Let \( \triangle ABC \) be a triangle such that \( BC > AC > AB \). A point \( X \) is marked on side \( BC \) such that \( AX = XC \). Let \( Y \) be a point on segment \( AX \) such that \( CY = AB \). Prove that \( \angle CYX = \angle ABC \).

In Tigre there are $2024$ islands, some of them connected by a two-way bridge. It is known that it is possible to go from any island to any other island using only the bridges (possibly several of them). In $k$ of the islands there is a flag ($0 \le k \le 2024$). Ana wants to destroy some of the bridges in such a way that after doing so, the following two conditions are met: $\bullet$ If an island has a flag, it is connected to an odd number of islands. $\bullet$ If an island does not have a flag, it is connected to an even number of islands. Determine all values of $k$ for which Ana can always achieve her objective, no matter what the initial bridge configuration is and which islands have a flag.

Given a set $S$ of integers, an allowed operation consists of the following three steps: $\bullet$ Choose a positive integer $n$. $\bullet$ Choose $n+1$ elements $a_0, a_1, \dots, a_n \in S$, not necessarily distinct. $\bullet$ Add to the set $S$ all the integer roots of the polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$. Beto must choose an initial set $S$ and perform several allowed operations, so that at the end of the process $S$ contains among its elements the integers $1, 2, 3, \dots, 2023, 2024$. Determine the smallest $k$ for which there exists an initial set $S$ with $k$ elements that allows Beto to achieve his objective.

There are 4 countries: Argentina, Brazil, Peru and Uruguay. Each country consists of 4 islands. There are bridges going back and forth between some of the 16 islands. Carlos noted that whenever he travels between some of the islands using the bridges, without using the same bridge twice, and ending in the island where he started his journey, he will necessarily visit at least one island of each country. Determine the maximum number of bridges there can be.

Let $S = \{2, 3, 4, \dots\}$ be the set of positive integers greater than 1. Find all functions $f : S \to S$ that satisfy \[ \text{gcd}(a, f(b)) \cdot \text{lcm}(f(a), b) = f(ab) \]for all pairs of integers $a, b \in S$. Clarification: $\text{gcd}(a,b)$ is the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ is the least common multiple of $a$ and $b$.

Let $ABC$ be an acute triangle with $AB < AC$, and let $H$ be its orthocenter. Let $D$, $E$, $F$ and $M$ be the midpoints of $BC$, $AC$, and $AH$, respectively. Prove that the circumcircles of triangles $AHD$, $BMC$, and $DEF$ pass through a common point.

Ana draws a checkered board that has at least 20 rows and at least 24 columns. Then, Beto must completely cover that board, without holes or overlaps, using only pieces of the following two types: Each piece must cover exactly 4 or 3 squares of the board, as shown in the figure, without leaving the board. It is permitted to rotate the pieces and it is not necessary to use all types of pieces. Explain why, regardless of how many rows and how many columns Ana's board has, Beto can always complete his task.

Let $ABC$ be a triangle with $AB < AC$, incentre $I$, and circumcircle $\omega$. Let $D$ be the intersection of the external bisector of angle $\widehat{ BAC}$ with line $BC$. Let $E$ be the midpoint of the arc $BC$ of $\omega$ that does not contain $A$. Let $M$ be the midpoint of $DI$, and $X$ the intersection of $EM$ with $\omega$. Prove that $IX$ and $EM$ are perpendicular.

Let $a$, $b$, $c$ be positive integers. Prove that for infinitely many positive odd integers $n$, there exists an integer $m > n$ such that $a^n + b^n + c^n$ divides $a^m + b^m + c^m$.

Let $N$ be a positive integer. A non-decreasing sequence $a_1 \le a_2 \le \dots$ of positive integers is said to be $N$-rioplatense if there exists an index $i$ such that $N = \frac{i}{a_i}$. Show that every sequence $2024$-rioplatense is $k$-rioplatense for $k=1, 2, 3, \dots, 2023$.

Let $n$ be a positive integer. Ana and Beto play a game on a $2 \times n$ board (with 2 rows and $n$ columns). First, Ana writes a digit from 1 to 9 in each cell of the board such that in each column the two written digits are different. Then, Beto erases a digit from each column. Reading from left to right, a number with $n$ digits is formed. Beto wins if this number is a multiple of $n$; otherwise, Ana wins. Determine which of the two players has a winning strategy in the following cases: $\bullet$ (a) $n = 1001$. $\bullet$ (b) $n = 1003$.

Let $ABC$ be a triangle with $\angle BAC = 90^\circ$ and $AB > AC$. Let $D$ be the foot of the altitude from $A$ to $BC$, $M$ be the midpoint of $BC$ and $A'$ be the reflection of $A$ over $D$. Let the mediatrix of $DM$ intersect lines $AB$ and $A'C$ at $P$ and $Q$, respectively. Let $K$ be the intersection of lines $A'C$ and $AB$. Prove that $PQ$ is tangent to the circumcircle of triangle $QDK$.