$ be a sparse set where no $Q_i$ lies inside the axis-aligned rectangle with the other two points as diagonal vertices. Prove that the area of triangle $Q_1Q_2Q_3$ is at least $\frac{\sqrt{5}}{2}$.
(2) Let $P$ be a sparse set with no three collinear points, all lying inside an axis-aligned square of side length $a$. Prove that the number of points in $P$ does not exceed
$$
\frac{a^2}{\sqrt{5}} + 2a + 1.
$$
Given an odd integer $n \ge 3$ and a regular $n$-gon with vertex set $V$, let $\mathcal{P}$ denote the set of all regular polygons with vertices in $V$. For example, if $n = 15$, then $\mathcal{P}$ includes 1 regular 15-gon, 3 regular pentagons, and 5 equilateral triangles.
Two players, Alice and Bob, play the following game. Initially, all points in $V$ are uncolored. Alice moves first, and they take turns coloring an uncolored point in $V$: Alice colors red, and Bob colors blue. The game ends when all points are colored. A polygon in $\mathcal{P}$ is called **red-dominated** if it has more red vertices than blue ones.
Find the largest integer $k$ such that, no matter how Bob plays, Alice can guarantee at least $k$ red-dominated polygons in $\mathcal{P}$ at the end.
Let $P(x)$ and $Q(x)$ be non-constant real polynomials such that for every positive integer $m$, there exists a positive integer $n$ with $P(m) = Q(n)$.
(1) Prove that if $\deg(Q)$ divides $\deg(P)$, then there exists a real polynomial $h(x)$ such that $P(x) = Q(h(x))$.
(2) Prove that $\deg(Q)$ divides $\deg(P)$.
Find all positive integers $m$ such that there exists an infinite set $A$ of positive integers satisfying: for any $m$ distinct elements $a_1, a_2, \dots, a_m$ in $A$, both $a_1+a_2+\dots+a_m$ and $a_1a_2\cdots a_m$ are square-free.
*Note: A positive integer $n$ is called square-free if it is not divisible by the square of any prime number.*
Let $p_1, p_2, \dots, p_{2025}$ be real numbers, and let $\{a_n^{(1)}\}_{n \ge 0}$, $\{a_n^{(2)}\}_{n \ge 0}$, $\dots$, $\{a_n^{(2025)}\}_{n \ge 0}$ be $2025$ real sequences satisfying:
(1) $a_0^{(i)}$ ($1 \le i \le 2025$) are all zero;
(2) $a_1^{(i)}$ ($1 \le i \le 2025$) are **not** all zero;
(3) For $i = 1, 2, \dots, 2025$ and any positive integer $n$,
$$
a_{n-1}^{(i)} + a_n^{(i)} + a_{n+1}^{(i)} = p_i \cdot a_n^{(i+1)},
$$
where $a_n^{(2026)} = a_n^{(1)}$.
Prove that there exists a positive real number $r$ and infinitely many positive integers $n$ such that
$$
\max \{|a_n^{(1)}|, |a_n^{(2)}|, \dots, |a_n^{(2025)}|\} \gt r.
$$