Let there be a circle with center $O$, and three distinct points $A, B, X$ on the circle, where $A, B, O$ are not collinear. Let $\Omega$ be the circumcircle of triangle $ABO$. Segments $AX, BX$ intersect $\Omega$ at points $C(\neq A), D(\neq B)$, respectively. Prove that $O$ is the orthocenter of triangle $CXD$.

For a sequence of positive integers $\{x_n\}$ where $x_1 = 2$ and $x_{n + 1} - x_n \in \{0, 3\}$ for all positve integers $n$, then $\{x_n\}$ is called a "frog sequence". Find all real numbers $d$ that satisfy the following condition. (Condition) For two frog sequence $\{a_n\}, \{b_n\}$, if there exists a positive integer $n$ such that $a_n = 1000b_n$, then there exists a positive integer $m$ such that $a_m = d\cdot b_m$.

Let \( S \) be a set consisting of \( 2024 \) points on a plane, such that no three points in \( S \) are collinear. A line \( \ell \) passing through any two points in \( S \) is called a "weakly balanced line" if it satisfies the following condition: (Condition) When the line \( \ell \) divides the plane into two regions, one region contains exactly \( 1010 \) points of \( S \), and the other region contains exactly \( 1012 \) points of \( S \) (where each region contains no points lying on \( \ell \)). Let \( \omega(S) \) denote the number of weakly balanced lines among the lines passing through pairs of points in \( S \). Find the smallest possible value of \( \omega(S) \).

Find the smallest positive integer \( k \geq 2 \) for which there exists a polynomial \( f(x) \) of degree \( k \) with integer coefficients and a leading coefficient of \( 1 \) that satisfies the following condition: (Condition) For any two integers \( m \) and \( n \), if \( f(m) - f(n) \) is a multiple of \( 31 \), then \( m - n \) is a multiple of \( 31 \).

Find the smallest real number $M$ such that $$\sum_{k = 1}^{99}\frac{a_{k+1}}{a_k+a_{k+1}+a_{k+2}} < M$$for all positive real numbers $a_1, a_2, \dots, a_{99}$. ($a_{100} = a_1, a_{101} = a_2$)

For a positive integer $n$, let $g(n) = \left[ \displaystyle \frac{2024}{n} \right]$. Find the value of $$\sum_{n = 1}^{2024}\left(1 - (-1)^{g(n)}\right)\phi(n).$$

In an acute triangle $ABC$, let a line $\ell$ pass through the orthocenter and not through point $A$. The line $\ell$ intersects line $BC$ at $P(\neq B, C)$. A line passing through $A$ and perpendicular to $\ell$ meets the circumcircle of triangle $ABC$ at $R(\neq A)$. Let the feet of the perpendiculars from $A, B$ to $\ell$ be $A', B'$, respectively. Define line $\ell_1$ as the line passing through $A'$ and perpendicular to $BC$, and line $\ell_2$ as the line passing through $B'$ and perpendicular to $CA$. Prove that if $Q$ is the reflection of the intersection of $\ell_1$ and $\ell_2$ across $\ell$, then $\angle PQR = 90^{\circ}$.

On a blackboard, there are $10$ numbers written: $1, 2, \dots, 10$. Nahyun repeatedly performs the following operations. (Operation) Nahyun chooses two numbers from the 10 numbers on the blackboard that are not in a divisor-multiple relationship, erases them, and writes their GCD and LCM on the blackboard. If every two numbers on the blackboard form a divisor-multiple relationship, Nahyun stops the process. What is the maximum number of operations Nahyun can perform? (Note: $a, b$ are in a divisor-multiple relationship iff $a \mid b$ or $b \mid a$.)